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GEOMETRY 2.6 Proving Geometric Relationships October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS ESSENTIAL QUESTION How can we prove geometric relationships using proofs? October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS HOW ARE THEOREMS CREATED? A theorem is a statement that can be proven. This means proofs. Once a theorem is proven using a proof, it can be used as a reason in other proofs. October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS THEOREM 2.3 Right Angle Congruence Theorem All Right Angles are congruent. (This should be obvious: all right angles measure 90° and so they all have the same measure and hence are congruent. Let’s formally prove this.) All Rt s October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS PROOF: RIGHT ANGLE CONGRUENCE THEOREM Reasons Statements 1. Given 2 2. Def. of right angle 3. Transitive Property 4. Def. angles QED October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS THEOREM 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle, or to congruent angles, then they are congruent. Example mA + mB = 180° mA + mC = 180° B A C Thus, B C. They are both supplementary to the same angle, A. Supp Thm October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS PROOF: CONGRUENT SUPPLEMENTS THEOREM Statements Reasons 1. Given 2. Def. of supp. angles 3. Transitive Property 4. Subtraction Property 5. Def. congruent angles October 1, 2015 QED GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS THEOREM 2.5 Congruent Complements Theorem If two angles are complementary to the same angle, or to congruent angles, then the angles are congruent. The proof of this theorem is very similar to the proof we just wrote Example for Theorem 2.4. We will not work through this proof right now. mR + mS = 90° mT + mS = 90° R S T Thus, R T R and T are complementary to the same angle, S. Comp Thm October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS POSTULATE 2.8 Linear Pair Postulate The angles of a linear pair are supplementary. 1 2 m1 + m2 = 180° October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS THEOREM 2.6 Vertical Angles Congruence Theorem Vertical Angles are congruent. Prove: 1 2 1 2 October 1, 2015 3 For this proof, we will use the second type of proof called a paragraph proof. Write it in a natural style; like you’re explaining it to someone. But be accurate! GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS PROOF: VERTICAL ANGLES CONGRUENCE THEOREM Given: Vertical Angles are congruent. Prove: 1 2 1 2 October 1, 2015 1 and 3 form a linear pair and by Linear Pair Post., their sum is 180. 3 Similarly, the sum of 2 and 3 is 180. Thus, 1 2 by the ≅ Suppl. Th. because they are supplementary to the same angle. GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS EXAMPLE 1 Reasons Statements 1. Given October 1, 2015 QED 3. Substitution Property 4. Subtraction Property 5. Division Property GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS YOUR TURN. Practice using the theorems shown today to solve the following problems. 1. Find m1, m2, m3. 135° (supp. s) 45° 1 2 45° (vert. s ) 3 135° (vert. s ) October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS EXAMPLE 2 (5x + 8)° Solve for x. (3x + 20)° Statement Reason 1. 5x + 8 = 3x + 20 1. Vert. ∠𝑠 Thm 2. 2x = 12 2. Subtr. Prop. 3. x = 6 3. Div. Prop. October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS EXAMPLE 3 Solve for x, and find each angle. 4(15) + 20 = 80° (4x + 20)° 5(15) + 25 = 100° Subst. Prop. (5x + 25)° Statement Reason 1. 4x + 20 + 5x + 25 = 180 1. Linear Pair Thm 2. 9x + 45 = 180 2. Simplify 3. 9x = 135 3. Subtr. Prop. 4. x = 15 4. Div. Prop. October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS ASSIGNMENT October 1, 2015 GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS