Triangle Theorems & Proofs Cheat Sheet

A printable reference covering triangle congruence, similarity, angle sums, midsegments, inequalities, and proof reasoning for grades 9-10.

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Triangle theorems help students prove that triangles are congruent, similar, or related by angle and side measures. This cheat sheet gives a compact reference for the theorems most often used in Geometry proofs. Students need these results to justify each step clearly instead of relying on visual guesses. It is especially useful for two-column proofs, paragraph proofs, and diagram-based reasoning. The most important ideas include the triangle angle sum, exterior angle theorem, congruence shortcuts, similarity shortcuts, and proportional side relationships. Many triangle proofs begin by marking given information, then finding shared sides, vertical angles, or parallel-line angle relationships. Congruent triangles have matching sides and angles equal, while similar triangles have matching angles equal and side lengths proportional. Inequality theorems connect side length and angle size, helping students reason about which parts of a triangle are larger or smaller.

Key Facts

The Triangle Sum Theorem states that the interior angles of any triangle satisfy $m\angle A + m\angle B + m\angle C = 180^\circ$ .
The Exterior Angle Theorem states that an exterior angle equals the sum of the two remote interior angles, so $m\angle 1 = m\angle A + m\angle B$ .
Triangles are congruent by SSS when all three pairs of corresponding sides are equal, such as $AB = DE$ , $BC = EF$ , and $AC = DF$ .
Triangles are congruent by SAS when two pairs of corresponding sides and the included angle are equal, such as $AB = DE$ , $AC = DF$ , and $\angle A \cong \angle D$ .
Triangles are congruent by ASA or AAS when two pairs of corresponding angles and one corresponding side are equal.
Triangles are similar by AA when two pairs of corresponding angles are congruent, and corresponding sides have a constant scale factor $k$ .
If $\triangle ABC \sim \triangle DEF$ , then corresponding sides are proportional, so $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ .
The Triangle Inequality Theorem states that the sum of any two side lengths must be greater than the third side, so $a + b > c$ , $a + c > b$ , and $b + c > a$ .

Vocabulary

Congruent triangles: Congruent triangles have the same shape and size, so all corresponding sides and corresponding angles are equal.
Similar triangles: Similar triangles have the same shape, with corresponding angles equal and corresponding side lengths proportional.
Corresponding parts: Corresponding parts are matching sides or angles in two triangles that occupy the same relative position.
Included angle: An included angle is the angle formed between two given sides of a triangle.
Midsegment: A midsegment is a segment joining the midpoints of two sides of a triangle, and it is parallel to the third side with half its length.
Proof: A proof is a logical argument that uses definitions, given information, and theorems to show that a statement must be true.

Common Mistakes to Avoid

Using SSA to prove triangle congruence is wrong because SSA does not always determine one unique triangle.
Matching corresponding sides in the wrong order is wrong because proportions such as $\frac{AB}{DE} = \frac{BC}{EF}$ only work when the sides truly correspond.
Assuming triangles are congruent from a diagram is wrong because diagrams are not always drawn to scale and every claim needs a reason.
Using AA to prove congruence is wrong because AA proves similarity only, not equal size.
Forgetting the included angle in SAS is wrong because the equal angle must be between the two equal sides.

Practice Questions

1 In $\triangle ABC$ , $m\angle A = 48^\circ$ and $m\angle B = 67^\circ$ . Find $m\angle C$ .
2 Triangles $\triangle ABC$ and $\triangle DEF$ are similar with $AB = 6$ , $DE = 9$ , and $BC = 10$ . Find $EF$ .
3 Side lengths of a triangle are $7$ , $11$ , and $x$ . Write the compound inequality that describes all possible values of $x$ .
4 A proof shows that two pairs of corresponding angles in two triangles are congruent. Explain whether this proves the triangles are congruent, similar, or neither, and justify your answer.