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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. - PowerPoint PPT Presentation

Given: Prove: x = 101. __________ 1. ___________2. __________ 2. ___________3. __________ 3. ___________4. __________ 4. ___________StatementsReasonsx = 10GivenSubstitutionSubtractionMultiplication

Given: m4 + m6 = 180 Prove: m5 = m61. 1. 2. 2. 3. 3. 4. 4. 5. 5. StatementsReasonsGivenAngle Add. Post.SubstitutionReflexivem4 = m4m4 + m5 = m4 + m6m4 + m5 = 180m4 + m6 = 180m5 = m6Subtraction

Given: m1 = m3 m2 = m4Prove: mABC = mDEF1. 1. 2. 2. 3. 3. 4. 4. StatementsReasonsm1 = m3; m2 = m4mABC = mDEFm1 + m2 = m3 + m4m1 + m2 = mABCm3 + m4 = mDEFGivenAddition Prop.Angle Add. Post.SubstitutionABC1243DEF

Given: ST = RN; IT = RU Prove: SI = UN1. ST = RN 1. 2. 2. 3. SI + IT = RU + UN 3. 4. IT = RU 4. 5. 5. StatementsReasonsST = SI + ITRN = RU + UNSI = UNGivenSegment Add. Post.SubstitutionGivenSubtraction Prop.SITRUN

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.

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

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

Proof of the Midpoint TheoremGiven: M is the midpoint of ABProve: AM = AB; MB = AB 1.1.2.2.3.3.4.4.5.5.6.6.M is the midpoint of ABMB = AB AM = MB AM + MB = AB AM + AM = AB 2AM = ABAM = ABGivenDef. of a MidpointSegment Add. Post.SubstitutionDivision PropertySubstitution

If BX is the bisector of ABC, then mABX = mABC and mXBC = mABC. Prove: mABX = mABC and mXBC = mABC.

A ray that divides an angle into two congruent adjacent angles.XWY @YWZIf WY is the bisector of XWZ, then mXWY = mXWZ and mYWZ = mXWZ.

Proof of the Bisector ThmProve: mABX = mABC and mXBC = mABC. Given: BX is the bisector of ABC.X1. Given2. mABX = mXBC2. Def. of an angle bisector3. mABX + mXBC = mABC3. Angle Addition Postulate4. mABX + mABX = mABC 2mABX = mABC4. Substitution5. mABX = mABC 5. Division Property 6. mXBC = mABC 6. Substitution