Enter the sides and angles of two triangles to check congruence. Tests SSS, SAS, ASA, AAS, and HL criteria, identifies which criterion matched, solves missing values, and compares areas.
Two triangles are congruent when they have exactly the same shape and size — every corresponding side is equal and every corresponding angle is equal. Proving congruence does not require checking all six measurements; instead, five classic criteria provide shortcuts: 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).
Congruence is fundamental in Euclidean geometry, architecture, and engineering. Structural trusses rely on congruent triangles for uniform load distribution. CNC machining checks that cut parts are congruent to the master template. In academic settings, students must identify the correct criterion in proof-based problems.
This calculator lets you enter up to three sides and three angles for each triangle, then automatically tests every applicable criterion and reports which one (if any) confirms congruence. It also solves for any missing values using the law of cosines and the angle-sum property, compares both areas, and displays a visual side-by-side bar chart. Presets load common textbook examples so you can explore criteria quickly without manual entry. A reference table summarizes all five criteria with their requirements.
This congruent triangles calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Side a₁, Side b₁, Side c₁, Angle A₁ (°) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
SSS: a₁ = a₂, b₁ = b₂, c₁ = c₂ SAS: two sides equal and included angle equal ASA: two angles equal and included side equal AAS: two angles equal and a non-included side equal HL: right triangle with hypotenuse and one leg equal Law of cosines: c² = a² + b² − 2ab·cos(C) Angle sum: A + B + C = 180°
Result: For a1=3, b1=4, c1=5, the tool returns the solved congruent triangles outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in congruent triangles formulas and reports derived values, checks, and classifications automatically.
This page is tailored to congruent triangles, with outputs tied directly to the form fields (Side a₁, Side b₁, Side c₁, Angle A₁ (°)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
Congruent triangles are identical in shape AND size (all sides and angles equal). Similar triangles have the same shape but may differ in size (angles equal, sides proportional).
SSA (two sides and a non-included angle) can produce two different triangles (the ambiguous case), so it does not guarantee a unique triangle. Use this as a practical reminder before finalizing the result.
HL stands for Hypotenuse-Leg. It applies only to right triangles: if the hypotenuse and one leg of one right triangle equal those of another, the triangles are congruent.
You need exactly three correctly chosen measurements for each triangle — matching one of the five criteria (SSS, SAS, ASA, AAS, or HL). Keep this note short and outcome-focused for reuse.
Yes. Congruent triangles can be reflections (mirror images) of each other. Congruence requires equal corresponding parts, regardless of orientation.
Yes. Since all corresponding sides and angles are equal, congruent triangles always have identical area, perimeter, and every other measurement.