Given: and are TEST 534

The triangle in Figure 2.24 is known as , or etc. order of letters A, B, and C being unimportant. Each point A, B, and C is a vertex of the triangle; collectively, these three points are the vertices of the triangle. , , and are the sides of the triangle. Point D is in the interior of the triangle; point E is on the triangle; and point F is in the exterior of the triangle. Triangles may be categorized by the lengths of their sides. Table 2.1 presents each type of triangle, the relationship among its sides, and a drawing in which congruent sides are marked. AC BC AB 䉭BCA, 䉭ABC A triangle symbol 䉭 is the union of three line segments that are determined by three noncollinear points. DEFINITION TABLE 2.1 Triangles Classified by Congruent Sides Type Number of Congruent Sides Scalene None Isosceles Two Equilateral Three C A B F E D Figure 2.24 TABLE 2.2 Triangles Classified by Angles Type Angles Type Angles Acute All angles acute Right One right angle Obtuse One obtuse angle Equiangular All angles congruent Triangles may also be classified according to their angles See Table 2.2. EXAMPLE 1 In not shown, , , and . Describe completely the type of triangle represented. Solution is a right isosceles triangle, or is an isosceles right triangle. 쮿 䉭HJK 䉭HJK m ∠J = 90° JK = 4 HJ = 4 䉭HJK Exs. 1–7 A theorem that follows directly from a previous theorem is known as a corollary of that theorem. Corollaries, like theorems, must be proved before they can be used. These proofs are often brief, but they depend on the related theorem. Some corollaries of Theorem 2.4.1 are shown on page 95. We suggest that the student make a drawing to illustrate each corollary. Discover From a paper triangle, cut the angles from the “corners.” Now place the angles together at the same vertex as shown. What is the sum of the measures of the three angles? ANSWER 180° 1 3 2 1 2 3 In a triangle, the sum of the measures of the interior angles is 180°. THEOREM 2.4.1 The first statement in the following “picture proof ” establishes the auxiliary line that is used. The auxiliary line is justified by the Parallel Postulate. PICTURE PROOF OF THEOREM 2.4.1 GIVEN: in Figure 2.25a PROVE: PROOF: Through C, draw . We see that . See Figure 2.25b. But and alternate interior angles are congruent. Then in Figure 2.25a. 쮿 At times, we use the notions of the equality and congruence of angles interchange- ably within a proof. See the preceding “picture proof.” m ∠A + m∠B + m∠C = 180° m ∠3 = m∠B m ∠1 = m∠A m ∠1 + m∠2 + m∠3 = 180° Í ED 7 AB m ∠A + m∠B + m∠C = 180° 䉭ABC A C B a Figure 2.25 b A C B 1 2 3 E D EXAMPLE 2 In not shown, and . Find . Solution In , , so . Thus, and . 쮿 m ∠T = 71° 109° + m ∠T = 180° 45° + 64° + m ∠T = 180° m ∠R + m∠S + m∠T = 180° 䉭RST m ∠T m ∠S = 64° m ∠R = 45° 䉭RST Technology Exploration Use computer software, if available. 1. Draw . 2. Measure , , and . 3. Show that Answer may not be “perfect.” m ∠C = 180° m ∠A + m∠B + ∠C ∠B ∠A 䉭 ABC In an earlier exercise, it was suggested that the sum of the measures of the three in- terior angles of a triangle is 180°. This is now stated as a theorem and proved through the use of an auxiliary or helping line. When an auxiliary line is added to the draw- ing for a proof, a justification must be given for the existence of that line. Justifications include statements such as There is exactly one line through two distinct points. An angle has exactly one bisector. There is only one line perpendicular to another line at a point on that line. When an auxiliary line is introduced into a proof, the original drawing is sometimes redrawn for the sake of clarity. Each auxiliary figure must be determined, but it must not be underdetermined or overdetermined. A figure is underdetermined when more than one possible figure is described. On the other extreme, a figure is overdetermined when it is impossible for all conditions described to be satisfied.