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**Proving Angles Congruent**

An Introduction to Geometric Proofs

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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

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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.

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**Sample Paragraph Proofs**

Ex. 3 page 98: Vertical angles are congruent.

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**Congruent Supplements Theorem**

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

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**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:

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**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)

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**All Right Angles are Congruent**

Draw a picture… Given….. Prove…..

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**Paragraph Proof By definition of right angles,**

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

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**If two angles are congruent and supplementary, then each is a right angle.**

Draw a picture… Given….. Prove…..

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**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!

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