1. Proving Angles Congruent

2.  Vertical Angles: Two angles whose sides form two pairs of opposite rays; form two pairs of congruent angles 1 2 3 4 <1 and <3 are Vertical angles <2 and <4 are Vertical angles

3. Proving Angles Congruent  Adjacent Angles: Two coplanar angles that share a side and a vertex 12 1 2 <1 and <2 are Adjacent Angles

4. 2.5 Proving Angles Congruent  Complementary Angles: Two angles whose measures have a sum of 90°  Supplementary Angles: Two angles whose measures have a sum of 180° 1 2 50° 40° 34 75° 105°

5. Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. Complementary b. Supplementary c. Vertical d. Adjacent 1 2 3 45

6. Making Conclusions Whether you draw a diagram or use a given diagram, you can make some conclusions directly from the diagrams. You CAN conclude that angles are  Adjacent angles  Adjacent supplementary angles  Vertical angles

7. Making Conclusions Unless there are markings that give this information, you CANNOT assume  Angles or segments are congruent  An angle is a right angle  Lines are parallel or perpendicular

8. Theorems About Angles Theorem 2-1Vertical Angles Theorem Vertical Angles are Congruent Theorem 2-2Congruent Supplements If two angles are supplements of the same angle or congruent angles, then the two angles are congruent

9. Theorems About Angles Theorem 2-3Congruent Complements If two angles are complements of the same angle or congruent angles, then the two angles are congruent Theorem 2-4 All right angles are congruent Theorem 2-5 If two angles are congruent and supplementary, each is a right angle

10. Proving Theorems Paragraph Proof: Written as sentences in a paragraph Given: <1 and <2 are vertical angles Prove: <1 = <2 Paragraph Proof: By the Angle Addition Postulate, m<1 + m<3 = 180 and m<2 + m<3 = 180. By substitution, m<1 + m<3 = m<2 + m<3. Subtract m<3 from each side. You get m<1 = m<2, which is what you are trying to prove. 1 2 3

11. Proving Theorems Given: <1 and <2 are supplementary <3 and <2 are supplementary Prove:<1 = <3 Proof: By the definition of supplementary angles, m<___ + m<____ = _____ and m<___ + m<___ = ____. By substitution, m<___ + m<___ = m<___ + m<___. Subtract m<2 from each side. You get __________.