Triangles Cheat Sheet

A printable reference covering triangle angle sums, area, perimeter, triangle types, congruence, similarity, and the Pythagorean theorem for grades 6-8.

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Triangles are one of the most important shapes in geometry because they appear in measurement, construction, coordinate geometry, and proofs. This cheat sheet helps students quickly review triangle types, angle rules, side relationships, and key formulas. It is useful for solving homework problems, checking work, and preparing for quizzes or tests. Students in grades 6-8 need these facts to build a strong foundation for later geometry topics.

Key Facts

The angles inside every triangle add to $180^\circ$ , so $m\angle A + m\angle B + m\angle C = 180^\circ$ .
The area of a triangle is $A = \frac{1}{2}bh$ , where $b$ is the base and $h$ is the perpendicular height.
The perimeter of a triangle is $P = a + b + c$ , where $a$ , $b$ , and $c$ are the side lengths.
The triangle inequality says that a triangle can form only if $a + b > c$ , $a + c > b$ , and $b + c > a$ .
In a right triangle, the Pythagorean theorem is $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
An exterior angle of a triangle equals the sum of the two nonadjacent interior angles, so $m\angle 1 = m\angle A + m\angle B$ .
Similar triangles have equal corresponding angles and proportional corresponding sides, such as $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$ .
Congruent triangles have the same size and shape, so all corresponding sides and angles are equal.

Vocabulary

Triangle: A triangle is a polygon with exactly three sides, three vertices, and three angles.
Right Triangle: A right triangle is a triangle with one angle measuring $90^\circ$ .
Hypotenuse: The hypotenuse is the longest side of a right triangle and is opposite the $90^\circ$ angle.
Congruent Triangles: Congruent triangles are triangles with equal corresponding side lengths and equal corresponding angle measures.
Similar Triangles: Similar triangles are triangles with equal corresponding angles and proportional corresponding side lengths.
Altitude: An altitude is a perpendicular segment from a vertex of a triangle to the opposite side or its extension.

Common Mistakes to Avoid

Using a slanted side as the height in $A = \frac{1}{2}bh$ is wrong because the height must be perpendicular to the base.
Forgetting that triangle angles add to $180^\circ$ leads to incorrect missing angle values, especially when one angle is outside the triangle.
Applying $a^2 + b^2 = c^2$ to every triangle is wrong because the Pythagorean theorem works only for right triangles.
Choosing the wrong side as the hypotenuse is a common error because $c$ must be the side opposite the $90^\circ$ angle and the longest side.
Assuming triangles are congruent just because they look alike is wrong because congruence requires matching side lengths and angle measures, not visual appearance.

Practice Questions

1 A triangle has angles $48^\circ$ and $67^\circ$ . What is the measure of the third angle?
2 Find the area of a triangle with base $b = 14\text{ cm}$ and height $h = 9\text{ cm}$ .
3 A right triangle has legs $a = 6$ and $b = 8$ . Use $a^2 + b^2 = c^2$ to find the hypotenuse $c$ .
4 Two triangles have the same angle measures but different side lengths. Explain why they are similar but not necessarily congruent.

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