Draw a segment from A perpendicular to m. 18. Draw A surveyor knows that a lot has the shape of an isosceles

In Exercises 35 to 37, complete each proof.

35. Given:

Prove: is isosceles HINT: First show that .

36. Given:

M is the midpoint of and Prove:

37. Given: Isosceles

with vertex P Isosceles with vertex Q Prove:

38. In isosceles triangle BAT, . Also,

. If and , find the perimeter of: a b c

39. In , ,

and . Find: a b c

40. In , .

bisects , and bisects . If , name all isosceles triangles shown in the drawing. m ∠P = 36° ∠PNM NA ∠PMN MB PM ⬵ PN 䉭PMN m ∠A m ∠ARB m ∠T m ∠RBT = 20° BR ⬵ BT ⬵ AR 䉭BAT 䉭RBT 䉭ARB 䉭BAT AR = 7.6 AB = 12.3 BR ⬵ BT ⬵ AR AB ⬵ AT 䉭MQP ⬵ 䉭NQP 䉭MNQ 䉭MNP MX ⬵ MT MT ⬜ WZ MX ⬜ WY YZ WY ⬵ WZ 䉭 RUS ⬵ 䉭VUT 䉭STU RU ⬵ VU ∠1 ⬵ ∠3 In Exercises 42 to 44, explain why each statement is true.

42. The altitude from the vertex of an isosceles triangle is also

the median to the base of the triangle.

43. The bisector of the vertex angle of an isosceles triangle

bisects the base.

44. The angle bisectors of the base angles of an isosceles

triangle, together with the base, form an isosceles triangle.

45. Given: In the figure,

, and Z is the midpoint of . XW XZ ⬵ YZ U V R S T 1 2 3 Y Z M X T W M N P Q B T R A Exercises 38, 39 M N A B Q P B C A Y X Z b e f a d c W

41. lies in the structural support system of the Ferris

wheel. If and , find the measures of and . ∠C ∠B AB = AC = 20 ft m ∠A = 30° 䉭ABC Prove: is a right triangle with . HINT: Let .

46. Given: In the figure,

. Also, . If YW = 14.3 in. and YZ = 7.8 in., find the perimeter of to the nearest tenth of an inch. 䉭XYW YZ ⬵ ZW a = e = 66° m ∠X = a m ∠XYW = 90° 䉭XYW Justifying Constructions Basic Constructions Justified 3.4 KEY CONCEPTS In earlier sections, construction methods were introduced that appeared to achieve their goals; however, the methods were presented intuitively. In this section, we justify the construction methods and apply them in further constructions. The justification of the method is a “proof ” that demonstrates that the construction accomplished its purpose. See Example 1. EXAMPLE 1 Justify the method for constructing an angle congruent to a given angle. GIVEN: by construction by construction PROVE: ∠B ⬵ ∠S DE ⬵ TR BD ⬵ BE ⬵ ST ⬵ SR ∠ABC PROOF Statements Reasons 1. ; 1. Given 2. 2. Given 3. 3. SSS 4. 4. CPCTC ∠B ⬵ ∠S 䉭EBD ⬵ 䉭RST DE ⬵ TR BD ⬵ BE ⬵ ST ⬵ SR ∠ABC S T R B D C A E Figure 3.38 In Example 2, we will apply the construction method that was justified in Exam- ple 1. Our goal is to construct an isosceles triangle that contains an obtuse angle. It is necessary that the congruent sides include the obtuse angle. A a a Figure 3.39 A C B a a b EXAMPLE 2 Construct an isosceles triangle in which obtuse is included by two sides of length a [see Figure 3.39a]. Solution Construct an angle congruent to . From A, mark off arcs of length a at points B and C as shown in Figure 3.39b. Join B to C to complete . 䉭ABC ∠A ∠A 쮿 쮿