Basic Proofs

Proving Two Triangles are Congruent

From the last lesson, we can use SAS, ASA, SSS, or SAA to prove two triangles are congruent.  Sometimes we have to do a little more, such as use vertical angles, the definition of midpoint or bisects, or the Isosceles Triangle Theorem to arrive at the necessary conditions for one of these. 

The following problems illustrate the ideas.  The proofs are given in statement-reason format in order to emphasize the details.  They could be rewritten in an easier-to-read "paragraph" format as the first problem illustrates.

Study these problems carefully to be sure you thoroughly understand the details as well as the "flow" of the proof.  For example, if a proof ends with SAS, then a pair of sides should be established as congruent before a pair of angles, and finally another pair of sides (so both side-pairs include the angle-pair) before SAS is used.

Problem 1

Given:    is the midpoint of  and .

Prove:   

Proof:

In paragraph form, we could write the proof like this:

Since is the midpoint of  and ,  and .   and  are vertical angles, so they are congruent.  Therefore  by SAS.

Problem 2

Given:    and

Prove: 

Proof:

Problem 3

Given:   bisects  and  bisects

Prove: 

Proof:

Problem 4

Given:  ,  , and  bisects

Prove: 

Note:  The symbol  means "is perpendicular to"

Proof:

Problem 5

Given:  , , and

Prove: 

Proof:

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