Proving Triangles Congruent: Explanation & Examples

Instructor: David Karsner

David holds a Master of Arts in Education

If the three sides of one triangle are the same length as the three sides of another triangle and the three angles of the first triangle have the same measure as the angles of the second triangle, then the two triangles are said to be congruent. You can measure all six sides and all six angles to see if they are the same. An easier means of determining if triangles are congruent is to use one of the five triangle congruence theorems.

The Vacant Lot

A woman owned a vacant lot that was in the shape of a rectangle 75 by 125 feet. The neighbors on either side of the lot liked to garden there. Both neighbors thought the other neighbor took up more than their share of the lot. Each neighbor also assumed that the owner was partial to the other neighbor. To end their bickering, the owner built a fence diagonally across the rectangle, creating two triangles. The neighbors didn't know what to think. Would these triangles be the same size? As it turned, out these triangles were the same size, and the owner was being fair. You can prove this using a triangle congruence theorem. This lesson will show you the five different shortcut theorems to prove that two triangles are congruent.

Congruence versus Similarity

There are two words in the world of triangles that seem a lot alike. Those words are congruence and similarity. Triangles can be congruent or similar, and there is only a small difference in their definition. Congruent triangles are triangles that have exactly the same lengths for their sides and the same measure for their angles. They are the same size and shape. Similar triangles have the same size angles, and the length of their sides are proportional. The have the same shape (but not necessarily the same size). The words congruent and equal also have similar definitions. Congruent means that two things are the same size. Congruence is represented using the symbol (=). Equal means that two things have the same number.

SSS Theorem (side-side-side)

If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. There is no need to find the measurements of the angles.

SAS Theorem (side-angle-side)

If two sides of one triangle and the angle between them are congruent to two sides and the angle between them of another triangle, then the two triangles are congruent. There is no need to find the value of the third side or the other two angles.

ASA Theorem (angle-side-angle)

If two angles and the side between them of one triangle are congruent to two angles and the side between of another triangle, then the triangles are congruent. There is no need to check the value of the third angle or the other two sides.

AAS Theorem (angle-angle-side)

If two angles and the side opposite of the second angle of one triangle are congruent to two angles and the side opposite of the second angle of another triangle, then the triangles are congruent. There is no need to check the value of the other angle or the other two sides.

HL Theorem (hypotenuse-leg)

This theorem is only to be used with right triangles. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another triangle, then the triangles are congruent.

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