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WARM UP Please pick up the worksheet on the cart and complete the 2 proofs.

WARM UP Please pick up the worksheet on the cart and complete the 2 proofs. Given: Prove: x = 10. Statements. Reasons. 1. __________ 1. ___________ 2. __________ 2. ___________ 3. __________ 3. ___________ 4. __________ 4. ___________. Given. Substitution.

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WARM UP Please pick up the worksheet on the cart and complete the 2 proofs.

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  1. WARM UP Please pick up the worksheet on the cart and complete the 2 proofs.

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

  3. Given: mÐ4 + mÐ6 = 180 Prove: mÐ5 = mÐ6 Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Given mÐ4 + mÐ6 = 180 mÐ4 + mÐ5 = 180 Angle Add. Post. Substitution mÐ4 + mÐ5 = mÐ4 + mÐ6 Reflexive mÐ4 = mÐ4 Subtraction mÐ5 = mÐ6

  4. Given: mÐ1 = mÐ3 mÐ2 = mÐ4 Prove: mÐABC = mÐDEF C F 2 4 1 3 A B E D Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. Given mÐ1 = mÐ3; mÐ2 = mÐ4 Addition Prop. mÐ1 + mÐ2 = mÐ3 + mÐ4 mÐ1 + mÐ2 = mÐABC Angle Add. Post. mÐ3 + mÐ4 = mÐDEF Substitution mÐABC = mÐDEF

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

  6. Section 2.3 Proving Midpoint and Angle Bisector Theorems

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

  9. If M is the midpoint of AB, then AM @ MB. DEFINITION VS. THEOREM Definition of Midpoint: the point that divides a segment into two congruent segments. A M B 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 A M B 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. M is the midpoint of AB Given AM = MB Def. of a Midpoint AM + MB = AB Segment Add. Post. AM + AM = AB 2AM = AB Substitution AM = ½AB Division Property Substitution MB = ½AB

  11. Angle Bisector Theorem Given: BX is the bisector of ÐABC. If BX is the bisector of ÐABC, then mÐABX = ½mÐABC and mÐXBC = ½mÐABC. Prove: mÐABX = ½mÐABC and mÐXBC = ½mÐABC.

  12. DEFINITION VS. THEOREM A ray that divides an angle into two congruent adjacent angles. Y ÐXWY @ÐYWZ X If WY is the bisector of ÐXWZ, then mÐXWY = ½mÐXWZ and mÐYWZ = ½mÐXWZ. W Z

  13. Proof of the Ð Bisector Thm 1. BX is the bisector of ÐABC. A Given: BX is the bisector of ÐABC. Prove: mÐABX = ½mÐABC and mÐXBC = ½mÐABC. X B C 1. Given 2. mÐABX = mÐXBC 2. Def. of an angle bisector 3. mÐABX + mÐXBC = mÐABC 3. Angle Addition Postulate 4. mÐABX + mÐABX = mÐABC 2mÐABX = mÐABC 4. Substitution 5. mÐABX = ½mÐABC 5. Division Property 6. mÐXBC = ½mÐABC 6. Substitution

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