1. 2.6 Proven Angles Congruent

2. Objective: To prove and apply theorems about angles. 2.6 Proven Angles Congruent

3. Vocabulary: Theorem Vertical Prove Apply Paragraph Proof Congruent 2.6 Proven Angles Congruent

4. Solve it! 2.6 Proven Angles Congruent

5. Vertical Angles:  Theorem 2.1: 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 and 2.6 Proven Angles Congruent

6. Vertical Angles: 2.6 Proven Angles Congruent

7. Vertical Angles: 2.6 Proven Angles Congruent

8. Adjacent Angles:  Two coplanar angles that share a side and a vertex 12 1 2 <1 and <2 are Adjacent Angles 2.6 Proven Angles Congruent

9. Complementary Angles:  Two angles whose measures have a sum of 90°  Two angles whose measures have a sum of 180° 1 2 50° 40° 34 75° 105° Supplementary Angles: 2.6 Proven Angles Congruent

10. 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 2.6 Proven Angles Congruent <2 and <3 <4 and <5 <3 and <5 <1 and <2 <2 and <3 <3 and <4 <4 and <5

11. 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 2.6 Proven Angles Congruent

12. 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 2.6 Proven Angles Congruent

13. Theorems About Angles 2.6 Proven Angles Congruent

14. Theorems About Angles 2.6 Proven Angles Congruent

15. Theorems About Angles All these theorems can be proven using Deductive Reasoning. 2.6 Proven Angles Congruent

16. 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 2.6 Proven Angles Congruent

17. 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 __________. 1 2 180 23 1 2 2 3 <1 = < 3 You might want to draw a picture first! 2.6 Proven Angles Congruent

18. Lesson Check 2.6 Proven Angles Congruent

19. Lesson Check 2.6 Proven Angles Congruent