2. Given: Prove: x = 10 1. __________ 1. ___________ 2. __________ 2. ___________ 3. __________ 3. ___________ 4. __________ 4. ___________ StatementsReasons x = 10 Given Substitution Subtraction Multiplication

3. Given: m  4 + m  6 = 180 Prove: m  5 = m  6 1. 2. 3. 4. 5. StatementsReasons Given Angle Add. Post. Substitution Reflexive m  4 = m  4 m  4 + m  5 = m  4 + m  6 m  4 + m  5 = 180 m  4 + m  6 = 180 m  5 = m  6 Subtraction

4. Given: m  1 = m  3 m  2 = m  4 Prove: m  ABC = m  DEF 1. 2. 3. 4. StatementsReasons m  1 = m  3; m  2 = m  4 m  ABC = m  DEF m  1 + m  2 = m  3 + m  4 m  1 + m  2 = m  ABC m  3 + m  4 = m  DEF Given Addition Prop. Angle Add. Post. Substitution AB C 1 24 3 DE F

5. Given: ST = RN; IT = RU Prove: SI = UN 1. ST = RN 1. 2. 3. SI + IT = RU + UN 3. 4. IT = RU 4. 5. Statements Reasons ST = SI + IT RN = RU + UN SI = UN Given Segment Add. Post. Substitution Given Subtraction Prop. S IT RUN

7. Postulate – A statement accepted without proof. Theorem – A statement that can be proven using other definitions, properties, and postulates. In this class, we will prove many of the Theorems that we will use.

8. If M is the midpoint of AB, then AM = ½AB and MB = ½AB. Hypothesis: M is the midpoint of AB Conclusion: AM = ½AB and MB = ½AB Write these pieces of the conditional statement as your “given” and “prove” information. Given: Prove:

9. Definition of Midpoint: the point that divides a segment into two congruent segments. If M is the midpoint of AB, then AM  MB. A B M Midpoint Theorem: If M is the midpoint of AB, then AM = ½AB and MB = ½AB. The theorem proves properties not given in the definition.

10. Proof of the Midpoint Theorem Given: M is the midpoint of AB Prove: AM = ½AB; MB = ½AB 1.2.3.4.5.6. M is the midpoint of AB MB = ½AB AM = MB AM + MB = AB AM + AM = AB 2AM = AB AM = ½AB Given Def. of a Midpoint Segment Add. Post. Substitution Division Property Substitution AMB

11. If BX is the bisector of  ABC, then m  ABX = ½m  ABC and m  XBC = ½m  ABC. Prove: m  ABX = ½m  ABC and m  XBC = ½m  ABC. Given: BX is the bisector of  ABC.

12. A ray that divides an angle into two congruent adjacent angles. X Y ZW  XWY  YWZ If WY is the bisector of  XWZ, then m  XWY = ½m  XWZ and m  YWZ = ½m  XWZ.

13. Proof of the  Bisector Thm Prove: m  ABX = ½m  ABC and m  XBC = ½m  ABC. Given: BX is the bisector of  ABC. X B C A 1. BX is the bisector of  ABC. 1. Given 2. m  ABX = m  XBC2. Def. of an angle bisector 3. m  ABX + m  XBC = m  ABC3. Angle Addition Postulate 4. m  ABX + m  ABX = m  ABC 2m  ABX = m  ABC 4. Substitution 5. m  ABX = ½m  ABC5. Division Property 6. m  XBC = ½m  ABC6. Substitution