Check if two triangles are congruent by entering their sides and angles. Determines which congruence criterion applies — SSS, SAS, ASA, AAS, or HL — with detailed explanations, preset pairs, and a...
Two triangles are congruent when they have exactly the same shape and size — every corresponding side and angle matches. In geometry there are five standard congruence criteria that let you prove congruence without measuring all six parts: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL, for right triangles only).
This calculator lets you enter the three sides and three angles of two triangles and automatically checks every criterion. It reports which tests pass, highlights the matching parts, and gives a plain-English explanation of why the triangles are or are not congruent. If a criterion does not apply, the tool explains what is missing.
Preset buttons load well-known pairs — congruent right triangles, reflected isosceles triangles, scaled (similar but not congruent) pairs, and deliberately non-congruent examples — so you can explore every case interactively. A reference table summarizes each criterion with its requirements and a visual icon, and proportion bars compare corresponding sides at a glance. This is an ideal study companion for geometry students, a quick verification tool for engineers, and a teaching aid for instructors demonstrating congruence proofs.
This calculator is useful when you need to separate true congruence from data that only looks similar at first glance. It checks every standard criterion side by side, so you can see whether the match comes from SSS, SAS, ASA, AAS, or HL instead of relying on guesswork. That makes it practical for proof writing, homework checking, exam review, and any coordinate or construction problem where corresponding parts must be confirmed precisely.
SSS: a₁=a₂, b₁=b₂, c₁=c₂. 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. HL: right triangles with equal hypotenuse and one leg.
Result: Congruent by SSS, SAS, ASA, AAS, and HL
Triangle 1: sides 3, 4, 5 with angles 36.87°, 53.13°, 90°. Triangle 2: sides 3, 4, 5 with angles 36.87°, 53.13°, 90°. The SSS criterion is satisfied (all three corresponding sides are equal), so the triangles are congruent.
Most triangle proof mistakes come from choosing a criterion too quickly. SSS works when all three sides truly correspond, SAS only works with the included angle, and ASA or AAS depend on whether the known side lies between the known angles. Seeing each test evaluated separately helps you identify which theorem actually justifies the conclusion instead of mixing together partial facts that do not prove congruence.
Students often expect a side-side-angle pattern to be enough, but SSA can describe two different triangles or no triangle at all. That is the classic ambiguous case. By omitting SSA from the passing criteria and highlighting only the valid theorems, the calculator reinforces an important proof habit: not every set of matching measurements guarantees a unique triangle.
Congruence is about matching corresponding parts, not just having the same numbers somewhere in the figure. When you enter both triangles, keep the side and angle labels aligned so the reported tests mean what you think they mean. Used that way, the tool becomes a strong check on diagram labeling, coordinate proofs, and classroom constructions where orientation changes but size and shape should stay identical.
Congruent triangles have exactly the same shape and size. Every corresponding side has the same length and every corresponding angle has the same measure.
SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), and HL (hypotenuse-leg for right triangles). Use this as a practical reminder before finalizing the result.
Given two sides and a non-included angle, two different triangles might exist (the ambiguous case), so SSA alone does not guarantee congruence. Keep this note short and outcome-focused for reuse.
Ideally yes for full checking, but the calculator can still test criteria that only require three or four known matching parts. Apply this check where your workflow is most sensitive.
Side lengths are considered equal within 0.01 units and angles within 0.1°, accounting for typical rounding.
This tool checks congruence (same size). For similarity (same shape, different size), use a dedicated triangle similarity calculator.