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Transcript
Warm-up 2.5 Fill in the blank 1. Perpendicular lines are two lines that intersect to form _______. 2. An angle is formed by two rays with the same endpoint. The endpoint is called the _________ of the angle. 2.5 Proving Angles Congruent Gonzales Geometry Definitions • Vertical Angles – Two angles whose sides form two pairs of opposite rays definitions • Adjacent angles – Two coplanar angles with a common side, a common vertex, and no common interior points Definitions • Complementary Angles – Two angles whose measures have a sum of 90 degrees. – Each angle is called the complement of the other Definitions • Supplementary Angles – Two angles whose measures have a sum of 180 degrees – Each angle is called the supplement of the other. definitions • Theorem – Statements that have been mathematically proven true. • Paragraph proof – Written as sentences in a paragraph Theorems • Vertical Angle Theorem – Vertical angles are congruent • Congruent Supplements theorem – If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. • Congruent Complements theorem – If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorems • Theorem 2-4 – All right angles are congruent • Theorem 2-5 – If two angles are congruent and supplementary, then each is a right angle. Proving Angles Congruent LESSON 2-5 Additional Examples Find the value of x. The angles with labeled measures are vertical angles because their sides are opposite rays. Apply the Vertical Angles Theorem to find x. 4x – 101 = 2x + 3 Vertical Angles Theorem 4x = 2x + 104 Addition Property of Equality 2x = 104 Subtraction Property of Equality x = 52 Division Property of Equality Quick Check Proving Angles Congruent LESSON 2-5 Additional Examples Write a paragraph proof using the given, what you are to prove, and the diagram. Given: WX = YZ Prove: WY = XZ Proof: It is given that WX = YZ. Adding the middle segment, XY, to each smaller segment gives WX + XY = YZ + XY. By the Segment Addition Postulate, WX + XY = WY, and XY + YZ = XZ. Using substitution, you get WY = XZ. Quick Check Examples Example Assignment • Pg. 100 • 1-22 47-52