Proving Angle Relationships

Protractor Postulate - Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.

Angle Congruence Congruence of angles is reflexive, symmetric, and transitive: 1. Reflexive  1   1 2. SymmetricIf  1   2, then  2   1 3. Transitive If  1   2 and  2   3, then  1   3

Angle Addition Postulate If R is in the interior of  PQS, then m  PQR + m  RQS = m  PQS If m  PQR + m  RQS = m  PQS, then R is in the interior of  PQS Q P R S

Angle Addition If and, find A B D C

Right Angle Theorems List facts that you observe about the perpendicular lines below:

Right Angle Theorems Perpendicular lines intersect to form four right angles All right angles are congruent Perpendicular lines form congruent adjacent angles If two angles are congruent and supplementary, then each angle is a right angle If two congruent angles form a linear pair, then they are right angles

Theorems Supplementary Theorem – if two angles form a linear pair, then they are supplementary angles. Complementary Theorem – if the non- common sides of two adjacent angles form a right angle, then the angles are complementary angles.

Theorems 2.6 Angles supplementary to the same angle or to congruent angles are congruent. 2.7 Angles complementary to the same angle or to congruent angles are congruent. 2.8 Vertical angles theorem: If two angles are vertical angles, then they are congruent.

Supplementary Angles Angles supplementary to the same angle or to congruent angles are congruent  s suppl. to same  or   s are  Example: m  1 + m  2 = 180 m  2 + m  3 = 180 Then,  1  

Complementary Angles Angles complementary to the same angle or to congruent angles are congruent  s compl. to same  or   s are  Example: m  1 + m  2 = 90 m  2 + m  3 = 90 Then,  1  