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Published byMadison O’Connor’ Modified over 8 years ago
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Proof Geometry 4-6: Some Theorems About Angles
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Warm-Up
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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
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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.
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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.
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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
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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.
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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
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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.
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The Complement Theorem Complements of congruent angles are congruent Food for thought: How will this proof be different from the previous?
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Homework Pg. 103 -104: #1-6,9,10
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