Practice Congruence in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

challenge
Points A(0,0)A(0,0), B(6,0)B(6,0), C(6,8)C(6,8) form a right triangle. Points D(1,1)D(1,1), E(7,1)E(7,1), F(7,9)F(7,9) form another. Are the triangles congruent?

Example 5

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

Example 6

easy
Two circles both have radius 66 cm. Are they congruent?

Example 7

easy
△ABC≅△DEF\triangle ABC \cong \triangle DEF with ∠B=75∘\angle B = 75^\circ. Find ∠E\angle E.

Example 8

hard
Two triangles have AB=DE=10AB=DE=10, BC=EF=10BC=EF=10, and ∠B=∠E=90∘\angle B = \angle E = 90^\circ. Find ACAC and confirm congruence.

Example 9

easy
Two segments both have length 4.24.2 cm. Are they congruent?

Example 10

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

Example 11

challenge
Points AA and BB are fixed. Explain why every point PP with PA=PBPA = PB lies on the perpendicular bisector of ABAB, using congruent triangles.

Example 12

medium
A rectangle ABCDABCD has diagonal BDBD. Show △ABD≅△CDB\triangle ABD \cong \triangle CDB.

Example 13

medium
A figure is translated 5 units right and rotated 90∘90^\circ. Is the image congruent to the original? Why?

Example 14

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

Example 15

medium
Triangles share two pairs of equal sides 6,86, 8 and a non-included angle of 30∘30^\circ opposite the side of length 66. Are they necessarily congruent?

Example 16

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

Example 17

easy
△ABC≅△DEF\triangle ABC \cong \triangle DEF and AC=9AC = 9. Find DFDF.

Example 18

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

Example 19

hard
In △ABC\triangle ABC, the angle bisector from AA meets BCBC at DD, and AB=ACAB=AC. Prove BD=DCBD = DC.

Example 20

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