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3-3 Adjacent and Vertical Angles

3-3 Adjacent and Vertical Angles. A basic property of angles is that the measure of an angle formed by the outside rays of adjacent angles is the sum of the measures of the adjacent angles. 3-3 Adjacent and Vertical Angles.

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3-3 Adjacent and Vertical Angles

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  1. 3-3 Adjacent and Vertical Angles • A basic property of angles is that the measure of an angle formed by the outside rays of adjacent angles is the sum of the measures of the adjacent angles.

  2. 3-3 Adjacent and Vertical Angles • Two nonstraight and nonzero angles are adjacent angles if and only if a common side is in the interior of the angles formed by the noncommon sides.

  3. Q R 50o 50o P V 3-3 Adjacent and Vertical Angles • VR is the angle bisector of ÐPVQ if and only if VR (except for point V) is in the interior of ÐPVQ and mÐPVR = mÐRVQ.

  4. 3-3 Adjacent and Vertical Angles • Angle Measure Postulate • Angle Addition Assumption • If angles AVC and CVB are adjacent angles, then mÐAVC + mÐCVB = mÐAVB. C A B V

  5. 3-3 Adjacent and Vertical Angles • If the measures of two angles are r and s, then the angles are: complementary angles iff r + s = 90; supplementary angles iff r + s = 180. C A B

  6. 3-3 Adjacent and Vertical Angles Equal Angle Measures Theorem • If two angles have the same measure, their complements have the same measure. • If two angles have the same measure, their supplements have the same measure.

  7. 3-3 Adjacent and Vertical Angles Two adjacent angles of a linear pair if and only if their noncommon sides are opposite rays. D A B C

  8. 3-3 Adjacent and Vertical Angles Linear Pair Theorem • If two angles form a linear pair, then they are supplementary. D A B C

  9. 1 2 3 3-3 Adjacent and Vertical Angles Two nonstraight angles are vertical angles if and only if the union of their side is two lines. 4

  10. 1 2 3 3-3 Adjacent and Vertical Angles Vertical Angle Theorem • If two angles are vertical angles, then their measures are equal. 4

  11. 3-3 Adjacent and Vertical Angles Tell why the indicated angles of these figures are not adjacent. • ÐA and ÐC • Ð1 and Ð2 B K C A J 1 L 2 M

  12. 3-3 Adjacent and Vertical Angles What is mÐGKL? What is mÐJKL? G J 60o 50o I K 90o L

  13. 3-3 Adjacent and Vertical Angles Suppose you are give two angles M and N such that mÐM = mÐN. If the measure of a complement of ÐM equals 18x + 10 and the measure of a complement of ÐN equals 8x + 30, what is the measure of a supplement of ÐG?

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