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The Congruent Triangle Made Simple (Finally!)

Marilyn L. Brown

Marilyn L. Brown

There are two things that can be done with a congruent triangle pair. First they can be proven congruent. Next the parts can be used as congruent angles and sides.

A triangle is made of six parts—three angles (abbreviated A) and three sides (abbreviated S).

Postulates that state two triangles are congruent.

SSS

If the three sides are the same length the triangles will have to be congruent. You may want to try to draw two triangles with sides of 2 cm, 3 cm and 4 cm that have different angles. It cannot be done. Rotate both triangles so the 4 cm side is on the left and you will see how they are the same. This is the Side-Side-Side postulate. It is often abbreviated as SSS. Formally stated it says, “If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.”

AAA

If three angles of one triangle are congruent to three angles of another triangle, then the triangles are congruent. This is Angle-Angle-Angle

SAS

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. This is the Side-Angle-Side postulate.

ASA

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. This is the Angle-Side-Angle postulate.

AAS

If two angles and a side not included in the two angles of one triangle are congruent to the two angles and a side not include in the two angles of another triangle, then the triangles are congruent. This is the Angle-Angle-Side postulate. Recognize that this may appear as a SAA; but it is still called AAS.

Some students immediately conclude that to prove the congruent triangles you must prove that any three parts in the first triangle are congruent to any three parts of the other triangle. This is not true. So you must memorize the postulates.

How do you prove a pair of sides or angles is congruent?

1. Reflexive property-a segment or angle is congruent to itself
2. Substitution property- if two things are congruent to the same thing then they are congruent to each other
3. Vertical angles are congruent
4. They are certain parts of a special triangle (isosceles or equilateral)
5. They are in the same relationship to a known quantity- complimentary or supplementary to same angle.
6. They have been bisected.

What do you do with congruent triangles?

If two triangles have been proven congruent (SSS, AAA, SAS, ASA, AAS), then the corresponding parts of congruent triangles are congruent. (CPCTC)

This means that if two triangles have been proven congruent then all parts are congruent. Then they can be substituted with the related part. (Substitution)

Hints

1. Mark the drawings according to what is given.

a. Use tracing paper, if you are not to write in your text.
b. If two or more triangles overlap, draw or trace them apart, so you can see each separately.

2. Study the drawing and information given.

Be sure you understand what you are trying to prove. What are some ways this can be proven?

3. Start the geometry proof.

a. Work forward as far as you can. Then work backward. See if you can fill in the middle.
b. Mark your drawing as you go. Have a reason for every mark you make and be sure you write it in your proof.
c. Use every bit of given information. Make certain you don’t overlook a given fact.
d. Read the most recently taught concepts again. Often proofs are placed in the text in certain places because of the skills they require.
e. Read the instructions again. Sometimes the instructions contain hints such as, “Complete these proofs using the SSS, SAS, ASA and AAS postulates.”

Does the congruent triangle make sense now? Click here for more tips.

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