Unit 4

Acute Triangle

A triangle containing three distinct angles that each measure less than $90^\circ$ .

Obtuse Triangle

A triangle containing exactly one interior angle that measures greater than $90^\circ$.

Right Triangle

A triangle containing exactly one interior angle that measures exactly $90^\circ$.

Equiangular Triangle

A triangle where all three interior angles are equal in measure (each is always exactly $60^\circ$).

Equilateral Triangle

A triangle where all three sides are equal in length.

Isosceles Triangle

A triangle with at least two congruent sides.

Scalene Triangle

A triangle where all three sides have completely different lengths.

Triangle Angle-Sum Theorem

The sum of the measures of the interior angles of any triangle is always exactly $180^\circ$.

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote (non-adjacent) interior angles.

Triangle Angle-Sum Corollaries

Direct logical extensions of the angle-sum rule: (1) The two acute angles of a right triangle are complementary ($90^\circ$), and (2) A triangle can contain at most one right angle or one obtuse angle.

Congruent Polygons

Geometric figures that have the exact same size and shape, meaning all corresponding sides and corresponding angles match perfectly.

Third Angle Theorem

If two angles of one triangle are congruent to two angles of another triangle, then their remaining third angles must also be congruent.

Side-Side-Side (SSS) Congruence Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Side-Angle-Side (SAS) Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Angle-Side-Angle (ASA) Congruence Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Congruence Theorem

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides (the base angles) are also congruent.

Equilateral Triangle Theorem

A triangle is equilateral if and only if it is entirely equiangular.