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 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!