# Congruence in Overlapping Triangles

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide an introduction to how to determine if triangles that overlap are congruent to each other. Definitions of congruent triangles and the five ways to prove triangles congruent are explained.

## Congruent Triangles

Two triangles are congruent if they are exactly the same size and shape, which means they have the same angle measures and the same side lengths. If we know that all the sides and all the angles are congruent in two triangles, then we know that the two triangles are congruent. To help show that two triangles are congruent and to understand which angles and which sides are congruent, we use tick marks to show corresponding sides and angles.

We don't necessarily need to know that all the sides and angles are congruent to prove two triangles are congruent.

## Ways To Prove Triangles are Congruent

Now, let's explore the different ways you can tell if two triangles are congruent and look at examples demonstrating these methods.

#### Side-Side-Side (SSS)

If we know that three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

#### Side-Angle-Side (SAS)

If we know that two sides and the angle between them are congruent to the two sides and the angle between them in another triangle, then the two triangles are congruent. It is important that the angle be the included angle, which means that it is between the two sides.

#### Angle-Side-Angle (ASA)

If we know that two angles and the side between them are congruent to the two angles and the side between them in another triangle, then the two triangles are congruent. In this case, the side is the included side, because it is between the two angles.

#### Angle-Angle-Side (AAS)

If we know that two angles and a side are congruent to the two angles and a side in another triangle, then the two triangles are congruent. The side here is the non-included side, which means it is not the side between the two angles.

#### Hypotenuse-Leg (HL)

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

Ways that do not work: There is no way to prove two triangles congruent if you have two sides and an angle that is not between the two sides. You also cannot prove two triangles congruent just by knowing all three angles are congruent.

## Congruence in Triangles with Overlapping Parts

#### Example 1

In this example, the two triangles are next to each other and they share a common side, AC. Since AC is the same side in both triangles, we know that it must be congruent to itself. This property is called the reflexive property of congruence, which says that any segment or angle is congruent to itself.

Since the triangles have three sides congruent to each other, then these two triangles are congruence by SSS.

#### Example 2

Triangle ABC and triangle ADC share a common side, AC. Again, by the reflexive property, we know that side AC is congruent to itself.

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