If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

- Notifications 0

- Add Friend ($5)

As a registered member you can:

- View all solutions for free
- Request more in-depth explanations for free
- Ask our tutors any math-related question for free
- Email your homework to your parent or tutor for free
- Grade 7 McGraw Hill Glencoe - Answer Keys

## Explanation:

Find the value of x.

Model with Mathematics Refer to the graphic novel below. Classify the triangle formed by the cabin, ropes course, and mess hall by its angles and sides.

Yes, email page to my online tutor. ( if you didn't add a tutor yet, you can add one here )

Thank you for doing your homework!

## Submit Your Question

## Chapter 4, Lesson 2: Angles of Triangles

- Extra Examples
- Personal Tutor
- Self-Check Quizzes

The resource you requested requires you to enter a username and password below:

Please read our Terms of Use and Privacy Notice before you explore our Web site. To report a technical problem with this Web site, please contact the site producer .

- Texas Go Math
- Big Ideas Math
- Engageny Math
- McGraw Hill My Math
- enVision Math
- 180 Days of Math
- Math in Focus Answer Key
- Math Expressions Answer Key
- Privacy Policy

## McGraw Hill My Math Grade 5 Chapter 12 Lesson 2 Answer Key Sides and Angles of Triangles

All the solutions provided in McGraw Hill My Math Grade 5 Answer Key PDF Chapter 12 Lesson 2 Sides and Angles of Triangles will give you a clear idea of the concepts.

## McGraw-Hill My Math Grade 5 Answer Key Chapter 12 Lesson 2 Sides and Angles of Triangles

A triangle is a polygon with three sides and three angles.

Talk About It

Question 1. Compare the side lengths of each pair of triangles above. What do you notice? Answer: Now we have to compare the above diagram and its measurements: Pair A: In pair A, the two triangles have equal sides. So it is said to be all sides are congruent. In pair B, two sides are congruent. In pair C, no sides are congruent.

Question 2. Compare the angle measures of each pair of triangles above. What do you notice? Answer: From the above diagram and the angles measured we noticed that: In pair A: Each has one right angle In pair B: Each has one obtuse angle. Obtuse triangle: An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90°. An obtuse-angled triangle has one of its vertex angles as obtuse and other angles as acute angles i.e. if one of the angles measures more than 90°, then the sum of the other two angles is less than 90°. The side opposite to the obtuse angle is considered the longest. In pair C: All are acute angles. Acute triangle: An acute triangle is a triangle in which all three interior angles are less than 90º. Although the three interior angles of the acute triangle lie between 0° to 90°, their sum is always 180 degrees.

Practice It

Measure the sides of each triangle to the nearest tenth of a centimetre. Then describe the number of congruent sides.

Measure the angles of each triangle to the nearest degree. Then describe the number of acute, right, or obtuse angles.

Write About It

Question 16. How are all triangles the same and how can they be different? Answer: Two triangles will be similar if the angles are equal (corresponding angles) and the sides are in the same ratio or proportion (corresponding sides). Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides must be the same. They can differ in the number of congruent sides and angles measures.

## McGraw Hill My Math Grade 5 Chapter 12 Lesson 2 My Homework Answer Key

Problem Solving

## Leave a Comment Cancel Reply

You must be logged in to post a comment.

## Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles

Middle School Students who are struggling with the concepts of Big Ideas Math Grade 8 Ch 3 Angles and Triangles need not worry anymore. Big Ideas Math Book 8th Grade Chapter 3 Angles and Triangles help you finish your homework in time and gives you assistance whenever you need it. Detailed Solutions provided in Big Ideas Math Grade 8 Answers Chapter 3 make it easy for you to attain knowledge regarding the particular topic in a matter of seconds. Access the BIM Book Grade 8 Ch 3 Angles and Triangles Answer Key via direct links and prepare effectively.

## Big Ideas Math Book 8th Grade Answer Key Chapter 3 Angles and Triangles

Access the Big Ideas Math Grade 8 Ch 3 Angles and Triangles Answers includes questions from the Lessons, Chapter Tests, Assessment Tests, Cumulative Practice, etc. Kick start your preparation and attempt the exam with confidence and stand out from the rest of the crowd. Refer to the direct links available for Grade 8 BIM Book Chapter 3 Angles and Triangles Solution Key and prepare the respective topic accordingly in no time.

Performance

## Angles and Triangles STEAM Video/Performance

Angles and triangles getting ready for chapter 3.

Lesson: 1 Parallel Lines and Transversals

## Lesson 3.1 Parallel Lines and Transversals

Parallel lines and transversals homework & practice 3.1.

Lesson: 2 Angles of Triangles

## Lesson 3.2 Angles of Triangles

Angles of triangles homework & practice 3.2.

Lesson: 3 Angles of Polygons

## Lesson 3.3 Angles of Polygons

Angles of polygons homework & practice 3.3.

Lesson: 4 Using Similar Triangles

## Lesson 3.4 Using Similar Triangles

Using similar triangles homework & practice 3.4.

Chapter 3 – Angles and Triangles

## Angles and Triangles Connecting Concepts

Angles and triangles chapter review, angles and triangles practice test, angles and triangles cumulative practice.

STEAM Video

Watch the STEAM Video “Honeycombs.” Then answer the following questions.

Answer: The sum of interior angles of the equilateral triangle = 180° x + x + x = 180° 3x° = 180° x = 180/3 x° = 60°

Performance Task

Turtle Shells

Chapter Exploration

Answer: 8 angles are formed by the parallel lines and the transversal b. Which of these angles have equal measures? Explain your reasoning.

EXPLORATION 1

Use the ﬁgure to ﬁnd the measure of the angle. Explain your reasoning

Question 1. ∠1

Answer: 63°

Question 2. ∠2

Answer: 117°

Answer: ∠1 and 59° are the supplementary angles ∠1 + 59° = 180° ∠1 = 180° – 59° ∠1 = 121° ∠2 and 59° are vertical angles. They are congruent. So, the measure of ∠1 is 121° ∠3 and 59° are supplementary angles. ∠3 + 59° = 180° ∠3 = 180° – 59° ∠3 = 121° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel side. the measure of ∠4 is 121° the measure of ∠5 is 59° the measure of ∠6 is 121° the measure of ∠7 is 59°

In Example 3, the measure of ∠4 is 84°. Find the measure of the angle. Explain your reasoning.

Question 4. ∠3

Question 5. ∠5

Answer: ∠4 and ∠5 are alternate interior angles formed by transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 84°

Question 6. ∠6

Answer: ∠3 and ∠6 are alternate exterior angles formed by transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠6 is 96°

Self-Assessment for Concepts & Skills Solve each exercise. Then rate your understanding of the success criteria in your journal.

FINDING ANGLE MEASURES Use the ﬁgure to ﬁnd the measures of the numbered angles.

Answer: ∠1 and 120° are the supplementary angles. ∠1 + 120° = 180° ∠1 = 180 – 120 ∠1 = 60° Thus the measure of ∠1 is 60° ∠2 and 120° are the vertical angles. They are congruent. Thus the measure of ∠2 is 120° ∠3 and 120° are the supplementary angles. ∠3 + 120° = 180° ∠3 = 180 – 120 ∠3 = 60° ∠4, ∠5, ∠6, ∠7 are corresponding angles are formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 60° Thus the measure of ∠5 is 120° Thus the measure of ∠6 is 120° Thus the measure of ∠7 is 60°

Answer: ∠1 and 35° are the supplementary angles. ∠1 + 35° = 180° ∠1 = 180 – 35 ∠1 = 145° Thus the measure of ∠1 is 145° ∠2 and 35° are the supplementary angles. ∠2 + 35° = 180° ∠2 = 180 – 35 ∠2 = 145° Thus the measure of ∠2 is 145° ∠3 and 35° are the vertical angles. They are congruent. Thus the measure of ∠3 is 35° ∠4, ∠5, ∠6, ∠7 are corresponding angles are formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 35° Thus the measure of ∠5 is 145° Thus the measure of ∠6 is 145° Thus the measure of ∠7 is 35°

Answer: ∠2, ∠6 are corresponding angles are formed by transversal intersecting parallel lines. ∠6, ∠8 are vertical angles are formed by transversal intersecting parallel lines. ∠5 does not belong to the other three because all the other three measure are equal.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Answer: The angle a and the angle of 70 degrees are complementary angles because they belong to a right triangle, where the third angle is the right angle. ∠a + 70 = 90 ∠a = 90 – 70 ∠a = 20°

Answer: The lines AB and CD are parallel. ABC and BCD are the corresponding angles formed by transversal intersecting parallel lines. ∠BCD = 55° ∠BAC + ∠ABC + ∠ACB = 180° The sum of the angles in a triangle is 180° ∠BAC + 55°+ 52° = 180° ∠BAC + 107° = 180° ∠BAC = 180° – 107° ∠BAC = 73° So, the head tube angle of a bike is 73°

Review & Refresh

Find the values of the ratios (red to blue) of the perimeters and areas of the similar ﬁgures.

Answer: perimeter of red hexagon/perimeter of blue hexagon = \(\frac{3}{5}\) The values of the ratios of the perimeter is \(\frac{3}{5}\) Area of red hexagon/Area of blue hexagon = (\(\frac{3}{5}\))² = \(\frac{9}{25}\) The values of the ratios of the area is \(\frac{9}{25}\)

Answer: perimeter of red trapezium /perimeter of blue trapezium = \(\frac{7}{6}\) The values of the ratios of the perimeter is \(\frac{7}{6}\) Area of red hexagon/Area of blue hexagon = (\(\frac{7}{6}\))² = \(\frac{49}{36}\) The values of the ratios of the area is \(\frac{49}{36}\)

Evaluate the expression.

Question 3. 4 + 3 2

Answer: 4 + 9 = 13

Question 4. 5(2) 2 – 6

Answer: 5(4) – 6 20 – 6 = 14

Question 5. 11 + (-7) 2 – 9

Answer: 11 + 49 – 9 11 + 40 = 50

Concepts, Skills, & Problem Solving EXPLORING INTERSECTIONS OF LINES Use a protractor to determine whether lines a and b are parallel. ( See Exploration 1, p. 103.)

Answer: Use a protractor to measure ∠1 and ∠2 ∠1 ≈ 60° ∠2 ≈ 60° ∠1 and ∠2, it means the two angles are congruent. The angles are exterior alternate angles. According to the converse of the exterior alternate angles theorem, the two lines are parallel. a || b

Answer: Use a protractor to measure ∠1 and ∠2 ∠1 ≈ 50° ∠2 ≈ 60° ∠1 and ∠2, it means the two angles are not congruent. The angles are exterior alternate angles. According to the converse of the exterior alternate angles theorem, the two lines are not parallel.

FINDING ANGLE MEASURES Use the ﬁgure to ﬁnd the measures of the numbered angles. Explain your reasoning.

Answer: ∠1 and 107° are corresponding angles. They are congruent. So, the measure of ∠1 is 107°. ∠1 and ∠2 are supplementary angles. ∠1 + ∠2 = 180° 107° + ∠2 = 180° ∠2 = 180° – 107° ∠2 = 73° So, the measure of ∠2 is 73°

Answer: ∠3 and 95° are corresponding angles. They are congruent. Thus the measure of ∠3 is 95° ∠3 and ∠4 are supplementary angles. ∠3 + ∠4 = 180° 95° + ∠4 = 180° ∠4 = 180 – 95 ∠4 = 85° So the measure of ∠4 is 85°

Answer: ∠5 and 49° are corresponding angles. They are congruent. So, the measure of ∠5 is 49° ∠5 and ∠6 are supplementary angles. ∠5 + ∠6 = 180° 49° + ∠6 = 180° ∠6 = 180° – 49° ∠6 = 131° So, the measure of ∠6 is 131°

Answer: Since the two lines are not parallel. Hence ∠5 is not congruent to ∠6. By this, we can say that your friend is not correct.

Answer: ∠1 and ∠2 are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. The measure of ∠1 is 60° so the measure of ∠2 is 60°

Question 13. OPEN-ENDED Describe two real-life situations that use parallel lines.

Answer: Example 1: The railroad tracks and the tram tracks are parallel lines. Example 2: The shelves of a bookcase.

USING CORRESPONDING ANGLES Use the ﬁgure to ﬁnd the measures of the numbered angles.

Answer: ∠1 and 60° are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. ∠1 and ∠2 are supplementary angles. ∠1 + ∠2 = 180° 60° + ∠2 = 180° ∠2 = 180° – 60° ∠2 = 119° So, the measure of ∠2 is 119° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 61° ∠4 and ∠2 are vertical angles. They are congruent. ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠5 is 119° So, the measure of ∠6 is 61° So, the measure of ∠7 is 119°

Answer: ∠1 and 99° are supplementary angles. ∠1 + 99° = 180° ∠1 = 180° – 99° ∠1 = 81° Thus the measure of ∠1 is 81° ∠2 and 99° are vertical angles. They are congruent. The measure of ∠2 is 99° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 81° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠4 is 99° So, the measure of ∠5 is 81° So, the measure of ∠6 is 99° So, the measure of ∠7 is 81°

Answer: ∠1 and 90° are supplementary angles. ∠1 + 90° = 180° ∠1 = 180° – 90° ∠1 = 90° Thus the measure of ∠1 is 90° ∠2 and 90° are vertical angles. They are congruent. Thus the measure of ∠2 is 90° ∠3 and ∠1 are vertical angles. They are congruent. So, the measure of ∠3 is 90° ∠4, ∠5, ∠6, ∠7 corresponding angles are congruent because they are formed by a transversal intersecting parallel lines. So, the measure of ∠4 is 90° So, the measure of ∠5 is 90° So, the measure of ∠6 is 90° So, the measure of ∠7 is 90°

USING CORRESPONDING ANGLES Complete the statement. Explain your reasoning.

Answer: ∠1 and ∠8 are corresponding angles. They are congruent. The measure of ∠1 = 124°, then the measure of ∠8 is 124° ∠8 and ∠4 are supplementary angles. ∠8 + ∠4 = 180° 124° + ∠4 = 180° ∠4 = 180° – 124° ∠4 = 56° So, the measure of ∠4 is 56°

Answer: ∠2 and ∠7 are corresponding angles. They are congruent. The measure of ∠2 = 48°, then the measure of ∠7 is 48° ∠7 and ∠3 are supplementary angles. ∠7 + ∠3 = 180° 48° + ∠3 = 180° ∠3 = 180° – 48° ∠3 = 132° Thus the measure of ∠3 = 132°

Answer: ∠4 and ∠2 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠2 is 55°

Answer: ∠6 and ∠8 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠8 is 120°

Answer: ∠7 and ∠2 are corresponding angles. They are congruent. The measure of ∠7 is 50.5°, so the measure of ∠2 is 50.5° ∠2 and ∠6 are supplementary angle. ∠2 + ∠6 = 180° 50.5° + ∠6 = 180° ∠6 = 180° – 50.5° ∠6 = 129.5° So, the measure of ∠6 is 129.5°

Answer: ∠3 and ∠6 are corresponding angles. They are congruent. The measure of ∠3 is 118.7° So, the measure of ∠6 is 118.7° ∠6 and ∠2 are supplementary angles. ∠6 + ∠2 = 180° 118.7° + ∠2 = 180° ∠2 = 180° – 118.7° ∠2 = 61.3° So, the measure of ∠2 is 61.3°

Answer: ∠4 and ∠5 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠1 is 40°

Question 24. REASONING Is there a relationship between exterior angles that lie on the same side of a transversal? interior angles that lie on the same side of a transversal? Explain.

Question 25. REASONING When a transversal is perpendicular to two parallel lines, all the angles formed measure 90°. Explain why.

Question 26. REASONING Two horizontal lines are cut by a transversal. What is the least number of angle measures you need to know to ﬁnd the measure of every angle? Explain your reasoning.

Answer: ∠1 and ∠7 are alternate exterior angles formed by a transeversal intersecting parallel lines. So, ∠1 and ∠7 are congruent. ∠1 and ∠5 are corresponding angles formed by a transeversal intersecting parallel lines. So, ∠1 and ∠5 are congruent. ∠5 and ∠7 are vertical angles so they are congruent. Hence ∠1 and ∠7 are congruent.

FINDING A VALUE Find the value of x.

Answer: ∠1 and 50° are alternate interior angles. They are congruent. So, the measure of ∠1 is 50° ∠2 and ∠1 are corresponding angles. They are congruent. So, the measure of ∠2 is 50° ∠2 and x are supplementary angle. ∠2 + x = 180° 50° + x = 180° x = 180° – 50° x = 130° So, the measure of x is 130°

Answer: ∠1 and 115° are corresponding angles. They are congruent. So, the measure of ∠1 is 115° ∠1 and x are alternate exterior angles. They are congruent. So, the measure of x is 115°

Answer: 180° rotation, translation about line t.

Question 31. OPEN-ENDED Refer to the ﬁgure. a. Do the horizontal lines appear to be parallel? Explain.

Answer: The three horizontal lines seem to spread apart, even though in reality they are parallel.

Answer: As the lines AB and CD are parallel and ∠BCD are alternate interior angles transversal BC, they are congruent. ∠ABC ≅ ∠BCD x = 64 b. How does the angle the puck hits the edge of the table relate to the angle it leaves the edge of the table?

Answer: m∠MBA + m∠ABC + m∠CBN = 180° 58 ° + 64° + m∠CBN = 180° 122 ° + m∠CBN = 180° m∠CBN = 180° – 122 ° m∠CBN = 58°

Find the measures of the interior angles of the triangle.

Answer: 81°, 25°, 74°

Explanation: Sum of all the angles in a triangle = 180° x° + 81° + 25° = 180° x° = 180° – 81° – 25° x = 74° Thus the measure of the interior angle is 74°

Answer: 43°, 51°, 86°

Explanation: Sum of all the angles in a triangle = 180° x° + (x – 35)° + 43° = 180° x° + x° – 35° + 43° = 180° 2x° + 8° = 180° 2x° = 180° – 8° 2x° = 172° x° = 172°/2 x° = 86° The measure of the interior angle of the triangle (x – 35)° = 86 – 35 (x – 35)° = 51° x° = 51° + 35° x° = 86°

Self-Assessment for Concepts & Skills

Question 4. VOCABULARY How many exterior angles does a triangle have at each vertex? Explain.

FINDING ANGLE MEASURES Find the value of x.

Answer: Sum of all the angles in a triangle = 180° x° + 25° + 40° = 180° x° + 65° = 180° x° = 180° x° = 180° – 65° x° = 115° Thus the value of x is 115°

Answer: x° = 50° + 55° x° = 105° Thus the value of x is 105°

Question 7. The Historic Triangle in Virginia connects Jamestown, Williamsburg, and Yorktown. The interior angle at Williamsburg is 120°. The interior angle at Jamestown is twice the measure of the interior angle at Yorktown. Find the measures of the interior angles at Jamestown and Yorktown. Explain your reasoning.

Answer: Given, A helicopter travels from point C to point A to perform a medical supply drop. The helicopter then needs to land at point B. A = 90° + 32° A = 122° Thus the helicopter should turn 122° at point A to travel towards point B.

Use the ﬁgure to ﬁnd the measure of the angle. Explain your reasoning.

Question 1. ∠2

Answer: 82°

∠2 and 82° are alternate exterior angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠2 is 82°

Question 2. ∠6

∠6 and 82° are vertical angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 82°

Question 3. ∠4

∠4 and 82° are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 82°

Question 4. ∠1

Answer: 98°

∠4 and 82° are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠4 is 82° ∠4 and ∠1 are supplementary angles ∠4 + ∠1 = 180° 82° + ∠1 = 180° ∠1 = 180° – 82° ∠1 = 98°

You spin the spinner shown.

Question 5. What are the favorable outcomes of spinning a number less than 4?

Answer: 1, 2, 3

Explanation: The favorable outcome of spinning a number less than 4 is 1, 2, and 3.

Question 6. In how many ways can spinning an odd number occur?

Answer: two ways Odd numbers = 1 and 3 So, in two ways spinning an odd number can occur.

Concepts, Skills, & Problem Solving

USING PARALLEL LINES AND TRANSVERSALS Consider the ﬁgure below. (See Exploration 2, p. 111.)

Question 7. Use a protractor to ﬁnd the measures of the labeled angles.

Answer: Use a protractor to determine the measures of the angles A, B, C. m∠A = 30° m∠B = 105° m∠C = 45° m∠D = 150° m∠E = 75° m∠F = 105° m∠G = 30°

Question 8. Is ∠F an exterior angle of Triangle ABC ? Justify your answer.

Answer: An exterior angle is the angle between one side of a triangle and the extension of an adjacent side. ∠F is not an exterior angle of triangle ABC because it has a side of triangle ABC, but not the extension of the adjacent side DF.

USING INTERIOR ANGLE MEASURES Find the measures of the interior angles of the triangle.

Answer: Sum of all the angles in a triangle = 180° x° + 90° + 30° = 180° x° + 120° = 180° x° = 180° – 120° x° = 60°

Answer: Sum of all the angles in a triangle = 180° x° + 65° + 40° = 180° x° + 105° = 180° x° = 180° – 105° x° = 75°

Answer: Sum of all the angles in a triangle = 180° x° + 35° + 45° = 180° x° + 80° = 180° x° = 180° – 80° x° = 100°

Answer: Sum of all the angles in a triangle = 180° x° + (x + 65)° + 25° = 180° x° + x° + 65° + 25° = 180° 2x° + 90° = 180° 2x° = 180° – 90° 2x° = 90° x° = 90°/2 x° = 45° (x + 65)° = 45 + 65 = 110 x° = 25°

Answer: Sum of all the angles in a triangle = 180° x° + (x – 44)° + 48° = 180° x° + x° – 44° + 48° = 180° 2x° + 4° = 180° 2x° = 180° – 4° 2x° = 176° x° = 176°/2 x° = 88° (x – 44)° = 88 – 44 = 44 x° = 44°

Answer: Sum of all the angles in a triangle = 180° x° + (x – 11)° + 73° = 180° x° + x° – 11° + 73° = 180° 2x° + 62° = 180° 2x° = 180° – 62° 2x° = 118° x° = 118°/2 x° = 59° (x – 11)° = 59 – 11 = 48 x° = 48°

FINDING EXTERIOR ANGLE MEASURES Find the measure of the exterior angle.

Answer: x° = 38° + 90° x° = 128° The measure of exterior angle is 128°

Answer: k° = 64° + 76° k° = 140° The measure of an exterior angle is 140°

Answer: 2a° = (a + 10°) + 44° 2a° = a + 54° 2a° – a° = 54 a° = 54 The measure of the exterior angle = 2a = 2(54°) = 108°

Answer: Sum of all the angles in a triangle = 180° x° + 75° + 75° = 180° x° + 150° – 150° = 180° – 150° x° = 30° Thus the angle that tornado direction needs to change is 30°.

Answer: Your friend is not correct because the measure of the exterior angle is equal to the sum of two non-adjacent interior angles.

Question 20. REASONING The ratio of the interior angle measures of a triangle is 2 : 3 : 5. What are the angle measures?

Answer: Sum of all the angles in a triangle = 180° 2x° + 3x° + 5x° = 180° 10x° = 180° x = 180/10 x = 18° 2x° = 2(18°) = 36° 3x° = 3(18) = 54° 5x° = 5(18) = 90°

Question 21. PROBLEM SOLVING The support for a window air-conditioning unit forms a triangle and an exterior angle. What is the measure of the exterior angle?

Answer: The measure of the exterior angle DBC is m∠DBC = m∠ABC + m∠ACB m∠ABC + m∠ACB = 90° 5x – 6 + 3x = 90 8x – 6 = 90 8x = 90 + 6 8x = 96 x = 96/8 x = 12 m∠DBC = m∠BAC+ m∠ACB = 90° + 3(12)° = 126°

Question 22. REASONING A triangle has an exterior angle with a measure of 120°. Can you determine the measures of the interior angles? Explain.

Answer: A triangle has an exterior angle with a measure of 120° m∠ACB = m∠A + m∠B m∠A + m∠B = 120° According to the exterior angles We have m∠C + m∠ACD = 180° m∠C + 120° = 180° m∠C = 180° – 120° m∠C = 60°

ANGLES OF TRIANGLES

Determine whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 23. Given three angle measures, you can construct a triangle.

Answer: We can construct a triangle if the sum of the measure of the 3 angles is 180°. As a matter of fact, if the sum of the measures of the 3 angles is 180° We can build an infinity of triangles that are similar.

Question 24. The acute interior angles of a right triangle are complementary.

Answer: Let A, B, C be the angles of a right triangle with m∠A = 90° m∠A + m∠B + m∠C = 180° 90° + m∠B + m∠C = 180° m∠B + m∠C = 180° – 90° m∠B + m∠C = 90° This means ∠B and ∠C are complementary.

Question 25. A triangle has more than one vertex with an acute exterior angle.

Answer: An exterior angle of a triangle and the adjacent triangle’s angle are complementary. If an exterior angle is acute, it means the adjacent triangle’s angle is obtuse. Since we are given that more than one exterior angle is acute, it means the triangle would have more than one obtuse angle, which is impossible. The statement is never true.

Answer: The angles z and w are supplementary z + w = 180° The sum of a triangle is 180° x + y + w = 180° z = 180° – w x + y = 180° – w z = x + y

Find the sum of the interior angle measures of the green polygon.

Answer: S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720°

Question 5. WRITING Explain how to ﬁnd the sum of the interior measures of a polygon.

Answer: Steps to find the sum of the interior measurements of the polygon: 1. Count the number of sides of the polygon. 2. Subtract the number of sides by 2. 3. Multiply the result of the subtraction by 180°

Question 6. FINDING THE SUM OF INTERIOR ANGLE MEASURES Find the sum of the interior angle measures of the green polygon.

Answer: S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360°

FINDING AN INTERIOR ANGLE MEASURE

Find the value of x.

Answer: S = (n – 2) . 180° S = (5 – 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° x° + 160° + 110° + 105° + 95° = 540° x° + 470° = 540° x° = 540° – 470° x° = 70° Thus the value of x is 70°.

Answer: S = (n – 2) . 180° S = (9 – 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260° x° + 165° + 155° + 150° + 140° + 135° + 130° + 125° + 110° = 1260° x° + 1105° = 1260° x° = 1260° – 1105° x° = 155° Thus the value of x is 155°

Question 9. A company installs an octagonal swimming pool. a. Find the value of a for the pool shown at the left.

Answer: S = (n – 2) . 180° S = (8 – 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° a° + 120° + a° + 120° + a° + 120° + a° + 120° = 1080° 4a° + 480° = 1080° 4a° = 1080° – 480° 4a° = 600° a° = 600/4 a° = 150° Thus the value of x is 150°.

Answer: No for any octagon the sum of the interior angles is 1080 degrees.

Question 10. DIG DEEPER! A Bronze Star Medal A is shown. a. How many interior angles are there?

Answer: 10 interior angles are there

Answer: S = (n – 2) . 180° S = (10 – 2) . 180° S = 8 . 180° S = 1440° Thus the sum of the interior angle measure is 1440°

Answer: 60°

Explanation: Sum of all the angles = 180° x° + 60° + 60° = 180° x° + 120° = 180° x° = 180° – 120° x° = 60°

Answer: 45°

Explanation: Sum of all the angles = 180° x° + x° + 90° = 180° 2x° + 90° = 180° 2x° = 180° – 90° 2x° = 90° x° = 45°

Answer: 113°

Explanation: x° = 65° + 48° x° = 113° Thus the measure of an exterior angle is 113°

Solve the proportion.

Explanation: \(\frac{x}{12}\) = \(\frac{3}{4}\) 12 . \(\frac{x}{12}\) = \(\frac{3}{4}\) . 12 x = 3 . 3 x = 9

Explanation: \(\frac{14}{21}\) = \(\frac{x}{3}\) 3 . \(\frac{14}{21}\) = \(\frac{x}{3}\) . 3 x = 2

Explanation: \(\frac{9}{x}\) = \(\frac{6}{2}\) 2. \(\frac{9}{x}\) = 6 18 = 6x x = 3

Concepts, Skills, & Problem Solving EXPLORING INTERIOR ANGLES OF POLYGONS Use triangles to ﬁnd the sum of the interior angle measures of the polygon. (See Exploration 1, p. 117.)

Answer: 360°

Explanation: Number of sides = 4 Number of interior triangles in the given figure = 2 The Sum of the measures of the interior angles using triangle = 2 . 180° = 360°

Answer: 1260°

Explanation: Number of sides = 9 Number of interior triangles in the given figure = 7 The Sum of the measures of the interior angles using triangle = 7 . 180° = 1260°

Answer: 540°

Explanation: Number of sides = 5 Number of interior triangles in the given figure = 3 The Sum of the measures of the interior angles using triangle = 3 . 180° = 540°

FINDING THE SUM OF INTERIOR ANGLE MEASURES Find the sum of the interior angle measures of the polygon.

Explanation: S = (n – 2) . 180° S = (4- 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360°

Answer: 1080°

Explanation: S = (n – 2) . 180° S = (8- 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080°

Explanation: S = (n – 2) . 180° S = (9- 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260°

Answer: To find the sum of the interior angle measures he should subtract 2 from the number of sides of the polygon and then multiply by 180° S = (n – 2) . 180° By this, we can say that your friend is not correct.

FINDING AN INTERIOR ANGLE MEASURE Find the value of x.

Answer: S = (n – 2) . 180° S = (4- 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360° x° + 155° + 25° + 137° = 360° x° + 317° = 360° x° = 360° – 317° x° = 43°

Answer: S = (n – 2) . 180° S = (6- 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° x° + x° + x° + x° + 90° + 90° = 720° 4x° + 180° = 720° 4x° = 720° – 180° 4x° = 540° x° = 540/4 x° = 135°

Answer: S = (n – 2) . 180° S = (6- 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° 3x° + 45° + 135° + x° + 135° + 45° = 720° 4x° + 360° = 720° 4x° = 720° – 360° 4x° = 360° x° = 360/4 x° = 90°

FINDING A MEASURE Find the measure of each interior angle of the regular polygon.

Answer: S = (n – 2) . 180° S = (3- 2) . 180° S = 1 . 180° S = 180° Thus the sum of the interior angle measure is 180° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 3. 180 ÷ 3 = 60°

S = (n – 2) . 180° S = (9 – 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 9. 1260 ÷ 9 = 140°

S = (n – 2) . 180° S = (12 – 2) . 180° S = 10 . 180° S = 1800° Thus the sum of the interior angle measure is 1800° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 12. 1800 ÷ 12 = 150°

Answer: No, my friend is not correct because to find the measure of each interior angle of a regular 20-gon, he should divide the sum of the measured interior angles by the number of interior angles, in this case, 20 but your friend divide it by 18 so he is not correct.

Answer: S = (n – 2) . 180° S = (5- 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 5. 540÷ 5 = 108°

b. RESEARCH Why are ﬁrehydrants made this way?

Question 22. PROBLEM SOLVING The interior angles of a regular polygon each measure 165°. How many sides does the polygon have?

Answer: (n – 2) . 180 = 165 . n 180n – 360 = 165n 180n – 360 + 360 – 165n = 165n + 360 – 165n 15n = 360 n = 360/15 n = 24 Therefore the polygon has 24 sides

Question 23. STRUCTURE A molecule can be represented by a polygon with interior angles that each measure 120°. What polygon represents the molecule? Does the polygon have to be regular? Justify your answers.

Answer: (n – 2) . 180 = 120 . n 180n – 360 = 120n 180n – 120n = 360 60n = 360 n = 6

Question 24. PROBLEM SOLVING The border of a Susan B. Anthony dollar is in the shape of a regular polygon. a. How many sides does the polygon have?

Answer: The polygon has 11 sides.

Answer: S = (n – 2) . 180° S = (11 – 2) . 180° S = 9 . 180° S = 1620° Thus the sum of the interior angle measure is 1620° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 11. 1620 ÷ 11 = 147°

Question 25. REASONING The center of the stained glass window is in the shape of a regular polygon. What are the measures of the interior angles of the green triangle?

Answer: (n-2)180°/n = (8-2)180°/8 = 135° m∠OAB = m∠OBA = 135/2 = 67.5° m∠AOB + m∠OAB + m∠OBA = 180° m∠AOB + 67.5° + 67.5° = 180° m∠AOB + 135° = 180° m∠AOB = 180° – 135° m∠AOB = 45°

Answer: The given polygon has 7 sides. S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900° 4 . 135° + 3 . x° = 900° 540° + 3x° = 900° 3x° = 900° – 540° 3x° = 360° x° = 360/3 x° = 120°

Using Similar Triangles

Tell whether the triangles are similar. Explain.

Answer: Yes

Explanation: x° + 66° + 90° = 180° x° + 156° = 180° x° = 180° – 156° x° = 24° y° + 24° + 90° = 180° y° + 114° = 180° y° = 180° – 114° y° = 66° The triangles have two pairs of congruent angles. Thus the triangles are similar.

Explanation: We are not given any information about the lengths of the sides either, therefore with only a pair of congruent angles, we cannot tell whether the triangles are similar.

Explanation: x° + 54° + 63° = 180° x° + 107° = 180° x° = 180° – 107° x° = 63°

Answer: Option B

Explanation: ΔPQR and ΔTSR are congruent as TS || PQ leads to two pairs of correspondent congruent angles. ΔPQR is a dilation of ΔTSR because their sides are proportional, the constant of proportionality being greater than 1. ΔPQR is a scale drawing of ΔTSR because their sides are proportional. The question that does not fit is “Are ΔPQR and ΔTSR the same size and shape?” because the triangles do not have the same size, but they have the same shape.

Answer: Aqueduct/2.6 = 5/1 Aqueduct = 5 × 2.6 Aqueduct = 13 Thus the length of the Aqueduct is 13 km.

Answer: a/10 = 3/6 6 × a = 3 × 10 6a = 30 a = 30/6 a = 5 The length from point Z to point Y is 5 miles. Time to travel from point Z to point Y = 5/3.5 = 1.56 hour

Find the measure of each interior angle of the regular polygon.

Question 1. octagon

Answer: The measure of each interior angle is 135°

Explanation: S = (n – 2) . 180° S = (8- 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 8. 1080÷ 8= 135°

Question 2. decagon

Answer: The measure of each exterior angle is 144°

Explanation: S = (n – 2) . 180° S = (10 – 2) . 180° S = 8 . 180° S = 1440° Thus the sum of the interior angle measure is 1440° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 10. 1440÷ 10= 144°

Question 3. 18-gon

Answer: The measure of each interior angle is 160°

Explanation: S = (n – 2) . 180° S = (18- 2) . 180° S = 16 . 180° S = 2880° Thus the sum of the interior angle measure is 2880° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 18. 2880 ÷ 18= 160°

Solve the equation. Check your solution.

Question 4. 3.5 + y = -1

Answer: Given the equation 3.5 + y = -1 y = -1 – 3.5 y = -4.5

Question 5. 9x = 54

Answer: Given the equation 9x = 54 x = 54/9 x = 6

Question 6. -4 = \(\frac{2}{7}\)p

Answer: Given the equation -4 = \(\frac{2}{7}\)p -4 × 7 = 2p 2p = -28 p = -28/2 p = -14

Concepts, Skills, & Problem Solving CREATING SIMILAR TRIANGLES Draw a triangle that is either larger or smaller than the one given and has two of the same angle measures. Explain why the new triangle is similar to the original triangle. (See Exploration 1, p. 123.)

IDENTIFYING SIMILAR TRIANGLES Tell whether the triangles are similar. Explain.

Answer: The triangles have two pairs of congruent angles. So, the third angles are congruent, and the triangles are similar.

Answer: x° + 36° + 72° = 180° x° + 108° = 180° x° = 180° – 108° x° = 72° y° + 33° + 72° = 180° y° + 105° = 180° y° = 180° – 105° y° = 72° The triangles do not have two pairs of congruent angles. Therefore the triangles are not similar.

Answer: x° + 64° + 85° = 180° x° + 149° = 180° x° = 180° – 149° x° = 31° y° + 26° + 85° = 180° y° + 111° = 180° y° = 180° – 111° y° = 69° The triangles do not have two pairs of congruent angles. Therefore the triangles are not similar.

Answer: x° + 48° + 81° = 180° x° + 129° = 180° x° = 180° – 129° x° = 51° y° + 48° + 51° = 180° y° + 99° = 180° y° = 180° – 99° y° = 81° The triangles have two pairs of congruent angles. Therefore the triangles are similar.

Answer: 2x + 90 = 180° 2x = 180 – 90° 2x = 90° x = 90/2 x = 45° The ruler on the left and the ruler on the right both have the shape of a right triangle with 45° angles, therefore they are similar in shape, while the middle ruler has 60°, 30° angles.

STRUCTURE Tell whether the triangles are similar. Explain.

Answer: m∠APB + m∠B = 90° m∠APB + 51° = 90° m∠APB = 90° – 51° m∠APB = 39° m∠APB + m∠BPD + m∠DPC = 180° 39° + 102° + m∠DPC = 180° m∠DPC + 141° = 180 m∠DPC = 180 – 141° m∠DPC = 39° m∠A = m∠C m∠APB = m∠DPC

Answer: ∠APB ≅ ∠CPD m∠APB = m∠CPD m∠APB = 29° m∠A + m∠B + m∠APB = 180° m∠A + 88° + 29° = 180° m∠A + 117° = 180° m∠A = 180° – 117° m∠A = 63° m∠PDC + m∠PDE = 180° m∠PDC + 91° = 180° m∠PDC = 180° – 91° m∠PDC = 89°

IDENTIFYING SIMILAR TRIANGLES Can you determine whether the triangles are similar? Explain.

Answer: PS || QR ∠PSQ and ∠SQR are interior angles using the transversal QS, thus they are congruent. ∠PSQ ≅ ∠SQR

Answer: As AB || DE there are two pairs of congruent alternate interior angles, using the transversals AE and BD. ∠A≅ ∠E ∠B≅ ∠D The two pairs of congruent angles are enough to prove that the triangles are similar. ΔABC ∼ ΔEDC

Answer: ΔAMN ∼ ΔABC MN/BC = AM/AB 1.5/d = 5/50 d = 1.5 × 10 d = 15 feet Therefore 15 feet is not an appropriate location.

Answer: The two triangles are similar because they are right triangles and ∠AXB ≅ ∠PXQ because they are vertical angles. PQ/300 = 80/240 240PQ = 24000 PQ = 24000/240 PQ = 100 steps

Answer: Given, A person who is 6 feet tall casts a 3-foot-long shadow. A nearby pine tree casts a 15-foot-long shadow. ΔXAB ∼ ΔXPQ AB/PQ = XB/XQ 6/PQ = 3/15 PQ = 30 ft

Question 21. OPEN-ENDED You place a mirror on the ground 6 feet from the lamppost. You move back 3 feet and see the top of the lamppost in the mirror. What is the height of the lamppost?

Question 22. DIG DEEPER! In each of two right triangles, one angle measure is two times another angle measure. Can you determine that the triangles are similar? Explain your reasoning.

Answer: We are given the right triangle ABC m∠A = 2m∠B Case 1: m∠A = 90° 90° = 2m∠B m∠B = 45° m∠C = 180° – 90° – 45° = 45° Case 2: m∠B = 90° m∠A = 2 × 90° = 180° Case 3: m∠C = 90° m∠A + m∠B = 180 – m∠C = 180° – 90° = 90° 2m∠B + m∠B = 90° 3m∠B = 90° m∠B = 30° m∠A = 2 . 30° = 60°

Answer: ΔABG ∼ ΔACF ΔACF ∼ ΔADE ΔABG ∼ ΔADE AB = BC = CD = BD/2 = 6.32/2 = 3.16 AB/CD = BG/DE 3BG = 6 BG = 2 feet ΔACF ∼ ΔADE AC/AD = CF/DE 2/3 = CF/6 3CF = 2(6) CF = 4 feet

Using the Problem-Solving Plan

Understand the problem You know two dimensions of a dog park and the ratio of the perimeter of the small dog section to the perimeter of the entire park. You are asked to ﬁnd the area of each section. Make a plan Verify that the small triangle and the large triangle are similar. Then use the ratio of the perimeters to ﬁnd the base or the height of each triangle and calculate the areas. Solve and check. Use the plan to solve the problem. Then check your solution.

Review Vocabulary

Choose and complete a graphic organizer to help you study the concept.

- interior angles formed by parallel lines and a transversal
- exterior angles formed by parallel lines and a transversal
- interior angles of a triangle
- exterior angles of a triangle
- similar triangles

Chapter Self-Assessment

3.1 Parallel Lines and Transversals (pp. 103–110)

Question 1. ∠8

Answer: 140°

Explanation: ∠8 and 140 degrees angle are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠8 is 140°

Question 2. ∠5

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140°

Question 3. ∠7

Answer: 40°

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140° ∠5 and ∠7 are supplementary angle. ∠5 + ∠7 = 180° 140° + ∠7 = 180° ∠7 = 180° – 140° ∠7 = 40° So, the measure of ∠7 is 40°

Question 4. ∠2

Explanation: 140 and ∠2 are supplementary angle. 140° + ∠2 = 180° ∠2 = 180° – 140° ∠2 = 40° So, the measure of ∠2 is 40°

Question 5. ∠6

Explanation: ∠5 and 140 degrees angle are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠5 is 140° ∠5 and ∠6 are supplementary angle. 140° + ∠6 = 180° ∠6 = 180° – 140° ∠6 = 40° So, the measure of ∠6 is 40°

Complete the statement. Explain your reasoning.

Answer: 123°

Explanation: ∠1 and ∠7 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠7 is 123°

Answer: 122°

Explanation: ∠2 and ∠6 are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠6 is 58° ∠5 and ∠6 are supplementary angle. ∠5 + ∠6 = 180° 58° + ∠5 = 180° ∠5 = 180° – 58° ∠5 = 122° So, the measure of ∠5 is 122°

Answer: 119°

Explanation: ∠3 and ∠5 are alternate interior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠3 is 119°

Explanation: ∠4 and ∠6 are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. So, the measure of ∠4 is 60°

Question 10. In Exercises 6–9, describe the relationship between ∠2 and ∠8.

Answer: ∠2 ≅ ∠8

Answer: ∠1 = 108°, ∠2 = 108°

Explanation: ∠3 and 72° are alternate interior angles. They are congruent. So, the measure of ∠3 is 72° ∠3 + ∠1 = 180° 72° + ∠1 = 180° ∠1 = 180° – 72° ∠1 = 108° So, the measure of ∠1 is 108° ∠1 and ∠2 are alternating interior angles. They are congruent.

3.2 Angles of Triangles (pp. 111 – 116)

Answer: Sum of all the angles in a triangle = 180° x° + 50° + 55° = 180° x° + 105° = 180° x° = 180° – 105° x° = 75°

Answer: Sum of all the angles in a triangle = 180° x° + (x + 8)° + 90° = 180° 2x° + 8° + 90° = 180° 2x° + 98° = 180° 2x° = 180° – 98° 2x° = 82 x° = 82/2 x° = 41° (x + 8)° = (41 + 8)° = 49°

Find the measure of the exterior angle.

Answer: s° = 50° + 75° s° = 125° Thus the measure of the exterior angle is 125°

Answer: Sum of all the angles in a triangle = 180° t° + (t + 10)° + (t + 20)° = 180° 3t° + 10° + 20° = 180° 3t° + 30° = 180° 3t° = 180° – 30° 3t° = 150° t° = 150/3 t° = 50° Exterior angle: t° + (t + 10)° t° + t° + 10° 2t° + 10° 2(50)° + 10° = 100° + 10° = 110° Thus the measure of the exterior angle is 110°.

Question 16. What is the measure of each interior angle of an equilateral triangle? Explain.

Answer: Sum of all the angles in a triangle = 180° x° + 30° + 56° = 180° x° + 86° = 180° x° = 180° – 86° x° = 94° Thus the measure of the interior angle of the triangle at Chertan = 94°

3.3 Angles of Polygons (pp. 117–122)

Find the sum of the interior angle measures of the polygon.

Answer: The polygon has 13 sides S = (n – 2) . 180° S = (13- 2) . 180° S = 11 . 180° S = 1980° Thus the sum of the interior angle measure is 1980°

Answer: The polygon has 9 sides S = (n – 2) . 180° S = (9- 2) . 180° S = 7 . 180° S = 1260° Thus the sum of the interior angle measure is 1260°

Answer: S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360° Thus the sum of the interior angle measure is 360° x° + 60° + 128° + 95° = 360° x° + 283° = 360° x° = 360° – 283° x° = 77° Thus the value of x is 77°.

Answer: S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900° x° + 135° + 125° + 135° + 105° + 150° + 140° = 900° x° + 790° = 900° x° = 900° – 790° x° = 110°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° x° + 120° + 140° + 92° + 125° + 130° = 720° x° + 607° = 720° x° = 720° – 607° x° = 113° The value of x° is 113°

Answer: The given polygon is an octagon. It has 8 sides. S = (n – 2) . 180° S = (8 – 2) . 180° S = 6 . 180° S = 1080° Thus the sum of the interior angle measure is 1080° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 8. 1080 ÷ 3 = 135°

3.4 Using Similar Triangles (pp. 123–128)

Answer: x° + 68° + 90° = 180° x° = 180° – 158° x° = 22° y° + 22° + 90° = 180° y° + 112° = 180° y° = 180° – 112° y° = 68° The triangles have two pairs of congruent angles. So, the triangles are similar.

Answer: x° + 100° + 30° = 180° x° + 130° = 180° x° = 180° – 130° x° = 50° y° + 100° + 50° = 180° y° + 150° = 180° y° = 180° – 150° y° = 30° The triangles have two pairs of congruent angles. So, the triangles are similar.

Answer: x° + 50° + 85° = 180° x° + 135° = 180° x° = 180° – 135° x° = 45° y° + 85° + 35° = 180° y° + 120° = 180° y° = 180° – 120° y° = 60° The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

Answer: ∠B ≅ ∠D ∠A ≅ ∠C ∠AXB ≅ ∠CXD ∠AXB and ∠CXD are vertical angles. ΔAXB ∼ ΔCXD

Question 28. A person who is 5 feet tall casts a shadow that is 4 feet long. A nearby building casts a shadow that is 24 feet long. What is the height of the building?

Answer: Given, A person who is 5 feet tall casts a shadow that is 4 feet long. A nearby building casts a shadow that is 24 feet long. Let the height of the building = x ft x/24 = 5/4 24 . x/24 = 5/4 . 24 x = 30 Thus the height of the building is 30 ft.

Practice Test

Question 1. ∠7

Answer: 47°

Explanation: ∠7 and 47° angles are alternate exterior angles formed by a transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠7 is 47°

Explanation: ∠6 and 47° angles are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 47°

Answer: 133°

Explanation: ∠4 and 47° are supplementary angles. 47° + ∠4 = 180° ∠4 = 180° – 47° ∠4 = 133° Thus the measure of ∠4 = 133°

Question 4. ∠5

Explanation: ∠6 and 47° angles are corresponding angles formed by transversal intersecting parallel lines. The angles are congruent. Thus the measure of ∠6 is 47° ∠6 + ∠5 = 180° 47° + ∠5 = 180° ∠5 = 180° – 47° ∠5 = 133° Thus the measure of ∠5 = 133°

Answer: 28°

Explanation: Sum of all the angles in a triangle = 180° x° + 129° + 23° = 180° x° + 152° = 180° x° = 180° – 152° x° = 28° Thus the value of x° is 28°

Answer: 68°

Explanation: Sum of all the angles in a triangle = 180° x° + (x – 24)° + 68° = 180° x° + x° – 24° + 68° = 180° 2x° + 44° = 180° 2x° = 180° – 44° 2x° = 136° x° = 68° (x – 24)° = (68 – 24)° = 44°

Answer: j° = 40° + 90° j° = 130° The measure of an exterior angle is 130°.

Answer: The coin has 7 sides. S = (n – 2) . 180° S = (7 – 2) . 180° S = 5 . 180° S = 900° Thus the sum of the interior angle measure is 900°

Answer: S = (n – 2) . 180° S = (5 – 2) . 180° S = 3 . 180° S = 540° Thus the sum of the interior angle measure is 540° 2x° + 125° + 90° + 2x° + 125° = 540° 4x° + 340° = 540° 4x° = 540° – 340° 4x° = 200° x° = 200/4 x° = 50° The value of x° is 50°

Answer: S = (n – 2) . 180° S = (6 – 2) . 180° S = 4 . 180° S = 720° Thus the sum of the interior angle measure is 720° In a regular polygon, each interior angle is congruent. So, divide the sum of the interior angle measures by the number of interior angles, 6. 720 ÷ 6 = 120°

Answer: To find x°: x° + 61° + 70° = 180° x° + 131° = 180° x° = 180° – 131° x° = 49° To find y°: x° + 39° + 70° = 180° x° + 109° = 180° x° = 180° – 109° x° = 71° The triangles do not have two pairs of congruent angles. So, the triangles are not similar.

Answer: ∠A ≅ ∠QPB ∠C ≅ ∠PQB ΔBPQ ∼ ΔBAC

Answer: One way: ∠3 and 65° are supplementary angles. ∠5 and ∠3 are alternate interior angles. Another way: ∠8 and 65° are alternate exterior angles. ∠5 and ∠8 are supplementary angles.

Answer: Given, You swim 3.6 kilometers per hour. d/105 = 80/140 105 . d/105 = 80/140 . 105 d = 60 The length of the pond is 60 m. Speed = 3.6 km per hour = 1 m sec Distance = d = 60m Time it will take to swim across the pond = distance/speed = 60/1 = 60 sec = 1 min

Answer: C = 11 + 1.6t C – 11 = 1.6t 1.6t = C – 11 t = (C – 11)/1.6 Thus the correct answer is option B.

Answer: 5(x – 4) = 3x 5x – 20 = 3x 5x – 3x = 20 2x = 20 x = 20/2 x = 10 Thus the correct answer is option I.

Answer: △PQR is similar to △STU PQ = 12 ST = 16 SU = 20 TU = 18 PQ/ST = QR/TU 12/16 = X/18 16X = 12 × 18 X = 216/16 X = 13.5 cm Thus the correct answer is option C.

Answer: ∠y and 125° are supplementary angles. 125° + ∠y = 180° ∠y = 180° – 125° ∠y = 55° So, the measure of ∠y = 55° ∠x and ∠y are alternate interior angles. They are congruent. So, the measure of ∠x = 55°

Answer: My friend made the error by multiplying both sides by –\(\frac{2}{5}\). To correct the error she should multiply both sides by –\(\frac{5}{2}\) instead of –\(\frac{2}{5}\) Thus the correct answer is option F.

Answer: Given, X(-6,-1) Y(-3,-5) X(-2,-3) Reflecting a point (x,y) in the y-axis. (x, y) = (-x, y) X(-6,-1) = X'(6, -1) Y(-3,-5) = Y'(3, -5) X(-2,-3) = Z'(2, -3) Thus the correct answer is option B.

Answer: S = (n – 2) . 180° Part B A quadrilateral has angles measuring 100°, 90°, and 90°. Find the measure of its fourth angle. Show your work and explain your reasoning.

Answer: The quadrilateral has 4 sides S = (n – 2) . 180° S = (4 – 2) . 180° S = 2 . 180° S = 360 ° Thus the sum of the interior angles is 360 ° x° + 100° + 90° + 90° = 360° x° + 280° = 360° x° = 360° – 280° x° = 80° Thus the value of x° is 80°

Answer: Number of sides = 3 The number of interior triangles in the given figure = 3 Sum of the interior angles measure using triangle = 3 × 180° = 540

Conclusion:

I wish the details prevailed in the above article is beneficial for all the 8th grade students. Hope our Big Ideas Math Answers Grade 8 Chapter 3 Angles and Triangles helped you a lot to overcome the difficulties in this chapter. Feel free to post your comments in the comment box. Stay tuned to our ccssmathanswers.com to get step by step explanation for all the Grade 8 chapters.

## Leave a Comment Cancel Reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

## IMAGES

## VIDEO

## COMMENTS

3. 17 x 4. 29 61 x 5. 41 37 x 6. 60 60 x Find the missing measure in each triangle with the given angle measures. 7. 45°, 35°, x° 8. 100°, x°, 40° 9. x°, 90°, 16° 10. Find the third angle of a right triangle if one of the angles measures 24°. 11. What is the third angle of a right triangle if one of the angles measures 51°? 12.

Lesson 3 Skills Practice Angles of Triangles Find the value of x in each triangle with the given angle measures. 1. 81 x 84 2. 24 36 x 3. 49 x 4. 38 38 x 5. 65 x 6. 71 x 45 7. 57˚, 51˚, x˚ 8. x˚, 126˚, 22˚ 9. 90˚, x˚, 50˚ Find the value of x in each triangle. 10. 37° 72° x° 11. 66° 33° x° 12. 50° 40° x° 13. 60° 60° x° 14 ...

Question 23. Given three angle measures, you can construct a triangle. Answer: We can construct a triangle if the sum of the measure of the 3 angles is 180°. As a matter of fact, if the sum of the measures of the 3 angles is 180°. We can build an infinity of triangles that are similar. Question 24.

Answer: The triangle C has all 3 angles that are less than a right angle. Question 6. Explain how a triangle is a special kind of polygon. ... McGraw Hill My Math Grade 3 Chapter 14 Lesson 3 My Homework Answer Key. Practice. Measure the sides of each triangle below to the nearest quarter of an inch. Then state the number of sides with equal ...

Created Date: 1/26/2017 7:58:59 AM

An exterior angle of a triangle is equal to the sum (far away) angles. x + y = w. P.3-L3 P.P. Key. 7) If the area of a trapezoid is 48, and the bases 1 are 10 and 6, find the height of the trapezoid. c) 6. 8) In triangle ABC, the measure of angle A is 60 degrees. If the measure of angle B is two times that of Angle C, we can conclude that:

Find step-by-step solutions and answers to Geometry: Homework Practice Workbook - 9780078908491, as well as thousands of textbooks so you can move forward with confidence. ... Angles of Triangles. Section 4-3: Congruent Triangles. Section 4-4: Proving Congruence: SSS, SAS. Section 4-5: Proving Congruence: ASA, AAS. Section 4-6: Isosceles and ...

Find step-by-step solutions and answers to Big Ideas Geometry - 9781635988031, as well as thousands of textbooks so you can move forward with confidence. ... Angles of Triangles. Section 5.2: Congruent Polygons. Section 5.3: Proving Triangle Congruence by SAS. ... With Expert Solutions for thousands of practice problems, you can take the ...

The 3 angles of a triangle has a sum of 180 so the 3rd angle is 180 - 64 - 65 = 51. Finally, straight line with 51 so it is 180 - 51 = 129. The angle that forms a straight line with 89 is 180 - 89 = 91. Then the 3 angles of the triangle I have shaded have a sum of 180. So x = 180 - 91 - 41 = 48.

Since the sum of the interior angles in a triangle is always 180 ∘ , we can use an equation to find the measure of a missing angle. Example: Find the value of x in the triangle shown below. 106 ∘ x ∘ 42 ∘. We can use the following equation to represent the triangle: x ∘ + 42 ∘ + 106 ∘ = 180 ∘. The missing angle is 180 ∘ minus ...

112 Chapter 3 Angles and Triangles 3.2 Lesson Lesson Tutorials Interior Angle Measures of a Triangle Words The sum of the interior angle xí yí zí measures of a triangle is 180°. Algebra x + y + z = 180 Key Vocabulary interior angles of a polygon, p. 112 exterior angles of a polygon, p. 112 EXAMPLE 1 Using Interior Angle Measures Find the ...

The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary. Use the corollary to set up and solve an equation. x° + 2x° = 90°. Corollary to the Triangle Sum Theorem. = 30 Solve for x. So, the measures of the acute angles are 30° and 2(30°) = 60°.

Classifying Triangles by Angles. a triangle with three acute angles. Acute Triangle. A triangle that has a 90 degree angle. Right Triangle. a triangle with one obtuse angle. ... See an expert-written answer! We have an expert-written solution to this problem! About us. About Quizlet; How Quizlet works; Careers; Advertise with us; Get the app ...

Email your homework to your parent or tutor for free; ... Independent Practice. Draw a triangle that satisfies each set of conditions. Then classify the triangle. ... Question 1 (request help) a triangle with three acute angles and three congruent sides. Type below: (show solution) Question 2 (request help) a triangle with one right angle and ...

LESSON/HOMEWORK. LECCIÓN/TAREA. LESSON VIDEO. ANSWER KEY. EDITABLE LESSON. EDITABLE KEY. SMART NOTEBOOK. Lesson 2 Angles and Their Measures. ... U09.AO.02 - Practice with Angles in a Triangle RESOURCE. ANSWER KEY. EDITABLE RESOURCE. EDITABLE KEY. Get Access to Additional eMath Resources.

2 Unit 10 - Lesson 1 Aim: I can determine the measure of Complementary, Supplementary, & Vertical Angles. Warm Up: Answer the following questions based off your prior knowledge of grade 7- (1) Which pairs of angles are complementary? a. 42° and 58° b. 100° and 80° c. 38° and 52° d. 300° and 60° (2) If angles x and y are supplementary, which diagram below illustrates that situation?

Standardized Test Practice Vocabulary Review Lesson Resources Extra Examples Personal Tutor Self-Check Quizzes. Common Core State Standards Supplement, SE Hotmath Homework Help Math Review Math Tools Multilingual eGlossary Study to Go Online Calculators. Mathematics. Home > Chapter 4 > Lesson 2. Oklahoma Geometry. Chapter 4, Lesson 2: Angles of ...

The triangle has 3 sides. so divide the perimeter and number of sides of the triangle to get the answer. S = 18 / 3 S = 6 Therefore, the measure of one side is 6 inches. Question 13. Mathematical PRACTICE Use Math Tools Measure the angles in the triangle shown. What type(s) of angles are in the triangle shown? Answer: Acute angle Explanation:

a theorem with a proof tat follows as a direct result of another theorem. corollary 4.1. the acute angles of a right triangle are complementary. corollary 4.2. ther ecan be at most one right or obtuse angle in a triangle. Study with Quizlet and memorize flashcards containing terms like Right Scalene (Diagram), Right Isosceles (Diagram), Acute ...

Question 14. Answer: ∠1 and 60° are corresponding angles formed by a transversal intersecting parallel lines. The angles are congruent. ∠1 and ∠2 are supplementary angles. ∠1 + ∠2 = 180° 60° + ∠2 = 180° ∠2 = 180° - 60° ∠2 = 119° So, the measure of ∠2 is 119° ∠3 and ∠1 are vertical angles. They are congruent.

Lesson 3 Angles Of Triangles Page 393 Answer Key. Math Workshop, Grade 4 Homework Helpers: Geometry Maths Plus: Ict Numeracy Link - Year 4 Geometry - Drill Sheets Gr. 6-8 Measures, Shape and Space Guided Math Made Easy, Grade 3 Geometry Iii Tm' 2001 Ed. Content Area Lessons Using Graphic Organizers, Grade 6 Introduction to Geometry Foundations ...

x = -7 m<ABC = 47. x = 60. x = 110. x = 65. x = 105. Practice applying triangle sum theorem and the exterior angle sum theorem Learn with flashcards, games, and more — for free.

Complete lesson 3 homework practice triangles answer key effortlessly on any device. Online document management has grown to be more popular with businesses and individuals. It provides a perfect eco-friendly replacement for standard printed out and signed paperwork, since you can get the proper form and safely store it online.