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(over Lesson 3-5). 1-1a. Slide 1 of 1. (over Lesson 3-5). 1-1b. Slide 1 of 1. §3.6 Congruent Angles. What You'll Learn. You will learn to identify and use congruent and vertical angles. Recall that congruent segments have the same ________. measure. Congruent angles.

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1-1a

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  1. (over Lesson 3-5) 1-1a Slide 1 of 1

  2. (over Lesson 3-5) 1-1b Slide 1 of 1

  3. §3.6 Congruent Angles What You'll Learn You will learn to identify and use congruent and vertical angles. Recall that congruent segments have the same ________. measure Congruent angles _______________ also have the same measure.

  4. 50° 50° B V §3.6 Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure B  V iff mB = mV

  5. 1 2 X Z §3.6 Congruent Angles arcs To show that 1 is congruent to 2, we use ____. To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: X  Z mX = mZ

  6. §3.6 Congruent Angles four When two lines intersect, ____ angles are formed. There are two pair of nonadjacent angles. vertical angles These pairs are called _____________. 1 4 2 3

  7. §3.6 Congruent Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 1 and 3 1 4 2 2 and 4 3

  8. 1 4 2 3 §3.6 Congruent Angles 1) On a sheet of paper, construct two intersecting lines that are not perpendicular. 2) With a protractor, measure each angle formed. 3) Make a conjecture about vertical angles. Consider: A. 1 is supplementary to 4. m1 + m4 = 180 Hands-On B. 3 is supplementary to 4. m3 + m4 = 180 Therefore, it can be shown that 1 3 Likewise, it can be shown that 24

  9. 1 4 2 3 §3.6 Congruent Angles 1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3 x = 4; 3 = 19° 2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4 x = 56; 4 = 65° 3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3 x = 9; 3 = 127° 4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4 x = 10; 4 = 97°

  10. §3.6 Congruent Angles Vertical angles are congruent. n m 2 1  3 3 1 2  4 4

  11. 130° x° §3.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°.

  12. §3.6 Congruent Angles Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° x – 10 = 125. 125° x = 135.

  13. §3.6 Congruent Angles Suppose two angles are congruent. What do you think is true about their complements? 1  2 2 + y = 90 1 + x = 90 y is the complement of 2 x is the complement of 1 y = 90 - 2 x = 90 - 1 Because 1  2, a “substitution” is made. y = 90 - 1 x = 90 - 1 x = y x  y If two angles are congruent, their complements are congruent.

  14. 60° 60° B A 1 2 3 4 §3.6 Congruent Angles If two angles are congruent, then their complements are _________. congruent The measure of angles complementary to A and B is 30. A  B If two angles are congruent, then their supplements are _________. congruent The measure of angles supplementary to 1 and 4 is 110. 110° 110° 70° 70° 4  1

  15. 3 1 2 §3.6 Congruent Angles If two angles are complementary to the same angle, then they are _________. congruent 3 is complementary to 4 5 is complementary to 4 4 3 5 5  3 If two angles are supplementary to the same angle, then they are _________. congruent 1 is supplementary to 2 3 is supplementary to 2 1  3

  16. 52° 52° A B §3.6 Congruent Angles Suppose A  B and mA = 52. Find the measure of an angle that is supplementary to B. 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128°

  17. §3.6 Congruent Angles If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25, What are m1 and m2 ? m1 + m3 = 90 Definition of complementary angles. m1 = 90 - m3 Subtract m3 from both sides. m1 = 90 - 25Substitute 25 in for m3. m1 = 65Simplify the right side. You solve for m2 m2 + m3 = 90 Definition of complementary angles. m2 = 90 - m3 Subtract m3 from both sides. m2 = 90 - 25Substitute 25 in for m3. m2 = 65Simplify the right side.

  18. G D 1 2 A C 4 B 3 E H §3.6 Congruent Angles 1) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3 x = 17; 3 = 37° 2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC x = 29; EBC = 121° 3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4 x = 16; 4 = 39° 4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1 x = 18; 1 = 43°

  19. Suppose you draw two angles that are congruent and supplementary. What is true about the angles?

  20. 1 2 C A B §3.6 Congruent Angles If two angles are congruent and supplementary then each is a __________. right angle 1 is supplementary to 2 1 and 2 = 90 All right angles are _________. congruent A  B  C

  21. B A 2 E 3 1 4 C D §3.6 Congruent Angles If 1 is supplementary to 4, 3 is supplementary to 4, and m 1 = 64, what are m 3 and m 4? They are vertical angles. 1  3 m 1 = m3 m 3 = 64 3 is supplementary to 4 Given Definition of supplementary. m3 + m4 = 180 64 + m4 = 180 m4 = 180 – 64 m4 = 116

  22. End of Lesson

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