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.
Showing a random 20 of 50 problems.
Example 1
challenge
Two triangles satisfy AB=DE=5, BC=EF=8, ∠C=∠F=35°. Are they necessarily congruent?△ABC: AB=5, BC=8, ∠C=35° — SSA configuration; congruence is NOT guaranteed
Example 2
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 3
easy
How many independent pieces of information are needed to guarantee a triangle's congruence?
Example 4
medium
Triangle ABC has ∠A=40°, ∠B=60°, and AB = 8. Triangle DEF has ∠D=40°, ∠E=60°, DE = 8. Are they congruent, and by what criterion?△ABC: ∠A=40°, ∠B=60°, AB=8 — congruent to △DEF with ∠D=40°, ∠E=60°, DE=8 by ASA
Example 5
easy
Two triangles with sides 3,4,5 each. Which criterion proves congruence?
Example 6
medium
A diagonal divides a parallelogram into two triangles. Which criterion shows the two triangles are congruent?
Example 7
easy
Two triangles share the following information: AB=DE=5, BC=EF=7, AC=DF=9. Are the triangles congruent? State the congruence criterion used.△ABC with sides 5, 7, 9 — identify the congruence criterion
Example 8
easy
Triangle ABC has sides 5, 6, 7. Triangle DEF has sides 5, 6, 7. Are they congruent?Triangle with sides 5, 6, 7 — are two such triangles congruent?
Example 9
challenge
Explain why a triangle made of three fixed-length rods is rigid, while a quadrilateral of four fixed-length rods is not.
Example 10
medium
Two right triangles have equal hypotenuses and one equal leg. Which special criterion applies?Right triangle: equal hypotenuse + one equal leg — which criterion?
Example 11
easy
△ABC has AB=5,BC=7,∠B=40°. △DEF has DE=5,EF=7,∠E=40°. Which criterion?△ABC: AB=5, BC=7, ∠B=40° — identify the congruence criterion
Example 12
medium
Two triangles have angles 50°, 60°, 70° each, but one has a side of 4 and the other a side of 8. Are they congruent?Same angles 50°, 60°, 70° but different side lengths — similar but not congruent
Example 13
hard
In △ABC, the perpendicular bisector of BC meets BC at midpoint M and hits A. Triangles ABM and ACM are congruent by which criterion?Perpendicular bisector from A to midpoint M of BC creates two SAS-congruent triangles ABM and ACM
Example 14
easy
Which of these is NOT a valid triangle congruence criterion?
Example 15
medium
Are two equilateral triangles with the same side length necessarily congruent?Equilateral triangle: all sides equal s — two equilateral triangles with the same side length are congruent by SSS
Example 16
medium
An isosceles triangle has its apex bisected by a line to the base. The two resulting triangles are congruent by which criterion?Isosceles triangle: apex angle bisected creates two SAS-congruent triangles
Example 17
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 18
easy
After proving △ABC≅△DEF, which acronym lets us conclude the corresponding parts are equal?
Example 19
medium
In the proof that base angles of an isosceles triangle are equal, which criterion is typically used?
Example 20
easy
Match each situation to the correct congruence criterion (SSS, SAS, ASA, AAS, HL): Two triangles have two angles and the included side equal.