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 with AB=2x+1 and DE=x+7. Find x.△ABC ≅ △DEF: AB = 2x+1, DE = x+7; solve for x
Example 2
medium
△ABC≅△DEF has AB=5,BC=7,CA=9. Find the perimeter of △DEF.△ABC with AB
Example 3
medium
In △ABC, ∠A=50∘ and ∠B=70∘. In △DEF, ∠D=50∘ and ∠F=60∘. With one side equal, can they be congruent?△ABC with ∠A
Example 4
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
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 6 cm. Are they congruent?
Example 7
easy
△ABC≅△DEF with ∠B=75∘. Find ∠E.Triangle ABC with ∠B = 75°; find the corresponding ∠E in congruent △DEF
Example 8
hard
Two triangles have AB=DE=10, BC=EF=10, and ∠B=∠E=90∘. Find AC and confirm congruence.△ABC with AB = BC = 10, ∠B = 90°; find AC
Example 9
easy
Two segments both have length 4.2 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 A and B are fixed. Explain why every point P with PA=PB lies on the perpendicular bisector of AB, using congruent triangles.
Example 12
medium
A rectangle ABCD has diagonal BD. Show △ABD≅△CDB.
Example 13
medium
A figure is translated 5 units right and rotated 90∘. 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 6,8 and a non-included angle of 30∘ opposite the side of length 6. 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 and AC=9. Find DF.
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, the angle bisector from A meets BC at D, and AB=AC. Prove BD=DC.
Example 20
medium
Why does AAA (all three angles equal) NOT prove two triangles congruent?