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2.8 Notes: Proving Angle Relationships
How can you prove a mathematical statement?
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Vocab! Angle Addition Postulate Supplement Theorem Complement Theorem Reflexive Property of Angle Congruence Β D is the interior of β π΄π΅πΆ if and only if πβ π΄π΅π·+πβ π·π΅π·=πβ π΄π΅πΆ If two angles form a linear pair, then they are supplementary angles If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles β 1β
β 1
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Vocab! If β 1β
β 2 then β 2β
β 1 If β 1β
β 2 and β 2β
β 3 then β 1β
β 3
Symmetric Property of Angle Congruence Transitive Property of Angle Congruence Congruent Supplement Theorem Congruent Complement Theorem If β 1β
β 2 then β 2β
β 1 If β 1β
β 2 and β 2β
β 3 then β 1β
β 3 Angles supplementary to the same angle or to congruent angles are congruent Angles complementary to the same angle or to congruent angles are congruent
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Example 1 β π & β π are complementary β π & β π are complementary
Definition of complementary angles Definition of complementary angles Substitution Reflexive Property πβ π=πβ π β πβ
β π
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Example 2 In the figure, β 1 and β 4 form a linear pair, and m β 3 + m β 1 = 180Β°. Prove that β 3 and β 4 are congruent. πβ π+πβ π=πππ Given Supplement Theorem πβ π+πβ π=πππ β πβ
β π Congruent Supplement Theorem
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Vertical Angles Theorem
Vocab! Vertical Angles Theorem If two angles are vertical angles, then they are congruent.
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Example 3 ο1 and ο2 are vertical angles and mο1 = (d β 32)Β° and mο2 = (175 β 2d)Β°, find mο1 and mο2. Justify each step. β π & β π are vertical angles Vertical Angles Theorem Definition of Congruency π
βππ=πππβππ
ππ
βππ=πππ ππ
=πππ Addition Property Division Property πβ π=ππ Substitution
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Right Angles Congruent Theorem
Vocab! Right Angles Congruent Theorem Β Right angles are always congruent to one another 2.9 Β Perpendicular lines intersect to form four right angles 2.10 Β All right angles are congruent
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Vocab! 2.11 Β Perpendicular lines form congruent adjacent angles 2.12 Β If two angles are congruent and supplementary, then each angle is a right angle 2.13 Β If two congruent angles form a linear pair, then they are right angles.
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