Objectives To use the AA Postulate and the SAS and SSS Theorems To use similarity to find indirect measurements

Transcription

1 7-3 roving riangles imilar ontent tandards G..5 Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.G.5 rove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. Objectives o use the ostulate and the and heorems o use similarity to find indirect measurements re the triangles similar? How do you know? (Hint: Use a centimeter ruler to measure the sides of each triangle.) You ve already learned how to decide whether two polygons are similar. his is a special case of that problem. HI I In the olve It, you determined whether the two hat is, you needed information about all three pairs of angles and all three pairs of sides. In this lesson, you ll learn an easier way to determine whether two triangles are similar. esson Vocabulary V measurement ssential Understanding You can show that two triangles are similar when you know the relationships between only two or three pairs of corresponding parts. ostulate 7-1 ngle-ngle imilarity ( ) ostulate ostulate If two angles of one triangle are congruent to two angles of another triangle, then the If... and hen hapter 7 imilarity

2 What do you need to show that the triangles are similar? o use the ostulate, you need to prove that two pairs of angles are congruent. roblem 1 Using the ostulate re the two triangles similar? How do you know? W and V V because both angles measure 45. W V because vertical angles are congruent. o, W V by the ostulate. J and because both angles measure 70. y the riangle ngle-um heorem, m and m Only one pair of J angles is congruent. o, J and are not similar. 30 W V Got It? 1. re the two triangles similar? How do you know? a. 3 b Here are two other ways to determine whether two heorem 7-1 ide-ngle-ide imilarity ( ) heorem heorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the If... and hen... You will prove heorem 7-1 in ercise 35. heorem 7-2 ide-ide-ide imilarity ( ) heorem heorem If the corresponding sides of two triangles are proportional, then the If... hen... You will prove heorem 7-2 in ercise 3. esson

3 roof roof of heorem 7-1: ide-ngle-ide imilarity heorem Given: rove:, lan for roof: hoose X on so that X. raw XY. how that XY by the ostulate. hen use the proportion X Y and the given proportion to show that Y. hen prove that XY. Finally, prove that by the ostulate. X Y roblem 2 Verifying riangle imilarity re the triangles similar? If so, write a similarity statement for the triangles. V 12 W 10 U 15 X Use the side lengths to identify corresponding sides. hen set up ratios for each pair of corresponding sides. hortest sides XV 2 3 U 10 ongest sides WX U emaining sides VW U ll three ratios are equal, so corresponding sides are proportional. U XVW by the heorem. 10 V X W How can you make it easier to identify corresponding sides and angles? ketch and label two separate triangles by the efleive roperty of ongruence and o, by the heorem is the included angle between two known sides in each triangle. 452 hapter 7 imilarity

4 Got It? 2. re the triangles similar? If so, write a similarity statement for the triangles and eplain how you know the a. G b. W 12 F roof roblem 3 roving riangles imilar Given: FG GH, J, F J rove: FGH J F G H J he triangles are isosceles, so the base angles are congruent. tatements You need to show that the triangles are similar. easons Find two pairs of corresponding congruent angles and use the ostulate to prove the 1) FG GH, J 1) Given 2) FGH is isosceles. J is isosceles. 2) ef. of an isosceles 3) F H, J 3) ase of an isosceles are. 4) F J 4) Given 5) H J 5) ransitive roperty of ) H ) ransitive roperty of 7) FGH J 7) ostulate Got It? 3. a. Given: rove: b. easoning For the figure at the right, suppose you are given only that. ould you prove that the triangles are similar? plain. esson

5 ssential Understanding ometimes you can use similar triangles to find lengths that cannot be measured easily using a ruler or other measuring device. You can use indirect measurement to find lengths that are difficult to measure directly. One method of indirect measurement uses the fact that light reflects off a mirror at the same angle at which it hits the mirror. roblem 4 Finding engths in imilar riangles ock limbing efore rock climbing, arius wants to know how high he will climb. He places a mirror on the ground and walks backward until he can see the top of the cliff in the mirror. What is the height of the cliff? J ft H 5.5 ft ft V 34 ft efore solving for, verify that the HV JV by the ostulate because and HV JV. HV JV ostulate H V J V orresponding sides of triangles are proportional ubstitute. 17 ross roducts roperty 31.2 olve for. he cliff is about 31 ft high. Got It? 4. easoning Why is it important that the ground be flat to use the method of indirect measurement illustrated in roblem 4? plain. 454 hapter 7 imilarity

6 esson heck o you know HOW? re the triangles similar? If yes, write a similarity statement and eplain how you know they are similar Z 2 3. U G F F 15 o you U? 4. Vocabulary How could you use indirect measurement to find the height of the flagpole at your school? 5. rror nalysis Which solution for the value of in the figure at the right is not correct? plain.... a. ompare and ontrast How are the imilarity heorem and the ongruence ostulate alike? How are they different? b. How are the imilarity heorem and the ongruence ostulate alike? How are they different? HI I 4 ractice and roblem-olving ercises HI I ractice etermine whether the If so, write a similarity ee roblems 1 and 2. statement and name the postulate or theorem you used. If not, eplain. 7. F.. J H G J U G Y H 12 esson

7 13. Given: 14. Given: 2, roof rove: roof 2 rove: ee roblem 3. Indirect easurement plain why the hen find the distance represented by ee roblem ft 0 ft 135 ft 5 ft in. 4 ft 10 ft irror 17. Washington onument t a certain time of day, a 1.-m-tall person standing net to the Washington onument casts a 0.7-m shadow. t the same time, the Washington onument casts a 5.-m shadow. How tall is the Washington onument? pply an you conclude that the triangles are similar? If so, state the postulate or theorem you used and write a similarity statement. If not, eplain F G 21. a. re two isosceles triangles always similar? plain. b. re two right isosceles triangles always similar? plain. 22. hink bout a lan On a sunny day, a classmate uses indirect measurement to find the height of a building. he building s shadow is 12 ft long and your classmate s shadow is 4 ft long. If your classmate is 5 ft tall, what is the height of the building? an you draw and label a diagram to represent the situation? What proportion can you use to solve the problem? 23. Indirect easurement 2-ft vertical post casts a 1-in. shadow at the same time a nearby cell phone tower casts a 120-ft shadow. How tall is the cell phone tower? 45 hapter 7 imilarity

8 lgebra For each pair of similar triangles, find the value of Given:,, roof rove: V is isosceles. V 2. Given:, G roof rove: G V G 2. easoning oes any line that intersects two sides of a triangle and is parallel to the third side of the triangle form two similar triangles? Justify your reasoning. 30. onstructions raw any with m 30. Use a straightedge and compass to construct J so that J. 31. easoning In the diagram at the right, W. and are altitudes. he scale factor of to W is 4 3. What is the ratio of to? plain how you know. W 32. oordinate Geometry has vertices (0, 0), (2, 4), and roof (4, 2). has vertices (0, 3), (1, 5), and (2, 4). rove that. (Hint: Graph and in the coordinate plane.) 33. Write a proof of the following: ny two nonvertical parallel roof lines have equal slopes. Given: onvertical lines 1 and 2, 1 2, F and are to the -ais rove: F F y 1 O F 2 hallenge 34. Use the diagram in ercise 33. rove: ny two nonvertical lines with equal slopes roof are parallel. 35. rove the ide-ngle-ide imilarity heorem (heorem 7-1). roof Given:, rove: esson

9 3. rove the ide-ide-ide imilarity heorem (heorem 7-2). roof Given: rove: tandardized est rep / 37. omplete the statement. y which postulate or theorem are the triangles similar? ; ; ; ; and 2 are alternate interior angles formed by two parallel lines and a transversal. If m2, what is m1? he length of a rectangle is twice its width. If the perimeter of the rectangle is 72 in., what is the length of the rectangle? 12 in. 1 in. 24 in. 3 in. tended esponse 40. Graph (2, 4), (4, ), (, 4), and (4, 2). What type of polygon is? Justify your answer. ied eview ZY. Use the diagram at the right to find the following. 41. the scale factor of to ZY 42. m 43. Y Y 45 Z ee esson 7-2. Use a protractor to find the measure of each angle. lassify the angle as acute, right, obtuse, or straight. ee esson Get eady! o prepare for esson 7-4, do ercises lgebra Identify the means and etremes of each proportion. hen solve for m ee esson hapter 7 imilarity