Congruent

Congruent

Exactly equal in size and shape. Congruent sides or segments have the exact same length. Congruent angles have the exact same measure. For any set of congruent geometric figures, corresponding sides, angles, faces, etc. are congruent.

Note: Congruent segments, sides, and angles are often marked.

Congruent Segments

Congruent Angles

All four sides of this rhombus are marked to show they are congruent to each other.

The marked angles of this isosceles triangle are congruent to each other.

Congruent Triangles	These triangles are congruent. Corresponding sides and angles are congruent.
Congruent Plane Figures	These figures are congruent because their shapes and sizes are identical, even though there are no sides or angles to measure and compare.
Congruent Pyramids	These are congruent figures because their sizes and shapes are identical. Note that corresponding faces of these pyramids are congruent.

Key Formula

\triangle ABC \cong \triangle DEF

Where:

$\cong$ = The congruence symbol, meaning 'is congruent to'
$\triangle ABC$ = The first triangle with vertices A, B, and C
$\triangle DEF$ = The second triangle with vertices D, E, and F
$AB = DE,\; BC = EF,\; AC = DF$ = All three pairs of corresponding sides are equal in length
$\angle A = \angle D,\; \angle B = \angle E,\; \angle C = \angle F$ = All three pairs of corresponding angles are equal in measure

Worked Example

Problem: Triangle ABC has sides AB = 5 cm, BC = 7 cm, AC = 8 cm and angles ∠A = 60°, ∠B = 80°, ∠C = 40°. Triangle DEF has sides DE = 5 cm, EF = 7 cm, DF = 8 cm and angles ∠D = 60°, ∠E = 80°, ∠F = 40°. Are the triangles congruent?

Step 1: List the corresponding sides and check if they are equal.

AB = DE = 5\text{ cm},\quad BC = EF = 7\text{ cm},\quad AC = DF = 8\text{ cm}

Step 2: List the corresponding angles and check if they are equal.

\angle A = \angle D = 60°,\quad \angle B = \angle E = 80°,\quad \angle C = \angle F = 40°

Step 3: Since all three pairs of corresponding sides are equal and all three pairs of corresponding angles are equal, the triangles are congruent.

\triangle ABC \cong \triangle DEF

Answer: Yes, △ABC ≅ △DEF because all corresponding sides and angles are equal.

Another Example

This example uses a known congruence statement to find missing measurements in the second triangle, showing how the order of vertices in the congruence notation determines which parts correspond.

Problem: You know that △PQR ≅ △XYZ. In △PQR, side PQ = 10 cm, ∠P = 50°, and ∠Q = 70°. Find side XY, ∠X, and ∠Z in △XYZ.

Step 1: Because the triangles are congruent, corresponding parts are equal. The order of the letters tells you which parts correspond: P ↔ X, Q ↔ Y, R ↔ Z.

Step 2: Side PQ corresponds to side XY, so they have the same length.

XY = PQ = 10\text{ cm}

Step 3: Angle P corresponds to angle X.

\angle X = \angle P = 50°

Step 4: Find ∠R first using the triangle angle sum, then use the correspondence to find ∠Z.

\angle R = 180° - 50° - 70° = 60°,\quad \angle Z = \angle R = 60°

Answer: XY = 10 cm, ∠X = 50°, and ∠Z = 60°.

Frequently Asked Questions

What is the difference between congruent and equal?

Equal typically refers to numbers or measurements having the same value (e.g., two lengths are both 5 cm). Congruent refers to geometric figures that have the same size and shape overall. You say two segments are congruent, but their lengths are equal. The symbol for congruence is ≅, while equality uses =.

What is the difference between congruent and similar?

Congruent figures have the same shape and the same size — all corresponding sides are equal and all corresponding angles are equal. Similar figures have the same shape but can be different sizes — corresponding angles are equal, but corresponding sides are only proportional, not necessarily equal. Every pair of congruent figures is also similar, but similar figures are not necessarily congruent.

How do you prove two triangles are congruent?

You do not need to check all six parts. There are shortcut tests: SSS (three pairs of sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), AAS (two angles and a non-included side equal), and HL (hypotenuse and one leg equal in right triangles). Any one of these is sufficient to prove congruence.

Congruent vs. Similar

	Congruent	Similar
Definition	Same shape and same size	Same shape but possibly different size
Corresponding sides	All equal in length	Proportional (equal ratios), not necessarily equal
Corresponding angles	All equal in measure	All equal in measure
Symbol	≅	~
Scale factor	Always 1	Can be any positive number
Relationship	All congruent figures are also similar	Similar figures are congruent only if the scale factor is 1

Why It Matters

Congruence is central to geometry proofs in courses from middle school through high school. When you prove two triangles congruent using tests like SSS, SAS, or ASA, you can immediately conclude that all their corresponding parts are equal (CPCTC), which lets you find unknown side lengths and angle measures. Outside the classroom, congruence underlies real-world precision — manufacturing identical parts, tiling floors with matching shapes, and ensuring structural symmetry in engineering all depend on figures being congruent.

Common Mistakes

Mistake: Confusing congruent with similar. Students sometimes say two figures are congruent when the figures have the same shape but different sizes.

Correction: Congruent requires both the same shape AND the same size. If corresponding sides are proportional but not equal, the figures are similar, not congruent.

Mistake: Writing the congruence statement with vertices in the wrong order, such as writing △ABC ≅ △FDE when the actual correspondence is A↔D, B↔E, C↔F.

Correction: The order of the vertices must match the correspondence. If A corresponds to D, B to E, and C to F, you must write △ABC ≅ △DEF. Mismatched vertex order leads to incorrect conclusions about which sides and angles are equal.

Related Terms

Similar — Same shape but not necessarily same size
Congruence Tests for Triangles — Shortcut methods to prove triangle congruence
CPCTC — Corresponding parts of congruent triangles are congruent
Corresponding — Matching parts between congruent figures
Angle — Congruent angles have equal measures
Line Segment — Congruent segments have equal lengths
Isosceles Triangle — Has two congruent sides and two congruent angles
Rhombus — Has four congruent sides