7.1 & 7.2 1/30/13

Bell Work 1. If ∆ QRS  ∆ ZYX, identify the pairs of congruent angles and the pairs of congruent sides. Solve each proportion x = 9x = 18 Q  Z; R  Y; S  X; QR  ZY; RS  YX; QS  ZX

Example 12: Story Problem Marta 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 Property 5(15) = x (12.5 ) Simplify. 75 = 12.5 x Divide both sides by 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 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 Property 9.2(996) = 6( x ) Simplify = 6 x Divide both sides by = x The height of the actual tower is 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 :996, or 9.2:6. So the ratios are equal, and the answer is correct.

7.2- Objectives Identify similar polygons Apply properties of similar polygons to solve problems

Vocab Similar Similar polygons Similarity ratio

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

T w o p o l y g o n s a r e s i m il a r p o l y g o n s if a n d o n l y if t h e ir c o rr e s p o n d i n g a n g l e s a r e c o n g r u e n t a n d t h e ir c o rr e s p o n d i n g si d e l e n g t h s a r e p r o p o rt i o n a l.

Example 1 Identify the pairs of congruent angles and corresponding sides. N  Q and P  R. By the Third Angles Theorem, M  T.

Example 2 Identify 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 3 Determine 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 4 Determine 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. mW = mS = 62° mT = 180° – 2(62°) = 56°

Example 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. Thus the similarity ratio is, and ∆LMJ ~ ∆PNS.Thus the similarity ratio is, and ∆LMJ ~ ∆PNS.

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

7.3 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 1 Explain 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 2 Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~.

Example 3 Verify that the triangles are similar. ∆DEF and ∆HJK D  H by the Definition of Congruent Angles. Therefore ∆DEF ~ ∆HJK by SAS ~.

Example 4 Verify that ∆TXU ~ ∆VXW. TXU  VXW by the Vertical Angles Theorem. Therefore ∆TXU ~ ∆VXW by SAS ~.

Example 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 continued Step 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.