1. Proof Geometry 4-6: Some Theorems About Angles

2. Warm-Up

3. Congruent Angles in Linear Pair Theorem If the angles in a linear pair are congruent, then each of them is a right angle Create a sketch Write the given and the prove Given:  1 and  2 are a linear pair Prove:  1 and  2 are right angles 12

4. Congruent Angles in Linear Pair Theorem If the angles in a linear pair are congruent, then each of them is a right angle StatementsReasons 1.  1 and  2 are a linear pair Given 2.  1 and  2 are supplementary Supplement Post 3.m  1 + m  2 = 180 Def of suppl 4.  1   2 Given 5.m  1 = m  2Def of  6.m  1 + m  1 = 180 Substitution of 5 into 3 7.2 m  1 =180 Simplify 8.m  1 = 90 Division prop. of eq. 9.  1 is a right angle Def. of right angle By repeating steps 6 to 9 with  2 we can show  2 is also a right angle Q.E.D.

5. Congruent Supplementary Angles Theorem If two angles are both congruent and supplementary, then each is a right angle. Proof similar to previous – angles just not necessarily adjacent.

6. Right Angles Theorem Any two right angles are congruent Create a sketch Write the given and the prove Given:  1 and  2 are right angles Prove:  1   2 12

7. Right Angles Theorem Any two right angles are congruent StatementsReasons 1.  1 and  2 are right angles Given 2. m  1 = 90, m  2 = 90 Def of right angle 3.m  1 = m  2 Transitive prop. of eq. 4.  1   2Def of  Q.E.D.

8. The Supplement Theorem Supplements of congruent angles are congruent. Create a sketch Write the given and the prove Given:  A and  B are a supplementary  C and  D are a supplementary  A   C Prove:  B   D AB CD

9. The Supplement Theorem Supplements of congruent angles are congruent. StatementsReasons 1.  A and  B are supplementary Given 2. m  A + m  B = 180 Def of suppl 3.  C and  D are supplementary Given 4.m  C + m  D = 180 Def of suppl 5.m  A + m  B = m  C + m  D Transitive prop. of eq. using 2 and 4 6.  A  C Given 7.m  A = m  CDef. of   s 8.m  A + m  B = m  A + m  D Substitute 7 into 5 9.m  B = m  D Subtraction prop. of eq. 10.  B  DDef. of   s Q.E.D.

10. The Complement Theorem Complements of congruent angles are congruent Food for thought: How will this proof be different from the previous?

11. Homework Pg. 103 -104: #1-6,9,10