Proving Angles Congruent

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Presentation transcript:

Proving Angles Congruent An Introduction to Geometric Proofs

Proof Is a set of deductive steps which are used to show that a conditional statement (or “Theorem”) is true. Each step is justified by a )postulate, ) property, )definition, )algebraic rules (“Simplify”), ) previously proved theorem or )as “Given” information

In a Paragraph Proof… Each step and reason is written as a sentence in a paragraph. “Given” info is explicitly stated or shown in a diagram * “Prove” the conclusion of the conditional; it is the last part of the proof.

Sample Paragraph Proofs Ex. 3 page 98: Vertical angles are congruent.

Congruent Supplements Theorem If 2 angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Complements Theorem If 2 angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Draw a picture….. Given: Prove:

Paragraph Proof By definition of congruence, By definition of complementary angles, _____________________________ By substitution (or transitive p of =), ____________________________ By subtraction p of =, ___________ By definition of congruence, q.e.d.! (quod est demonstratum)

All Right Angles are Congruent Draw a picture… Given….. Prove…..

Paragraph Proof By definition of right angles, _________________________ By substitution (or transitive p of =), ____________________________ By definition of congruence, ________________________ q.e.d.! (quod est demonstratum)

If two angles are congruent and supplementary, then each is a right angle. Draw a picture… Given….. Prove…..

Paragraph Proof By definition of congruent angles, __________________________ By definition of supplementary angles,____________________ Substituting one angle measure for the other yields______________ Which then simplifies into ________. Use the division p of = to get____________. Because the angles are congruent, ________________ and by the definition of right angles, both angles must be_____________ q.e.d!