Congruence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Congruence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Two geometric figures are congruent if they have exactly the same size and shape, so one can be placed on the other perfectly.

If you could pick up one shape and place it exactly on the other, they're congruent.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Two figures are congruent when one can be slid, flipped, or turned to land exactly on the other.

Common stuck point: The procedure for congruence is the easy part; the trap is calling scaled copies congruent. Asking "Can one figure be moved (slid, flipped, turned) to land exactly on the other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can one figure be moved (slid, flipped, turned) to land exactly on the other?

Worked Examples

Example 1

easy

Triangle ABC has sides 3 cm, 4 cm, 5 cm. Triangle DEF has sides 3 cm, 4 cm, 5 cm. Are they congruent?

Triangle ABC with sides 3 cm, 4 cm, 5 cm

Answer

Yes,

\triangle ABC \cong \triangle DEF

by SSS.

First step

Step 1: Congruent figures have exactly the same size and shape.

Full solution

2
Step 2: Compare corresponding sides: AB=DE=3, BC=EF=4, AC=DF=5.
3
Step 3: All three pairs of sides are equal, so by SSS (Side-Side-Side) congruence, the triangles are congruent.
4
Step 4: Write the congruence statement: $\triangle ABC \cong \triangle DEF$ .

SSS congruence states that if all three sides of one triangle equal all three sides of another, the triangles must be identical in shape and size. The angles are automatically determined by the side lengths.

Example 2

medium

Two rectangles: Rectangle 1 has dimensions 4 cm × 6 cm. Rectangle 2 has dimensions 6 cm × 4 cm. Are they congruent? Explain.

Example 3

medium

\triangle ABC \cong \triangle DEF

. The perimeter of

\triangle DEF

42

. If

AB = 10

and

BC = 14

, find

CA

Example 4

medium

A rectangle

ABCD

has diagonal

BD

. Show

\triangle ABD \cong \triangle CDB

Example 5

hard

\triangle ABC

, the angle bisector from

A

meets

BC

D

, and

AB=AC

. Prove

BD = DC

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Are a square with side 5 cm and a rhombus with side 5 cm necessarily congruent? Explain why or why not.

Example 2

hard

Triangle PQR:

\angle P = 50°

\angle Q = 60°

, PQ = 8 cm. Triangle XYZ:

\angle X = 50°

\angle Y = 60°

, XY = 8 cm. Are they congruent? Which postulate applies?

Triangle PQR: ∠P = 50°, ∠Q = 60°, PQ = 8 cm

Example 3

easy

Two figures are congruent. What can you say about their sizes and shapes?

Example 4

easy

\triangle ABC \cong \triangle DEF

. Which side corresponds to

AB

Example 5

easy

\triangle ABC \cong \triangle DEF

with

AB = 5

. What is the length of

DE

Example 6

easy

Are two squares with side

4

congruent?

Example 7

easy

Does flipping a shape over (reflecting it) keep it congruent to the original?

Example 8

easy

\triangle ABC \cong \triangle DEF

and angle

A = 40^\circ

. Find angle

D

Triangle ABC with ∠A = 40°; find the corresponding angle D in congruent △DEF

Example 9

easy

Two triangles have all three pairs of sides equal. Are they congruent?

Example 10

easy

Two figures have the same shape but one is twice as large. Are they congruent?

Example 11

medium

Two triangles share two equal sides and the equal angle BETWEEN them. Which congruence rule applies?

Example 12

medium

Why does AAA (all three angles equal) NOT prove two triangles congruent?

Example 13

medium

\triangle ABC \cong \triangle DEF

. The perimeter of

\triangle ABC

30

. What is the perimeter of

\triangle DEF

Example 14

medium

In triangle

ABC

AB = AC

(isosceles). The midpoint of

BC

M

. Explain why

\triangle ABM \cong \triangle ACM

Example 15

medium

Two triangles have a pair of equal angles, an adjacent pair of equal sides, and another pair of equal angles (in the order angle-side-angle). Which rule proves congruence?

Example 16

medium

A figure is translated 5 units right and rotated

90^\circ

. Is the image congruent to the original? Why?

Example 17

medium

Why is SSA (two sides and a non-included angle) not a valid congruence rule? Give the idea.

Example 18

medium

\triangle ABC \cong \triangle DEF

with

AB = 3x - 1

and

DE = 2x + 4

. Find

x

Example 19

challenge

In a parallelogram

ABCD

, prove that the diagonal

AC

splits it into two congruent triangles.

Example 20

challenge

Points

A

and

B

are fixed. Explain why every point

P

with

PA = PB

lies on the perpendicular bisector of

AB

, using congruent triangles.

Example 21

challenge

Two right triangles have equal hypotenuses and one pair of equal legs. Are they necessarily congruent? Name the rule and justify.

Example 22

challenge

A figure has rotational symmetry of order 3. Explain, using congruence, why its three 'arms' must be congruent to each other.

Example 23

easy

\triangle ABC \cong \triangle DEF

with

BC=7

. Find

EF

Example 24

easy

\triangle ABC \cong \triangle DEF

with

\angle B = 75^\circ

. Find

\angle E

Triangle ABC with ∠B = 75°; find the corresponding ∠E in congruent △DEF

Example 25

easy

Are two equilateral triangles with side length

8

congruent?

Equilateral triangle with side length 8

Example 26

easy

\triangle ABC \cong \triangle DEF

and

AC = 9

. Find

DF

Example 27

medium

\triangle ABC

and

\triangle DEF

AB = DE

\angle A = \angle D

, and

AC = DF

. Which postulate proves congruence?

△ABC: AB = DE, ∠A = ∠D (included), AC = DF — SAS postulate

Example 28

medium

\triangle ABC \cong \triangle DEF

with

AB = 2x+1

and

DE = x+7

. Find

x

△ABC ≅ △DEF: AB = 2x+1, DE = x+7; solve for x

Example 29

medium

\triangle ABC

\angle A = 50^\circ

and

\angle B = 70^\circ

. In

\triangle DEF

\angle D = 50^\circ

and

\angle F = 60^\circ

. With one side equal, can they be congruent?

△ABC with ∠A = 50°, ∠B = 70°, ∠C = 60°

Example 30

medium

Two right triangles each have legs

5

and

12

. Are they congruent?

Example 31

medium

\triangle ABC \cong \triangle DEF

with

\angle A = (3x)^\circ

and

\angle D = (x+40)^\circ

. Find

x

△ABC ≅ △DEF: ∠A = (3x)°, ∠D = (x+40)°; find x

Example 32

medium

Triangles share two pairs of equal sides

6, 8

and a non-included angle of

30^\circ

opposite the side of length

6

. Are they necessarily congruent?

Example 33

medium

\triangle ABC \cong \triangle DEF

has

AB=5, BC=7, CA=9

. Find the perimeter of

\triangle DEF

△ABC with AB = 5, BC = 7, CA = 9

Example 34

hard

\triangle ABC

AD

is the median to

BC

, and

AB = AC

. Prove

\triangle ABD \cong \triangle ACD

Example 35

hard

Quadrilateral

ABCD

has

AB=CD

and

AB \parallel CD

. Prove

\triangle ABC \cong \triangle CDA

Example 36

hard

Two triangles have

AB=DE=10

BC=EF=10

, and

\angle B = \angle E = 90^\circ

. Find

AC

and confirm congruence.

△ABC with AB = BC = 10, ∠B = 90°; find AC

Example 37

hard

\triangle ABC \cong \triangle DEF

AB = 3x+2

DE = 5x-8

, and

BC = 11

. Find

EF

△ABC ≅ △DEF: AB = 3x+2, DE = 5x−8, BC = EF = 11

Example 38

hard

Right

\triangle ABC

and right

\triangle DEF

have hypotenuses

AC = DF = 13

and legs

BC = EF = 5

. Are they congruent?

Example 39

challenge

In quadrilateral

ABCD

, the diagonals

AC

and

BD

bisect each other at

M

. Prove

\triangle AMB \cong \triangle CMD

Example 40

challenge

Equilateral

\triangle ABC

is rotated

60^\circ

about its centroid

G

. Explain why the image coincides with the original triangle.

Example 41

challenge

Points

A(0,0)

B(6,0)

C(6,8)

form a right triangle. Points

D(1,1)

E(7,1)

F(7,9)

form another. Are the triangles congruent?

Right △ABC: A(0,0), B(6,0), C(6,8); legs 6 and 8, hypotenuse 10

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Related Concepts

Basic Shapes Equal Similarity

Background Knowledge

These ideas may be useful before you work through the harder examples.

shapesequal