a All sides of are of the same length.

H I M 67° 23° ? 5000⬘

31. Given: and intersect

at K; Prove:

32. The sum of the measures of

two angles of a triangle equals the measure of the third largest angle. What type of triangle is described?

33. Draw, if possible, an

a isosceles obtuse triangle. b equilateral right triangle.

34. Draw, if possible, a

a right scalene triangle. b triangle having both a right angle and an obtuse angle.

35. Along a straight shoreline, two houses are located at points

H and M. The houses are 5000 feet apart. A small island lies in view of both houses, with angles as indicated. Find . m ∠I ∠P ⬵ ∠N ∠M ⬵ ∠Q PQ MN

39. A lamppost has a design such

that and . Find and .

40. For the lamppost of

Exercise 39, suppose that and that . Find , , and .

41. The triangular symbol on the

“PLAY” button of a DVD has congruent angles at M and N. If , what are the measures of angle M and angle N? m ∠P = 30° m ∠C m ∠B m ∠A m ∠C = 3 m∠A m ∠A = m∠B m ∠B m ∠A ∠A ⬵ ∠B m ∠C = 110° K P N M Q

36. An airplane has leveled off is flying horizontally at an

altitude of 12,000 feet. Its pilot can see each of two small towns at points R and T in front of the plane. With angle measures as indicated, find m . ∠R 12,000 65 37 ? R T

37. On a map, three Los Angeles

suburbs are located at points N Newport Beach, P Pomona, and B Burbank. With angle measures as indicated, determine and .

38. The roofline of a house shows

the shape of right triangle ABC with . If the measure of is 24° larger than the measure of , then how large is each angle? ∠CBA ∠CAB m ∠C = 90° m ∠P m ∠N 33 2 x P N B x A B C A B C M N P P N M Q

42. A polygon with four sides is called a quadrilateral.

Consider the figure and the dashed auxiliary line. What is the sum of the measures of the four interior angles of this or any other quadrilateral?

43. Explain why the following statement is true.

Each interior angle of an equiangular triangle measures 60°.

44. Explain why the following statement is true.

The acute angles of a right triangle are complementary. In Exercises 45 to 47, write a formal proof for each corollary.

45. The measure of an exterior angle of a triangle equals the

sum of the measures of the two nonadjacent interior angles.

46. If two angles of one triangle are congruent to two angles

of another triangle, then the third angles are also congruent.

47. Use an indirect proof to establish the following theorem:

A triangle cannot have more than one right angle.

48. Given:

, , and bisects bisects Prove: is a right angle ∠G ∠CFE FG ∠BCF CG Í AB 7 Í DE Í CF Í DE Í AB 2 1

3 4

A C B G F D E

49. Given: bisects

bisects Find: m ∠M m ∠Q = 42° ∠MPR PQ ∠MNP NQ Q M R P N b b a a A F E C D B

50. Given: In rt. ,

bisects and bisects . Find: m ∠FED ∠ABC BF ∠CAB AD 䉭ABC Convex Polygons Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon Concave Polygon Diagonals of a Polygon Regular Polygon Equilateral Polygon Equiangular Polygon Polygram Convex Polygons 2.5 KEY CONCEPTS A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoints. DEFINITION The polygons we generally consider in this textbook are convex; the angle measures of convex polygons are between 0° and 180°. Convex polygons are shown in Figure 2.28; those in Figure 2.29 are concave. A line segment joining two points of a concave poly- gon can contain points in the exterior of the polygon. Thus, a concave polygon always has at least one reflex angle. Figure 2.30 shows some figures that aren’t polygons at all Convex Polygons Figure 2.28 Concave Polygons Figure 2.29 Not Polygons W Z X Y R S T Figure 2.30