Congruence Examples: 46 Problems with Answers

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?

Answer

Yes, △ABC≅△DEF by SSS.

First step

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

Full solution

  1. 2
    Step 2: Compare corresponding sides: AB=DE=3, BC=EF=4, AC=DF=5.
  2. 3
    Step 3: All three pairs of sides are equal, so by SSS (Side-Side-Side) congruence, the triangles are congruent.
  3. 4
    Step 4: Write the congruence statement: △ABC≅△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
△ABC≅△DEF. The perimeter of △DEF is 42. If AB=10 and BC=14, find CA.

Example 4

medium
A rectangle ABCD has diagonal BD. Show △ABD≅△CDB.

Example 5

hard
In △ABC, the angle bisector from A meets BC at 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: ∠P=50°, ∠Q=60°, PQ = 8 cm. Triangle XYZ: ∠X=50°, ∠Y=60°, XY = 8 cm. Are they congruent? Which postulate applies?

Example 3

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

Example 4

easy
△ABC≅△DEF. Which side corresponds to AB?

Example 5

easy
△ABC≅△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
△ABC≅△DEF and angle A=40∘. Find angle D.

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
△ABC≅△DEF. The perimeter of △ABC is 30. What is the perimeter of △DEF?

Example 14

medium
In triangle ABC, AB=AC (isosceles). The midpoint of BC is M. Explain why △ABM≅△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∘. 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
△ABC≅△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
△ABC≅△DEF with BC=7. Find EF.

Example 24

easy
△ABC≅△DEF with ∠B=75∘. Find ∠E.

Example 25

easy
Are two equilateral triangles with side length 8 congruent?

Example 26

easy
△ABC≅△DEF and AC=9. Find DF.

Example 27

medium
In △ABC and △DEF, AB=DE, ∠A=∠D, and AC=DF. Which postulate proves congruence?

Example 28

medium
△ABC≅△DEF with AB=2x+1 and DE=x+7. Find x.

Example 29

medium
In △ABC, ∠A=50∘ and ∠B=70∘. In △DEF, ∠D=50∘ and ∠F=60∘. With one side equal, can they be congruent?

Example 30

medium
Two right triangles each have legs 5 and 12. Are they congruent?

Example 31

medium
△ABC≅△DEF with ∠A=(3x)∘ and ∠D=(x+40)∘. Find x.

Example 32

medium
Triangles share two pairs of equal sides 6,8 and a non-included angle of 30∘ opposite the side of length 6. Are they necessarily congruent?

Example 33

medium
△ABC≅△DEF has AB=5,BC=7,CA=9. Find the perimeter of △DEF.

Example 34

hard
In △ABC, AD is the median to BC, and AB=AC. Prove △ABD≅△ACD.

Example 35

hard
Quadrilateral ABCD has AB=CD and AB∥CD. Prove △ABC≅△CDA.

Example 36

hard
Two triangles have AB=DE=10, BC=EF=10, and ∠B=∠E=90∘. Find AC and confirm congruence.

Example 37

hard
△ABC≅△DEF. AB=3x+2, DE=5x−8, and BC=11. Find EF.

Example 38

hard
Right △ABC and right △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 △AMB≅△CMD.

Example 40

challenge
Equilateral △ABC is rotated 60∘ 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?
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Background Knowledge

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

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