2.6 Proving Geometric Relationships

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Presentation transcript:

2.6 Proving Geometric Relationships Geometry 2.6 Proving Geometric Relationships How can you prove angles congruent?

Geometry 2.6 Proving Geometric Relationships Topic/Objectives Use angle congruence properties. Prove theorems about special angle pairs. June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Theorem 2.2: Angle Congruence Angle congruence is reflexive, symmetric and transitive. Examples Reflexive: ABC  ABC Symmetric: If A  B, then B  A Transitive: If A  B, and B  C, then A  C The proofs are similar to those for segment congruence June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Example Given: 1  2, 3  4 and 2  3 Prove: 1  4 Statement Reason 1. 1  2 1. Given 2. 2  3 2. Given 3. 3  4 3. Given 4 1 4. 1  4 4. Transitive 2 3 June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Theorem 2.3 Right Angle Conguence 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. We won’t prove this.) All Rt s  June 1, 2018 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.  supp Example mA + mB = 180° mA + mC = 180° Thus, B  C. They are both supplementary to the same angle, A. June 1, 2018 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.  comp Example mR + mS = 90° mT + mS = 90° Thus, R  T R and T are complementary to the same angle, S. June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Postulate 2.8 Linear Pair Postulate The angles of a linear pair are supplementary. 1 2 m1 + m2 = 180° June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Vertical Angles are congruent. Prove: 1  2 1 2 3 June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Vertical Angles are congruent. Prove: 1  2 1 2 3 1 and 3 form a linear pair and by Post 12, their sum is 180. Similarly, the sum of 2 and 3 is 180. Thus, 1  2 since they are supplementary to the same angle. (See Theorem 2.4.) June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Vertical Angles are Congruent June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Practice Examples Find m1, m2, m3. 135° (supp. s) 45° 1 2 3 45° (vert. s ) 135° (vert. s ) June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Practice Solve for x. (3x + 20)° (5x + 8)° Vertical Angles are Congruent 5x + 8 = 3x + 20 2x = 12 x = 6 June 1, 2018 Geometry 2.6 Proving Geometric Relationships

Geometry 2.6 Proving Geometric Relationships Practice Solve for x, and find each angle. Linear Pair: Sum is 180 4x + 20 + 5x + 25 = 180 9x + 45 = 180 9x = 135 x = 15 4(15) + 20 = 80° 5(15) + 25 = 100° (4x + 20)° (5x + 25)° June 1, 2018 Geometry 2.6 Proving Geometric Relationships