Section 2.8 ~ Proving Angles Relationships

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Presentation transcript:

Section 2.8 ~ Proving Angles Relationships Be able to apply theorems to prove angles congruent! Given: AB = BC A B C

Solve for x. Fill in each missing reason: Given: LM bisects KLN M N LM bisects KLN mNLM = mMLK 4x = 2x + 40 2x = 40 x = 20 given L K def  bisector substitution subtraction division

Solve for x and justify each step: B Given: mAXC = 139° A mAXC = 139 mAXB + mBXC = mAXC x + 2x + 10 = 139 3x + 10 = 139 3x = 129 x = 43 given X C  addition postulate substitution simplify subtraction division

Try it on your own! Find the value of x and justify each step. 4x - 101 = 2x + 3 _________ ________ subtraction 2x = 104 _________ ________ division

Using the Transitive Property of Angle Congruence Statement: A ≅ B, B ≅ C mA = mB mB = mC mA = mC A ≅ C Reason: Given Def. Cong. Angles Transitive property of equality

Using the Transitive Property Given: m3 = 40, 1 ≅ 2, 2 ≅ 3 Prove: m1 = 40 1 4 2 3

Using the Transitive Property Statement: m3 = 40, 1 ≅ 2, 2 ≅ 3 1 ≅ 3 m1 = m 3 m1 = 40 Reason: Given Trans. Prop of Cong. Angles Def. Cong. Angles Substitution

Proving All rt <‘s are congruent Reason: Given Def. Right angle Transitive property Def. Cong. Angles Statement: 1 and 2 are right angles m1 = 90, m2 = 90 m1 = m2 1 ≅ 2

Proving Complement/Supplement Theorem Given: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 Prove: 2 ≅ 3 3 4 1 2

Proving complement/Supplement Theorem Reason: Given Def. Supplementary angles Transitive property of equality Def. Congruent Angles Substitution property Subtraction property Statement: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 m 1 + m 2 = 180; m 3 + m 4 = 180 m 1 + m 2 =m 3 + m 4 m 1 = m 4 m 1 + m 2 =m 3 + m 1 m 2 =m 3 2 ≅ 3

Using Linear Pairs In the diagram, m8 = m5 and m5 = 125. Explain how to show m7 = 55 7 8 5 6

Solution: Using the transitive property of equality m8 = 125. The diagram shows that m 7 + m 8 = 180. Substitute 125 for m 8 to show m 7 = 55.

Proving The Vertical Angles Theorem Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6

Proving the Vertical Angles Theorem Statement: 5 and 6 are a linear pair, 6 and 7 are a linear pair 5 and 6 are supplementary, 6 and 7 are supplementary 5 ≅ 7 Reason: Given Linear Pair postulate Congruent Supplements Theorem