Congruence Criteria Examples in Math

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

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

Concept Recap

Five sets of conditions that guarantee two triangles are congruent: SSS (three pairs of equal sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg for right triangles).

Imagine building a triangle from sticks and hinges. If you fix all three side lengths (SSS), there's only one triangle you can make. If you fix two sides and the angle between them (SAS), the triangle is locked in place. You don't need all six measurements—just the right three.

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: Congruence criteria are the minimal side-and-angle sets (SSS, SAS, ASA, AAS, HL) that force two triangles to be congruent.

Common stuck point: The procedure for congruence criteria is the easy part; the trap is using SSA (or 'AAA') as a criterion. Asking "Do the matched sides and angles equal (not just proportional to) those of the other triangle by a valid criterion?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the matched sides and angles equal (not just proportional to) those of the other triangle by a valid criterion?

Worked Examples

Example 1

easy
Two triangles share the following information: AB=DE=5AB = DE = 5, BC=EF=7BC = EF = 7, AC=DF=9AC = DF = 9. Are the triangles congruent? State the congruence criterion used.

Answer

△ABC≅△DEF\triangle ABC \cong \triangle DEF by SSS.

First step

1
Step 1: List what is known: all three pairs of corresponding sides are equal — AB=DEAB = DE, BC=EFBC = EF, AC=DFAC = DF.

Full solution

  1. 2
    Step 2: Identify the applicable congruence criterion. When all three sides of one triangle equal the corresponding sides of another, we use SSS (Side-Side-Side).
  2. 3
    Step 3: Conclude: By SSS, △ABC≅△DEF\triangle ABC \cong \triangle DEF.
SSS (Side-Side-Side) congruence states that if all three sides of one triangle are equal in length to the corresponding sides of another triangle, the triangles are congruent. This works because the shape of a triangle is completely determined by its three side lengths.

Example 2

medium
In right triangles △PQR\triangle PQR and △XYZ\triangle XYZ, both have a right angle. The hypotenuse PR=XZ=13PR = XZ = 13 and leg PQ=XY=5PQ = XY = 5. Are the triangles congruent? Which criterion applies?

Example 3

hard
In △ABC\triangle ABC, the perpendicular bisector of BCBC meets BCBC at midpoint MM and hits AA. Triangles ABMABM and ACMACM are congruent by which criterion?

Example 4

challenge
Prove: if a quadrilateral has both diagonals equal and bisecting each other, it is a rectangle. Identify the congruence step.

Practice Problems

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

Example 1

easy
Match each situation to the correct congruence criterion (SSS, SAS, ASA, AAS, HL): Two triangles have two angles and the included side equal.

Example 2

hard
Explain why SSA (two sides and a non-included angle) is NOT a valid congruence criterion by giving a counterexample or geometric explanation.

Example 3

easy
List the five standard triangle congruence criteria.

Example 4

easy
△ABC\triangle ABC has AB=5,BC=7,∠B=40°AB=5, BC=7, \angle B=40°. △DEF\triangle DEF has DE=5,EF=7,∠E=40°DE=5, EF=7, \angle E=40°. Which criterion?

Example 5

easy
Two triangles with sides 3,4,53, 4, 5 each. Which criterion proves congruence?

Example 6

medium
△ABC\triangle ABC: ∠A=30°,∠B=60°,AB=10\angle A=30°, \angle B=60°, AB=10. △DEF\triangle DEF: ∠D=30°,∠F=60°,DE=10\angle D=30°, \angle F=60°, DE=10. Which criterion?

Example 7

medium
An isosceles triangle has its apex bisected by a line to the base. The two resulting triangles are congruent by which criterion?

Example 8

medium
Diagonals of a rectangle bisect each other. Use a congruence criterion to identify which proves the two triangles formed by a diagonal congruent.

Example 9

medium
Two triangles with sides 5,12,135, 12, 13 are placed mirror-image. Are they still congruent? Answer 11 for yes.

Example 10

medium
△ABC\triangle ABC and △DEF\triangle DEF have ∠A=∠D=90°\angle A=\angle D=90°, hypotenuses BC=DF=10BC=DF=10, legs AB=DE=6AB=DE=6. Which criterion?

Example 11

medium
In the proof that base angles of an isosceles triangle are equal, which criterion is typically used?

Example 12

medium
Why does AAS work even though the side is not between the two angles?

Example 13

medium
Are two equilateral triangles with the same side length necessarily congruent?

Example 14

hard
Triangles ABCABC and DEFDEF satisfy AB=DE,BC=EF,∠A=∠DAB=DE, BC=EF, \angle A=\angle D. Is congruence guaranteed?

Example 15

hard
Two triangles with ∠A=∠D=90°\angle A=\angle D=90°, AB=DE=3AB=DE=3, AC=DF=4AC=DF=4. Find the hypotenuse and name the criterion.

Example 16

hard
A kite has its two pairs of adjacent equal sides. A diagonal splits it into two triangles. Which criterion proves these triangles congruent?

Example 17

hard
Show that in a parallelogram, opposite sides are equal by congruence. Which criterion is used on the two triangles formed by a diagonal?

Example 18

hard
Triangles share an angle and have proportional sides forming that angle (not equal). Are they congruent?

Example 19

challenge
Two triangles satisfy AB=DE=5AB=DE=5, BC=EF=8BC=EF=8, ∠C=∠F=35°\angle C=\angle F=35°. Are they necessarily congruent?
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Background Knowledge

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

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