Congruent Triangles

There are many different types of triangles. Congruent triangles are those with three pairs of coordinating sides and three pairs of coordinating angles. Basically, by translating, rotating, and/or flipping one triangle, you can get it to line up exactly with the other triangle. If that is the case, they are congruent.

Indicating congruence

If one triangle, $△ABC$ , and another triangle, $△DEF$ , are congruent, we write

$△ABC\cong △DEF$

In the example below,

$△ABC\cong △DEF$

But $△ABC$ is not congruent with $△GHI$ .

Also note, we write that $△ABC\cong △DEF$ , but not that $△ABC\cong △DFE$ . The corresponding vertices must come in the correct order.

Proving congruence in triangles

It's not necessary to measure all three sides and all three angles to determine if two triangles are congruent. There are a handful of ways you can determine congruence by measuring just three sides and/or angles:

1. SSS (side, side, side): Triangles are congruent if all three corresponding sides are equal in length.
2. SAS (side, angle, side): Triangles are congruent if a pair of corresponding sides and the included angle are equal.
3. ASA (angle, side, angle): Triangles are congruent if a pair of corresponding angles and the included side are equal.
4. AAS (angle, angle, side): Triangles are congruent if a pair of corresponding angles and a non-included side are equal.
5. RHS (hypotenuse leg of a right triangle): Two right-angled triangles are congruent if the hypotenuse and one leg are equal. this is a special case of SAS.
6. Caution: Don't use AAA. Two triangles can have all three angles equal and yet not be congruent, as they might have differently scaled side lengths .

Example 1

$△ABC$ measures: $\text{Side} a=7\mathrm{cm}, \text{Side} b=5\mathrm{cm}, \text{Side} c=6\mathrm{cm}$

$△DEF$ measures: $\text{Side} f=6\mathrm{cm}, \text{Side} b=5\mathrm{cm}, \text{Side} c=7\mathrm{cm}$

Are $△ABC$ and $△DEF$ congruent?

Yes, they are, because all three corresponding sides are the same.

Example 2

$△ABC$ measures: $\text{Side} a=3.6\mathrm{cm}, \text{Side} b=7.1\mathrm{cm}$ , and the included $\text{Angle} A=49°$

$△DEF$ measures: $\text{Side} d=7.1\mathrm{cm}, \text{Side} e=3.6\mathrm{cm}$ , and the included $\text{Angle} D=48°$

Are $△ABC$ and $△DEF$ congruent?

No, because the included angles A and D are not the same.

Example 3

$△ABC$ measures: $\text{Angle} A=23°\mathrm{, }\text{Angle} B=124°$ , and the included Side $a=8.9\mathrm{cm}$

$△DEF$ measures: $\text{Angle} D=124°\mathrm{, }\text{Angle} E=23°$ , and the included Side $b=7.9\mathrm{cm}$

Are $△ABC$ and $△DEF$ congruent?

No, because the included sides A and D are not the same.

Example 4

$△ABC$ measures: $\text{Angle} A=42°\mathrm{, }\text{Angle} B=76°$ , and the non-included Side $c=15\mathrm{cm}$

$△DEF$ measures: $\text{Angle} D=42°\mathrm{, }\text{Angle} E=75°$ , and the non-included Side $f=15\mathrm{cm}$

Are $△ABC$ and $△DEF$ congruent?

No, because angles B and E are not the same.

Example 5

$△ABC$ is a right-angled triangle that measures: $\mathrm{Hypotenuse}=5.2\mathrm{cm}$ , $\mathrm{Long\; Leg}=7.9\mathrm{cm}$

$△DEF$ is a right-angled triangle that measures: $\mathrm{Hypotenuse}=5.2\mathrm{cm}$ , $\mathrm{Long\; Leg}=7.9\mathrm{cm}$

Are $△ABC$ and $△DEF$ congruent?

Yes, because both hypotenuses and long legs are the same.

Example 6

$△ABC$ measures: $\text{Angle} A=40°\mathrm{, }\text{Angle} B=110°\mathrm{, }\text{Angle} C=30°$

$△DEF$ measures: $\text{Angle} D=110°\mathrm{, }\text{Angle} E=30°\mathrm{, }\text{Angle} F=40°$

Are $△ABC$ and $△DEF$ congruent?

Not enough information is given, because AAA is not a reliable proof for whether a triangle is congruent or not.

Practice proving congruent triangles

A.

$△ABC$ measures: $\text{Angle} A=35°, \text{Angle} B=113°$ , and the included $\text{Side} a=6.7\mathrm{cm}$

$△DEF$ measures: $\text{Angle} D=113°, \text{Angle} E=34°$ , and the included $\text{Side} d=6.7\mathrm{cm}$

Are $△ABC$ and $△DEF$ congruent?

No, because angles A and E are not the same.

B.

$△ABC$ measures: $\text{Angle} A=40°, \text{Angle} B=110°, \text{Angle} C=30°$

$△DEF$ measures: $\text{Angle} D=110°, \text{Angle} E=30°, \text{Angle} F=40°$

Are $△ABC$ and $△DEF$ congruent?

Not enough information is given, because AAA is not a reliable proof for whether a triangle is congruent or not.

Topics related to the Congruent Triangles

Triangles

AA Similarity

Congruent Figures

Flashcards covering the Congruent Triangles

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Congruent Triangles

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

Get help learning about congruent triangles

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