## 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: If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. Because if the two angle pairs are the same, then the third pair must also be equal when the three angle pairs are all equal, the three pairs of sides must be in proportion.

Question 3.Find RS in the figure at the left.Answer:

## 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: The corresponding angles of two similar triangles are always congruent but the corresponding angles of two analogous triangles are always harmonious but the corresponding sides of the two triangles dont have to be harmonious. In an analogous triangle, the corresponding sides are commensurable which means that the rates of corresponding sides are equal. However, also the corresponding sides are harmonious, but in other cases, if these rates are 1.

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.

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:

To estimate the height of the pole CDE can be proved similar to CAD asC

CED = CBA

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.CDE ~ CAB

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

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**Read Also: How To Find Kp In Chemistry **

## 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

Question 6.Decide whether the hexagons in Tile Design 1 are similar. Explain.

Answer: Both the hexagons are different. On the outer side of the hexagon, all the sides are equal. In the inside hexagon among the 6-sided 3 sides are different.

Question 7.Decide whether the hexagons in Tile Design 2 are similar. Explain.

## 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

**Also Check: What Is First Law Of Thermodynamics In Physics **

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

Get the handy preparation material free of cost and enhance your math skills with ease. To make this possible, students have to access the given links below and download the * Big Ideas Math Geometry Textbook Answers Ch 8 Similarity in Pdf.* In this guide, teachers, parents, and kids can find various questions from basic to complex levels. By solving the questions of chapter 8 similarity covered in BIM Geometry Ch 8 Answer key exercises, practice tests, chapter tests, cumulative assessments, etc. helps you to improve your skills & knowledge.

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.\

## 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:Among the four triangles the second triangle is different.The second triangle is not a right-angled triangle.Except for the second triangle all triangles are right angled triangles.

Monitoring progress and Modeling with Mathematics

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

Question 3.

Here the lengths of the sides that include T and L are not proportionalSince the triangles are not similar, thus the scale factor cannot be found.Hence the scale factor does not similar in this case.

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.

In Exercises 17 and 18, use XYZ.

Question 17.The shortest side of a triangle similar to XYZ is 20 units long. Find the other side lengths of the triangle.Answer:

**Read Also: Common Core Algebra 1 Unit 3 Lesson 4 Homework **

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## 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.

**Read Also: Compact Numerical Methods For Computers Linear Algebra And Function Minimisation **

## Exercise 81 Similar Polygons

Vocabulary and Core Concept Check

Question 1.**COMPLETE THE SENTENCE**For two figures to be similar, the corresponding angles must be ____________ . and the corresponding side lengths must be _____________ .

Answer:

\ = \ = \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.

Explanation:\ = \ = \66 x 4 = 3xx = \ = 88

Question 15.perimeter of larger polygon: 120 yd: rttio: \Answer:

perimeter of larger polygon: 85 m ratio: \

Answer:The perimeter of smaller polygon is 34 m.

Explanation:\ = \ = \85 x 2 = 5xx = \ = 34

Question 17.**MODELING WITH MATHEMATICS**A school gymnasium is being remodeled. The basketball court will be similar to an NCAA basketball court, which has a length of 94 feet and a width of 50 feet. The school plans to make the width of the new court 45 feet. Find the perimeters of ail NCAA court and of the new court in the school.Answer:

Question 18.**MODELING WITH MATHEMATICS**Your family has decided to put a rectangular patio in your backyard. similar to the shape of your backyard. Your backyard has a length of 45 feet and a width of 20 feet. The length of your new patio is 18 feet. Find the perimeters of your backyard and of the patio.

Question 19.

Question 25.