Angle Relationship Proofs

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Transcript Angle Relationship Proofs

Angle Relationship Proofs

Linear Pair Postulate  Angles which form linear pairs are supplementary.

Linear Pair Postulate  π‘šβˆ 1 + π‘šβˆ 2 = 180Β°  π‘šβˆ 2 + π‘šβˆ 3 = 180Β°

Vertical Angle Theorem  Vertical angles are congruent: π‘šβˆ 1 = π‘šβˆ 3

Vertical Angle Theorem  Vertical angles are congruent: ∠1 β‰… ∠3

Vertical Angle Theorem Proof  Given: Angles 1 and 2, and angles 2 and 3, are linear pairs.

 Prove: Angles 1 and 3 are congruent.

Vertical Angle Theorem Proof

Statements

∠1 & ∠2 π‘Žπ‘Ÿπ‘’ π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ; ∠2 & ∠3 π‘Žπ‘Ÿπ‘’ π‘Ž π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘π‘Žπ‘–π‘Ÿ π‘šβˆ 1 + π‘šβˆ 2 = 180Β° π‘šβˆ 3 + π‘šβˆ 2 = 180Β° π‘šβˆ 1 = π‘šβˆ 3 ∠1 β‰… ∠3

Reasons

Given Linear Pair Postulate Linear Pair Postulate Substitution Def. of Congruence

Corresponding Angle Postulate  If two lines crossed by a transversal are parallel, their corresponding angles are congruent.

Corresponding Angle Postulate

Other theorems  Using the Linear Pair Postulate and the Corresponding Angles Postulate, we can prove…

Other theorems  Alt. Ext. Angles are Congruent  Alt. Int. Angles are Congruent

Other theorems  Same Side Ext. Angles are Supplementary  Same Side Int. Angles are Supplementary

Corresponding Angle Postulate  If two lines crossed by a transversal are parallel, their corresponding angles are congruent.

Converse of Corresponding Angles Postulate  If two lines crossed by a transversal have congruent corresponding angles, they are parallel.