7.1 & 7.27.1 & 7.21/30/13

Bell WorkBell Work

1. If ∆QRS ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides.

Solve each proportion.

2. 3.

x = 9 x = 18

Q Z; R Y; S X; QR ZY; RS YX; QS ZX

Example 12: Story ProblemExample 12: Story ProblemMarta is making a scale drawing of her bedroom. Her rectangular room is 12 ½ feet wide and 15 feet long. On the scale drawing, the width of her room is 5 inches. What is the length?

The answer will be the length of the room on the scale drawing.

Understand the problem

Make a Plan

Let x be the length of the room on the scale drawing. Write a proportion that compares the ratios of the width to the length.

Solve

Cross Products Property5(15) = x(12.5)

Simplify.75 = 12.5x

Divide both sides by 12.5.x = 6

The length of the room on the scale drawing is 6 inches.

Look Back

Check the answer in the original

problem. The ratio of the width to the

length of the actual room is 12 :15,

or 5:6. The ratio of the width to the

length in the scale drawing is also

5:6. So the ratios are equal, and the

answer is correct.

Example 13: Problem StoryExample 13: Problem Story

Suppose the special-effects team made a different model with a height of 9.2 m and a width of 6 m. What is the height of the actual tower? The width of the actual tower is 996 m

Understand the Problem

The answer will be the height of the tower.

Make a Plan

Let x be the height of the tower. Write a proportion that compares the ratios of the height to the width.

Solve

Cross Products Property9.2(996) = 6(x)

Simplify.9163.2 = 6x

Divide both sides by 6.1527.2 = x

The height of the actual tower is 1527.2 feet.

Look Back

Check the answer in the original problem. The ratio of the height to the width of the model is 9.2:6. The ratio of the height to the width of the tower is 1527.2:996, or 9.2:6. So the ratios are equal, and the answer is correct.

7.2- Objectives7.2- Objectives

Identify similar polygons

Apply properties of similar polygons to solve problems

VocabVocab

SimilarSimilar polygonsSimilarity ratio

Similar PolygonsSimilar Polygons

Figures that are similar (~) have the same shape but not necessarily the same size.

Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.

Example 1Example 1Identify the pairs of congruent angles and corresponding sides.

N Q and P R. By the Third Angles Theorem, M T.

Example 2Example 2Identify the pairs of congruent angles and corresponding sides.

B G and C H. By the Third Angles Theorem, A J.

A similarity ratio is the ratio of the lengths of

the corresponding sides of two similar polygons.

The similarity ratio of ∆ABC to ∆DEF is , or .

The similarity ratio of ∆DEF to ∆ABC is , or 2.

Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.

Writing Math

Example 3Example 3Determine whether the polygons are similar. If so, write the similarity ratio and a similarity

statement.

rectangles ABCD and EFGH

Step 1 Identify pairs of congruent angles.

A E, B F, C G, and D H.

All s of a rect. are rt. s and are .

Step 2 Compare corresponding sides.

The similarity ratio is , and ABCD ~ EFGH.

Example 4Example 4Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement.

∆ABCD and ∆EFGH

Step 1 Identify pairs of congruent angles.

P R and S W isos. ∆

Step 2 Compare corresponding angles.

Since no pairs of angles are congruent, the triangles are not similar.mW = mS = 62°

mT = 180° – 2(62°) = 56°

Example 5Example 5

Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio and a similarity statement.

Step 1 Identify pairs of congruent angles.

N M, L P, S J

Step 2 Compare corresponding sides.

When you work with proportions, be sure the ratios compare corresponding measures.

Helpful Hint

7.37.3

There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

Example 1Example 1Explain why the triangles are similar and write a similarity statement.

Since , B E by the Alternate Interior Angles Theorem. Also, A D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

Example 2Example 2

Verify that the triangles are similar.

∆PQR and ∆STU

Therefore ∆PQR ~ ∆STU by SSS ~.

Example 3Example 3

Verify that the triangles are similar.

∆DEF and ∆HJK

D H by the Definition of Congruent Angles.

Therefore ∆DEF ~ ∆HJK by SAS ~.

Example 4Example 4

Verify that ∆TXU ~ ∆VXW.

TXU VXW by the Vertical Angles Theorem.

Therefore ∆TXU ~ ∆VXW by SAS ~.

Example 5Example 5

Explain why ∆ABE ~ ∆ACD, and then find CD.

Step 1 Prove triangles are similar.

A A by Reflexive Property of , and

Therefore ∆ABE ~ ∆ACD by AA ~.

B C since they are both right angles.

Ex 5 continuedEx 5 continuedStep 2 Find CD.

Corr. sides are proportional. Seg. Add. Postulate.

Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA.

Cross Products Prop. x(9) = 5(3 + 9)

Simplify. 9x = 60

Divide both sides by 9.