## : : Ans: Key: Study: Guide: And: Interventionpdf

## Lesson 81 Similar Polygons

**Monitoring Progress**

Question 1.In the diagram, JKL ~ PQR. Find the scale factor from JKL to PQR. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality.

Answer:The pairs of congruent angles are K = Q, J = P, L = RThe scale factor is \The ratios of the corresponding side lengths in a statement of proportionality are \

Explanation:

Scale factor = \= \= \Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion\ = \\ = \KM = \ x 35KM = 42

The two gazebos shown are similar pentagons. Find the perimeter of Gazebo A.

Answer:Perimeter of Gazebo A = 46 m

Explanation:Scale factor = \= \So, \ = \\ = \x = 12\ = \\ = \ED = 10\ = \\ = \DC = 8\ = \\ = \BC = 6Therefore, perimeter = 6 + 8 + 10 + 12 + 10 = 46

Question 5.In the diagram, GHJK ~ LMNP. Find the area of LMNP.Area of GHJK = 84m2

Area of LMNP = 756 m2

Explanation:As shapes are similar, their corresponding side lengths are proportional.Scale Factor k = \= \Area of LMNP = k² x Area of GHJK= 3² x 84

\ = \ = \The ratio of perimeter is \.

In Exercises 13-16, two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon.

Question 13.perimeter of smaller polygon: 48 cm: ratio: \Answer:

perimeter of smaller polygon: 66 ft: ratio: \

Answer:The perimeter of larger polygon is 88 ft.

## Proving Triangle Similarity By Aa

**Exploration 1**

Comparing Triangles

Work with a partner. Use dynamic geometry software.

a. Construct ABC and DEF So that mA = mD = 106°, mB = mE = 31°, and DEF is not congruent to ABC.Answer:mC mF

b. Find the third angle measure and the side lengths of each triangle. Copy the table below and record our results in column 1.Answer:

c. Are the two triangles similar? Explain.**CONSTRUCTING VIABLE ARGUMENTS**To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results in constructing arguments.Answer:

d. Repeat parts to complete columns 2 and 3 of the table for the given angle measures.Answer:

e. Complete each remaining column of the table using your own choice of two pairs of equal corresponding angle measures. Can you construct two triangles in this way that are not similar?Answer:

f. Make a conjecture about any two triangles with two pairs of congruent corresponding angles.Answer:

Communicate Your Answer

Question 2.What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent?Answer:

Find RS in the figure at the left.Answer:

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## Objectives Of 8th Grade Math

Below is the general list of math objectives that 8th Grade Students should attain

- Identify Rational and Irrational Numbers
- Calculate and Approximate Principal Square Roots
- Solve Linear Inequalities in Two Variables.
- Differentiate between Types of Sampling Techniques.
- Determine Mean, Median, Mode, and Range of a Set of Real World Data.

## Exercise 82 Proving Triangle Similarity By Aa

Vocabulary and Core Concept Check

Question 1.**COMPLETE THE SENTENCE**If two angles of one triangle are congruent to two angles of another triangle. then the triangles are __________ .Answer:

Question 2.**WRITING**Can you assume that corresponding sides and corresponding angles of any two similar triangles are congruent? Explain.Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 6. determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.

Question 3.

Describe and correct the error in finding the value of x.Answer:

Question 21.**MODELING WITH MATHEMATICS**You can measure the width of the lake using a surveying technique, as shown in the diagram. Find the width of the lake, WX. Justify your answer.Answer:

Question 22.**MAKING AN ARGUMENT**You and your cousin are trying to determine the height of a telephone pole. Your cousin tells you to stand in the poles shadow so that the tip of your shadow coincides with the tip of the poles shadow. Your Cousin claims to be able to use the distance between the tips of the shadows and you, the distance between you and the pole, and your height to estimate the height of the telephone pole. Is this possible? Explain. Include a diagram in your answer.Answer:

**REASONING**In Exercises 23 26, is it possible for JKL and XYZ to be similar? Explain your reasoning.

Question 23.mJ = 71°, mK = 52°, mX = 71°, and mZ = 57°Answer:

JKL is a right triangle and mX + mY= 150°.Answer:

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## Exercise 83 Proving Triangle Similarity By Sss And Sas

Vocabulary and Core Concept Check

Question 1.**COMPLETE THE SENTENCE**You plan to show that QRS is similar to XYZ by the SSS Similarity Theorem . Copy and complete the proportion that you will use:Answer:

**WHICH ONE DOESNT BELONG?**Which triangle does not belong with the other three? Explain your reasoning.Answer:

Monitoring progress and Modeling with Mathematics

In Exercises 3 and 4, determine whether JKL or RST is similar to ABC.

Question 3.

In Exercises 5 and 6, find the value of x that makes DEF ~ XYZ.

Question 5.

Question 6.Answer:

In Exercises 7 and 8, verify that ABC ~ DEF Find the scale factor of ABC to DEF

Question 7.ABC: BC = 18, AB = 15, AC = 12DEF: EF = 12, DE = 10, DF = 8Answer:

ABC: AB = 10, BC = 16, CA = 20DEF: DE = 25, EF = 40, FD =50Answer:

In Exercises 9 and 10. determine whether the two triangles are similar. If they are similar, write a similarity statement and find the scale factor of triangle B to triangle A.

Question 9.

Question 10.Answer:

In Exercises 11 and 12, sketch the triangles using the given description. Then determine whether the two triangles can be similar.

Question 11.In RST, RS = 20, ST = 32, and mS = 16°. In FGH, GH = 30, HF = 48, and mH = 24°.Answer:

Question 12.The side lengths of ABC are 24, 8x, and 48, and the side lengths of DEF are 15, 25, and 6x.

Answer:\ = \ = \\ = \x = 5

In Exercises 13 16. show that the triangles are similar and write a similarity statement. Explain your reasoning.

Question 13.

Question 3.

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## Big Ideas Math Book Geometry Answer Key Chapter 8 Similarity

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Tell whether the ratios form a proportion.

Question 1.\

Answer:Yes, the ratios \ form a proportion.

Explanation:A proportion means two ratios are equal.So, cross product of \ is 21 x 5 = 105 = 35 x 3Therefore, \ form a proportion.

Question 2.\

Answer:Yes, the ratios \ form a proportion.

Explanation:If the cross product of two ratios is equal, then it forms a proportion.So, 24 x 24 = 576 = 64 x 9Therefore, the ratios \ form a proportion.

Question 3.\

Answer:The ratios \ do not form a proportion.

Explanation:If the cross product of two ratios is equal, then it forms a proportion.So, 56 x 6 = 336, 28 x 8 = 224Therefore, the ratios \ do not form a proportion.

Question 4.\

Answer:The ratios \ do not form a proportion.

Explanation:If the cross product of two ratios is equal, then it forms a proportion.So, 9 x 18 = 162, 27 x 4 = 108Therefore, the ratios \ do not form a proportion.

Question 5.\

Answer:The ratios \ form a proportion.

Question 6.\

## Lesson 83 Proving Triangle Similarity By Sss And Sas

**Monitoring progress**

Which of the three triangles are similar? Write a similarity statement.

Answer:The ratios are equal. So, LMN, XYZ are similar.The ratios are not equal. So LMN, RST are not similar.

Explanation:Compare LMN, XYZ by finding the ratios of corresponding side lengthsShortest sides: \ = \ = \Longest sides: \ = \ = \Remaining sides: \ = \ = \The ratios are equal. So, LMN, XYZ are similar.Compare LMN, RST by finding the ratios of corresponding side lengthsShortest sides: \ = \ = \Longest sides: \ = \Remaining sides: \ = \ = \The ratios are not equal. So LMN, RST are not similar.

Question 2.The shortest side of a triangle similar to RST is 12 units long. Find the other side 1enths of the triangle.

Answer:The other side lengths of the triangle are 15 units, 16.5 units.

Explanation:The shortest side of a triangle similar to RST is 12 unitsScale factor = \ = \So, other sides are 33 x \ = 16.5, 30 x \ = 15.

Explain how to show that the indicated triangles are similar.

Question 3.

XZW and YZX are not proportional.

Explanation:The shorter sides: \Longer sides: \ = \The side lengths are not proportional. So XZW and YZX are not proportional.

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## Lesson 82 Proving Triangle Similarity By Aa

**Monitoring Progress**

Show that the triangles are similar. Write a similarity statement.

Question 1.

Suppose that \ \ in Example 2 part . Could the triangles still be similar? Explain.Answer:

Question 4.**WHAT IF?**A child who is 58 inches tall is standing next to the woman in Example 3. How long is the childs shadow?Answer:

Question 5.You are standing outside, and you measure the lengths 0f the shadows cast by both you and a tree. Write a proportion showing how you could find the height of the tree.Answer:

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## Proving Triangle Similarity By Sss And Sas

**Exploration 1**

Work with a partner: Use dynamic geometry software.

a. Construct ABC and DEF with the side lengths given in column 1 of the table below.Answer:

b. Copy the table and complete column 1.Answer:

c. Are the triangles similar? Explain your reasoning.Answer:

d. Repeat parts for columns 2 6 in the table.Answer:

e. How are the corresponding side lengths related in each pair of triangles that are similar? Is this true for each pair of triangles that are not similar?Answer:

f. Make a conjecture about the similarity of two triangles based on their corresponding side lengths.**CONSTRUCTING VIABLE ARGUMENTS**To be proficient in math, you need to analyze situations by breaking them into cases and recognize and use counter examples.Answer:

g. Use your conjecture to write another set of side lengths of two similar triangles. Use the side lengths to complete column 7 of the table.Answer:

Work with a partner: Use dynamic geometry software. Construct any ABC.a. Find AB, AC, and mA. Choose any positive rational number k and construct DEF so that DE = k AB, DF = k AC, and mD = mA.Answer:

b. Is DEF similar to ABC? Explain your reasoning.Answer:

c. Repeat parts and several times by changing ABC and k. Describe your results.Answer:

Communicate Your Answer

Question 3.What are two ways to use corresponding sides of two triangles to determine that the triangles are similar?Answer:

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