a Are any two equilateral hexagons similar?

In Exercises 21 to 24, .

21. Given: , ,

Find: BC

22. Given: , ,

Find: AB HINT: Find DB first.

23. Given: , ,

Find: BC

24. Given: , ,

Find: AE

25. Pentagon ABCDE pentagon GHJKL not shown,

, and . If the perimeter of ABCDE is 50, find the perimeter of GHJKL.

26. Quadrilateral MNPQ quadrilateral WXYZ not shown,

, and . If the longest side of MNPQ is of length 8, find the length of the longest side of WXYZ.

27. A blueprint represents the 72-ft length of a building by a

line segment of length 6 in. What length on the blueprint would be used to represent the height of this 30-ft-tall building?

28. A technical drawing shows the -ft lengths of the legs of

a baby’s swing by line segments 3 in. long. If the diagram should indicate the legs are ft apart at the base, what length represents this distance on the diagram? In Exercises 29 to 32, use the fact that triangles are similar.

29. A person who is walking away from a 10-ft lamppost

casts a shadow 6 ft long. If the person is at a distance of 10 ft from the lamppost at that moment, what is the person’s height? 2 1 2 3 1 2 YZ = 7 PQ = 5 GH = 9 AB = 6 DB = AE AC = 18 AD = 4 EC = BC AC = 20 DE = 4 DB = BC AD = 8 DE = 5 EC = BC AE = 6 DE = 4 䉭ADE 䉭ABC C A B D E Exercises 21–24

30. With 100 ft of string out, a kite is 64 ft above ground

level. When the girl flying the kite pulls in 40 ft of string, the angle formed by the string and the ground does not change. What is the height of the kite above the ground after the 40 ft of string have been taken in?

31. While admiring a rather tall tree, Fred notes that the

shadow of his 6-ft frame has a length of 3 paces. On the level ground, he walks off the complete shadow of the tree in 37 paces. How tall is the tree?

32. As a garage door closes, light is

cast 6 ft beyond the base of the door as shown in the accompanying drawing by a light fixture that is set in the garage ceiling 10 ft back from the door. If the ceiling of the garage is 10 ft above the floor, how far is the garage door above the floor at the time that light is cast 6 ft beyond the door?

33. In the drawing,

with transversals and m. If D and C are the midpoints of and , respectively, then is trapezoid ABCD similar to trapezoid DCFE?

34. In the drawing, .

Suppose that transversals and m are also parallel. D and C are the midpoints of and , respectively. Is parallelogram ABCD similar to parallelogram DCFE?

35. Given , a second triangle

is constructed so that and . a Is congruent to ? b Using intuition appearance, does it seem that is similar to ? 䉭ABC 䉭XTN ∠B ∠T ∠N ⬵ ∠C ∠X ⬵ ∠A 䉭XTN 䉭ABC BF AE Í AB 7 Í DC 7 Í EF BF AE Í AB 7 Í DC 7 Í EF 10 10 6 B C A T N X R S T U W

36. Given

, a second triangle is constructed so that , , and . a What is the constant value of the ratios , , and ? b Using intuition appearance, does it seem that is similar to ? 䉭RST 䉭UVW WU RT VW ST UV RS WU = 2RT VW = 2ST UV = 2RS 䉭UVW 䉭RST A D m E B C F Exercises 33, 34 For Exercises 37 and 38, use intuition to form a proportion based on the drawing shown.

37. has an inscribed rhombus ARST. If

and , find the length x of each side of the rhombus. AC = 6 AB = 10 䉭ABC C B A T R ␹ S E F G H 6 2 C D A B

38. A square with sides of length 2 in. rests as shown on a

square with sides of length 6 in. Find the perimeter of trapezoid ABCD. AAA AA CSSTP CASTC SAS SSS Proving Triangles Similar 5.3 KEY CONCEPTS Because of the difficulty of establishing proportional sides, our definition of similar polygons and therefore of similar triangles is almost impossible to use as a method of proof. Fortunately, some easier methods are available for proving triangles similar. If two triangles are carefully sketched or constructed so that their angles are congruent, they will appear to be similar, as shown in Figure 5.11. S T H R J K 䉭 HJK 䉭SRT Figure 5.11 If the three angles of one triangle are congruent to the three angles of a second triangle, then the triangles are similar AAA. POSTULATE 15 Corollary 5.3.1 of Postulate 15 follows from knowing that if two angles of one triangle are congruent to two angles of another triangle, then the third angles must also be con- gruent. See Corollary 2.4.4. Technology Exploration Use a calculator if available. On a sheet of paper, draw two similar triangles, ABC and 䉭DEF. To accomplish this, use your protractor to form three pairs of congruent corresponding angles. Using a ruler, measure , , , , , and . Show that . NOTE: Answers are not “perfect.” AB DE = BC EF = AC DF DF EF DE AC BC AB 䉭 If two angles of one triangle are congruent to two angles of another triangle, then the tri- angles are similar AA. COROLLARY 5.3.1 Rather than use AAA to prove triangles similar, we will use AA instead because it requires fewer steps.