2.2 Proofs Involving Congruence

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Presentation transcript:

2.2 Proofs Involving Congruence CHAPTER 2.2 Proofs Involving Congruence Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Congruent Segments Two segments that have the same length are called congruent segments. The symbol ≅ means congruent. We mark congruent segments in a figure with exactly the same number of tick marks. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definition Two angles that have the same measure are called congruent angles. Recall that the symbol ≅ means congruent. We mark congruent angles with exactly the same number of arcs, as shown in the figures below. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions Congruent figures have the exact same shape and size. All the figures below are congruent. As shown below, a flip, rotate (turn), or slide does not affect whether figures are congruent because they still have the same shape and size. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Definitions When figures are congruent, their corresponding sides are congruent, and their corresponding angles are congruent. Corresponding Angles & Sides Copyright © 2014 Pearson Education, Inc.

Naming Congruent Parts For the two figures, we are given that ABCD ≅ TRQS. Solution The figures help, but having the congruent angles listed in the corresponding order is all that is needed. ABCD ≅ TRQS Angles: Sides: Copyright © 2014 Pearson Education, Inc.

Proving Triangles Are Congruent Given: segments LM ≅ LO, MN ≅ ON, ∠M ≅ ∠O, ∠MLN ≅ ∠OLN Prove: ΔLMN ≅ ΔLON Copyright © 2014 Pearson Education, Inc.

Proving Triangles Are Congruent Statements Reasons 1. Given 2. Reflexive Property of Congruence 3. Given 4. Third Angles Theorem 5. Definition of congruent triangles Copyright © 2014 Pearson Education, Inc.

Postulate 4.3-2 Side-Angle-Side (SAS) Postulate Copyright © 2014 Pearson Education, Inc.

Proving Triangles Are Congruent Given: The figure with congruent segments shown by equal number of tick marks Prove: Statements Reasons 1. Given 2. Vertical Angles Theorem 3. SAS Postulate Copyright © 2014 Pearson Education, Inc.

Postulate 4.4-1 Angle-Side-Angle (ASA) Postulate Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Identifying ASA Multiple Choice: Choose two triangles that are congruent by the ASA Postulate. Explain why. a. b. c. d. Solution Choices b and d are congruent by ASA because for these two triangles, the sides marked congruent are the included sides of the two congruent angles. Copyright © 2014 Pearson Education, Inc.

Postulate 4.3-1 Side-Side-Side (SSS) Postulate Copyright © 2014 Pearson Education, Inc.

Proving Triangles Are Congruent Given: Prove: Statements Reasons 1. Given 2. Reflexive Property of Congruence 3. SSS Postulate Copyright © 2014 Pearson Education, Inc.