If two lines are cut by a transversal so that the exterior

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. Each angle of an equiangular triangle measures 60°. COROLLARY 2.4.2 The acute angles of a right triangle are complementary. COROLLARY 2.4.3 The following example illustrates Corollary 2.4.4. General Rule: The proof of a corollary is completed by using the theorem upon which the corollary depends. Illustration: Using of Example 3, the proof of Corollary 2.4.3 depends on the fact that . With , it follows that . m ∠N + m∠Q = 90° m ∠M = 90° m ∠M + m∠N + m∠Q = 180° 䉭NMQ STRATEGY FOR PROOF 왘 Proving a Corollary EXAMPLE 3 GIVEN: is a right angle in not shown; FIND: Solution Because the acute of a right triangle are complementary, 쮿 m ∠Q = 33° ‹ 57° + m ∠Q = 90° m ∠N + m∠Q = 90° ∠s m ∠Q m ∠N = 57° 䉭NMQ ∠M If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. COROLLARY 2.4.4 EXAMPLE 4 In and triangles not shown, . Also, . a Find . b Find . c Is Solution a b Using , we repeat part a to find . c Yes, both measure 69°. 쮿 ∠T ⬵ ∠Z m ∠Z = 69° m ∠X + m∠Y + m∠Z = 180° m ∠T = 69° 111° + m ∠T = 180° 52° + 59° + m ∠T = 180° m ∠R + m∠S + m∠T = 180° ∠T ⬵ ∠Z? m ∠Z m ∠T m ∠S = m∠Y = 59° m ∠R = m∠X = 52° 䉭XYZ 䉭RST Exs. 8–12 Figure 2.26 b 5

6 1

2 3 4 When the sides of a triangle are extended, each angle that is formed by a side and an extension of the adjacent side is an exterior angle of the triangle. With B-C-D in Figure 2.26a, is an exterior angle of ; for a triangle, there are a total of six exterior angles—two at each vertex. [See Figure 2.26b.] In Figure 2.26a, and are the two nonadjacent interior angles for exterior . These angles A and B are sometimes called remote interior angles for exterior . ∠ACD ∠ACD ∠B ∠A 䉭ABC ∠ACD A C B a D The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles. COROLLARY 2.4.5 EXAMPLE 5 GIVEN: In Figure 2.27, FIND: x Solution By Corollary 2.4.5, Check: , , and ; so , which satisfies the conditions of Corollary 2.4.5. 쮿 120 = 80 + 40 m ∠T = 40° m ∠S = 80° m ∠1 = 120° x = 10 2x = x + 10 x 2 + 2x = x 2 - 2x + 3x + 10 m ∠1 = m∠S + m∠T m ∠T = 3x + 10 m ∠S = x 2 - 2x m ∠1 = x 2 + 2x R T S V 1 2 Figure 2.27 Exs. 13–19 Exercises 2.4 In Exercises 1 to 4, refer to . On the basis of the information given, determine the measure of the remaining angles of the triangle.

1. and

2. and

3. 4.

and

5. Describe the auxiliary line segment as determined,

overdetermined, or underdetermined. a Draw the line through vertex C of . 䉭ABC m ∠A = m∠C m ∠B = 42° m ∠A = m∠C = 67° m ∠C = 82° m ∠B = 39° m ∠B = 42° m ∠A = 63° 䉭ABC A B C Exercises 1–6 b Through vertex C, draw the line parallel to . c With M the midpoint of , draw perpendicular to .

6. Describe the auxiliary line segment as determined,

overdetermined, or underdetermined. a Through vertex B of , draw . b Draw the line that contains A, B, and C. c Draw the line that contains M, the midpoint of . In Exercises 7 and 8, classify the triangle not shown by considering the lengths of its sides.

7. a All sides of are of the same length.

b In , DE ⫽ 6, EF ⫽ 6, and DF ⫽ 8. 䉭DEF 䉭ABC AB Í AB ⬜ AC 䉭ABC AB CM AB AB

8. a In , .

b In , RS ⫽ 6, ST ⫽ 7, and RT ⫽ 8. In Exercises 9 and 10, classify the triangle not shown by considering the measures of its angles.

9. a All angles of measure 60°.

b In , , , and .

10. a In , .

b In , , , and . In Exercises 11 and 12, make drawings as needed.

11. Suppose that for and

, you know that and . Explain why .

12. Suppose that T is a point on side of

. Also, bisects , and . If and are the angles formed when intersects , explain why . In Exercises 13 to 15, j 储 k and .

13. Given:

Find: , and

14. Given:

Find: and

15. Given: ,

Find: , , and

16. Given: and as

shown Find: x, y, and z ∠s MN ⬜ NQ m ∠4 m ∠3 m ∠2 m ∠5 = 41.5° m ∠1 = 122.3° m ∠5 m ∠1, m∠4, m ∠2 = 74° m ∠3 = 55° m ∠5 m ∠1, m∠2 m ∠4 = 72° m ∠3 = 50° 䉭ABC ∠1 ⬵ ∠2 PQ RT ∠2 ∠1 ∠P ⬵ ∠Q ∠PRQ RT 䉭PQR PQ ∠C ⬵ ∠Q ∠B ⬵ ∠N ∠A ⬵ ∠M 䉭MNQ 䉭ABC m ∠T = 70° m ∠S = 65° m ∠R = 45° 䉭RST m ∠X = 123° 䉭XYZ m ∠F = 90° m ∠E = 50° m ∠D = 40° 䉭DEF 䉭ABC 䉭RST XY ⬵ YZ 䉭XYZ

19. Given: with B-D-E-C

Find: m ∠B m ∠1 = m∠2 = 70° m ∠3 = m∠4 = 30° 䉭ABC k 1 2 4 3 5 6 j B C A Exercises 13–15 43 ° y z 65 ° 28 ° x R N M P Q

17. Given:

bisects Find: m ∠3 m ∠A = 110° ∠ADC DB AB 7 DC 1 2 3 A B D C Exercises 17, 18 B A 1 3 4 5 2 D E C Exercises 19–22

18. Given:

bisects Find: m ∠A m ∠1 = 36° ∠ADC DB AB 7 DC

20. Given: with B-D-E-C

Find: in terms of x

21. Given: with and

Find: x , , and

22. Given: with and

Find: x , , and

23. Consider any triangle and one exterior angle at each

vertex. What is the sum of the measures of the three exterior angles of the triangle?

24. Given: Right with

right Find: x

25. Given:

Find: x and y

26. Given: ,

Find: x

27. Given:

Find: x

28. Given:

Find: x

29. Given:

Find: x, y, and

30. Given: Equiangular

bisects Prove: is a right 䉭 䉭RVS ∠SRT RV 䉭RST m ∠5 m ∠4 = 2x - y - 40 m ∠3 = 2y m ∠2 = 4y m ∠1 = x m ∠5 = 5x + 1 - 2 m ∠3 = 5x - 3 m ∠1 = 8x + 2 m ∠2 = x 3 m ∠1 = x 2 m ∠2 = x 2 m ∠1 = x m ∠3 = 3x m ∠2 = y m ∠1 = x m ∠2 = 5x + 2 m ∠1 = 7x + 4 ∠C 䉭ABC m ∠B m ∠BAC m ∠BAC = x m ∠B = m∠C = x 2 䉭ABC m ∠DAE m ∠1 m ∠DAE = x 2 m ∠1 = m∠2 = x 䉭ADE m ∠B m ∠3 = x m ∠1 = 2x 䉭ABC 1 A C B 3 2 Exercises 24–27 1 3 2 4 5 Exercises 28, 29 R V S T H I M 67° 23° ? 5000⬘

31. Given: and intersect