Triangle Congruence and Similarity Checker

Math Geometry • Basic Shapes and Properties

Written by STEM Calculators Team Published January 21, 2026 Updated February 24, 2026

Check whether two triangles are congruent (SSS, SAS, ASA, AAS, HL) or similar (AA, SSS, SAS), find the scale factor, and optionally fill missing sides when similarity holds.

Extra option: you can input triangles using coordinates (x,y). The tool auto-tests all 6 vertex correspondences. Canvas shows both triangles side-by-side (no overlap).

Options

Input mode Check Solve missing sides

Input rule: you may enter mixed data (e.g., SAS / ASA / AAS). If you enter all 3 sides, angles are computed from sides. If you also typed angles, they must match (within tolerance) or an error is shown.

Triangle 1 (Sides / Angles)

Sides (a, b, c)

Angles (degrees) (A, B, C)

If you provide exactly 2 angles, the third is computed as \(180^\circ-(A+B)\).

Right-triangle hint (for HL)

Triangle 1 right?

Triangle 2 (Sides / Angles)

Sides (a, b, c)

Angles (degrees) (A, B, C)

Right-triangle hint (for HL)

Triangle 2 right?

Ready

Enter triangle data and click Verify.

Visualization: Triangle 1 (left) and Triangle 2 (right)

Drag to pan • Wheel/trackpad to zoom • “Reset view” restores framing.

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Theory

Congruence vs. Similarity

Two triangles are congruent if they have the same size and shape (they can be matched by rigid motions). Two triangles are similar if they have the same shape but may differ by a scale factor \(k\).

Congruent ⇒ Similar (with \(k=1\)).

Congruence criteria

SSS: all three corresponding sides match.
SAS: two corresponding sides and the included angle match.
ASA: two angles and the included side match.
AAS: two angles and a non-included side match.
HL (right triangles): hypotenuse and one leg match (both triangles must be right).

Similarity criteria and scale factor

AA: two angles match ⇒ triangles are similar.
SSS similarity: \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k\).
SAS similarity: two side ratios match and the included angle matches.

If triangles are similar, a corresponding side scales by \[ a_1 = k\,a_2,\quad b_1 = k\,b_2,\quad c_1 = k\,c_2. \]

Input rule used by this calculator

A triangle cannot freely “choose” angles once all three sides are fixed. Therefore:

If you enter all 3 sides, the calculator computes the angles using the Law of Cosines.
If you also typed angles, they are treated as a consistency check. If inconsistent, an error is shown.
If you enter angles with one side, the calculator can complete the triangle using the Law of Sines.

Coordinates option

With coordinates, side lengths come from distance: \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \] Then angles are computed from the three side lengths using the Law of Cosines.

Frequently Asked Questions

How do I know if two triangles are congruent or similar?

Congruent triangles match exactly in size and shape, while similar triangles match in shape but can differ by a scale factor k. This checker tests standard criteria (SSS, SAS, ASA, AAS, HL for congruence and AA, SSS, SAS for similarity) to decide which relationship applies.

What is the HL test for triangle congruence and when can I use it?

HL applies only to right triangles: if both triangles are right and their hypotenuse and one corresponding leg are equal, then the triangles are congruent. The calculator lets you auto-detect or manually mark triangles as right for this check.

How is the similarity scale factor found?

When similarity holds, corresponding sides are proportional so a1/a2 = b1/b2 = c1/c2 = k. The checker computes k from the matched correspondence and can use it to fill missing sides if that option is enabled.

Can I enter triangle coordinates instead of side lengths and angles?

Yes. In coordinate mode, the tool computes side lengths from distances between points and then derives angles from the side lengths, testing all 6 possible vertex correspondences to find the best match.

Why does the calculator show an error if I enter three sides and also enter angles?

Three side lengths determine a unique triangle, so the angles implied by those sides must match any angles you type. If the typed angles are inconsistent (beyond tolerance), the checker flags it as an input error.

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Frequently Asked Questions

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