Construct the Proof. This formal proof is

Several additional theorems are now stated, the proofs of which are left as exercises. This list contains theorems that are quite useful when cited as reasons in later proofs. A formal proof is provided only for Theorem 1.7.6. EXAMPLE 3 Give a formal proof for Theorem 1.7.1. If two lines meet to form a right angle, then these lines are perpendicular. Given: and intersect at E so that ⬔AEC is a right angle Figure 1.66 Prove: ⊥ PROOF Statements Reasons 1. and intersect so 1. Given that ⬔AEC is a right angle 2. m ⬔AEC = 90 2. If an ⬔ is a right ⬔, its measure is 90 3. ⬔AEB is a straight ⬔, so 3. If an ⬔ is a straight ⬔, its m ⬔AEB = 180 measure is 180 4. m ⬔AEC + m ⬔CEB = 4. Angle-Addition Postulate m ⬔AEB 2, 3, 4 5. 90 + m ⬔CEB = 180 5. Substitution 5 6. m ⬔CEB = 90 6. Subtraction Property of Equality 2, 6 7. m ⬔AEC = m ⬔CEB 7. Substitution 8. ⬔AEC ⬵ ⬔CEB 8. If two ⬔s have = measures, the ⬔s are ⬵ 9. ⊥ 9. If two lines form ⬵ adjacent ⬔s, these lines are › 쮿 Í CD Í AB Í CD Í AB Í CD Í AB Í CD Í AB Figure 1.66 If two angles are complementary to the same angle or to congruent angles, then these angles are congruent. THEOREM 1.7.2 If two angles are supplementary to the same angle or to congruent angles, then these angles are congruent. THEOREM 1.7.3 Any two right angles are congruent. THEOREM 1.7.4 See Exercise 25 for a drawing describing Theorem 1.7.2. See Exercise 26 for a drawing describing Theorem 1.7.3. Exs. 6–8 E B A C D If the exterior sides of two adjacent acute angles form perpendicular rays, then these an- gles are complementary. THEOREM 1.7.5 EXAMPLE 4 Study the formal proof of Theorem 1.7.6. 1 2 C B D A Figure 1.67 If the exterior sides of two adjacent angles form a straight line, then these angles are supplementary. THEOREM 1.7.6

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G F H E Figure 1.68 General Rule: The last reason explains why the last statement must be true. Never write “Prove” for any reason. Illustration: The final reason in the proof of Theorem 1.7.6 is the definition of supple- mentary angles: If the sum of measures of 2 angles is 180°, the angles are supplementary. STRATEGY FOR PROOF 왘 The Final Reason in the Proof Exs. 9–12 Technology Exploration Use computer software if available. 1. Draw containing point F. Also draw as in Figure 1.68. 2. Measure ⬔3 and ⬔4. 3. Show that m⬔3 + m⬔4 = 180°. Answer may not be perfect. FH Í EG PICTURE PROOF OF THEOREM 1.7.5 Given: › Prove: ⬔1 and ⬔2 are complementary Proof: We see that ⬔1 and ⬔2 are parts of a right angle. Then m ⬔1 + m ⬔2 = 90°, so ⬔1 and ⬔2 are complementary. BC BA For Theorem 1.7.5, we create an informal proof called a picture proof. Although such a proof is less detailed, the impact of the explanation is the same This is the first of several “picture proofs” found in this textbook. Given: ⬔3 and ⬔4 and Figure 1.68 Prove: ⬔3 and ⬔4 are supplementary Í EG PROOF Statements Reasons 1. ⬔3 and ⬔4 and 1. Given 2. m ⬔3 + m ⬔4 = m ⬔EFG 2. Angle-Addition Postulate 3. ⬔EFG is a straight angle 3. If the sides of an ⬔ are opposite rays, it is a straight ⬔ 4. m ⬔EFG = 180 4. The measure of a straight ⬔ is 180 5. m ⬔3 + m ⬔4 = 180 5. Substitution 6. ⬔3 and ⬔4 are supplementary 6. If the sum of the measures of two ⬔s is 180, the ⬔s are supplementary Í EG 쮿 Exercises 1.7 In Exercises 1 to 6, state the hypothesis H and the conclusion C for each statement.

1. If a line segment is bisected, then each of the equal

segments has half the length of the original segment.

2. If two sides of a triangle are congruent, then the triangle is

isosceles.

3. All squares are quadrilaterals. 4. Every regular polygon has congruent interior angles.

5. Two angles are congruent if each is a right angle. 6. The lengths of corresponding sides of similar polygons are proportional.

7. Name, in order, the five parts of the formal proof of a

theorem.

8. Which part hypothesis or conclusion of a theorem

determines the a Drawing? b Given? c Prove?

9. Which part Given or Prove of the proof depends upon the

a Hypothesis of Theorem? b Conclusion of Theorem?

10. Which of the following can be cited as a reason in a

proof? a Given c Definition b Prove d Postulate For each theorem stated in Exercises 11 to 16, make a Drawing. On the basis of your Drawing, write a Given and a Prove for the theorem.

11. If two lines are perpendicular, then these lines meet to

form a right angle.

12. If two lines meet to form a right angle, then these lines are

perpendicular.

13. If two angles are complementary to the same angle, then

these angles are congruent.

14. If two angles are supplementary to the same angle, then

these angles are congruent.

15. If two lines intersect, then the vertical angles formed are

congruent.

16. Any two right angles are congruent.

In Exercises 17 to 24, use the drawing at the right and apply the theorems of this section.

17. If m ⬔1

= 125°, find m ⬔2, m⬔3, and m⬔4.

18. If m ⬔2

= 47°, find m ⬔1, m⬔3, and m⬔4.

19. If m ⬔1

= 3x + 10 and m ⬔3 = 4x - 30, find x and m ⬔1.

20. If m ⬔2

= 6x + 8 and m ⬔4 = 7x, find x and m ⬔2.

21. If m ⬔1

= 2x and m ⬔2 = x , find x and m ⬔1.

22. If m ⬔2

= x + 15 and m ⬔3 = 2x, find x and m ⬔2.

23. If m ⬔2

= - 10 and m⬔3 = + 40, find x and m ⬔2.

24. If m ⬔1

= x + 20 and m ⬔4 = , find x and m ⬔4. In Exercises 25 to 33, complete the formal proof of each theorem.

25. If two angles are complementary to the same angle, then

these angles are congruent. Given: ⬔1 is comp. to ⬔3 ⬔2 is comp. to ⬔3 Prove: ⬔1 ⬵ ⬔2 x 3 x 3 x 2 3 2

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D A C B O If two line segments are congruent, then their midpoints separate these segments into four congruent segments. THEOREM 1.7.7 If two angles are congruent, then their bisectors separate these angles into four congru- ent angles. THEOREM 1.7.8 Exs. 13, 14 The final two theorems in this section are stated for convenience. We suggest that the student make drawings to illustrate Theorem 1.7.7 and Theorem 1.7.8. 2 3 1 PROOF Statements Reasons 1. ⬔1 is comp. to ⬔3 1. ? ⬔2 is comp. to ⬔3 2. m ⬔1 + m ⬔3 = 90 2. ? m ⬔2 + m ⬔3 = 90 3. m ⬔1 + m ⬔3 = 3. ? m ⬔2 + m ⬔3 4. m ⬔1 = m ⬔2 4. ? 5. ⬔1 ⬵ ⬔2 5. ?

27. If two lines intersect, the vertical angles formed are

congruent.

28. Any two right angles are congruent. 29. If the exterior sides of two adjacent acute angles form

perpendicular rays, then these angles are complementary. Given: ⊥ Prove: ⬔1 is comp. to ⬔2 BC BA

26. If two angles are supplementary to the same angle, then

these angles are congruent. Given: ⬔1 is supp. to ⬔2 ⬔3 is supp. to ⬔2 Prove: ⬔1 ⬵ ⬔3 HINT: See Exercise 25 for help. C D N B A M 2 1 3 Exercise 26

30. If two line segments are congruent, then their midpoints

separate these segments into four congruent segments. Given ⬵ M is the midpoint of N is the midpoint of Prove: NC ⬵ DN ⬵ MB ⬵ AM DC AB DC AB PROOF Statements Reasons 1. ⊥ 1. ? 2. ? 2. If two rays are ›, then they meet to form a rt. ⬔ 3. m ⬔ABC = 90 3. ? 4. m ⬔ABC = m ⬔1 + m ⬔2 4. ? 5. m ⬔1 + m ⬔2 = 90 5. Substitution 6. ? 6. If the sum of the measures of two angles is 90, then the angles are complementary BC BA 2 B C 1 A

31. If two angles are congruent, then their bisectors separate

these angles into four congruent angles. Given: ⬔ABC ⬔EFG bisects ⬔ABC bisects ⬔EFG Prove: ⬔1 ⬔2 ⬔3 ⬔4 ⬵ ⬵ ⬵ FH BD ⬵ B C A D 1 2 F G E H

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32. The bisectors of two adjacent supplementary angles form

a right angle. Given: ⬔ABC is supp. to ⬔CBD bisects ⬔ABC bisects ⬔CBD Prove: ⬔EBF is a right angle BF BE B A D C F 1 2 3 4 E

33. The supplement of an acute angle is an obtuse angle.

HINT: Use Exercise 28 of Section 1.6 as a guide.