Geometry 4.1 Notes

The measures of the three interior angles of a triangle add to 180°.

Triangle Sum Theorem

An exterior angle of a triangle equals the sum of the two remote interior angles.

Exterior Angle Theorem

If two sides of a triangle are congruent, the angles opposite those sides are congruent.

Isosceles Triangle Theorem

A triangle where all sides are equal and all angles measure 60°.

Equilateral Triangle

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Triangle Inequality Theorem

Corresponding Parts of Congruent Triangles are Congruent, used once triangles are proven congruent.

CPCTC

For right triangles, if the hypotenuse and one leg of a triangle are congruent to the hypotenuse and a corresponding leg of another triangle, the triangles are congruent.

Hypotenuse-Leg (HL) Theorem

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

SSS Postulate

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

SAS Postulate

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

ASA Postulate

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

AAS Postulate

Mark given information, choose a congruence postulate, prove triangles congruent step-by-step, then use CPCTC.

Congruence in Proofs

The measures of the three interior angles of a triangle add to 180°.

Triangle Sum Theorem

An exterior angle of a triangle equals the sum of the two remote interior angles.

Exterior Angle Theorem

If two sides of a triangle are congruent, the angles opposite those sides are congruent.

Isosceles Triangle Theorem

A triangle where all sides are equal and all angles measure 60°.

Equilateral Triangle

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Triangle Inequality Theorem

Corresponding Parts of Congruent Triangles are Congruent, used once triangles are proven congruent.

CPCTC

For right triangles, if the hypotenuse and one leg of a triangle are congruent to the hypotenuse and a corresponding leg of another triangle, the triangles are congruent.

Hypotenuse-Leg (HL) Theorem

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

SSS Postulate

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

SAS Postulate

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

ASA Postulate

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

AAS Postulate

Mark given information, choose a congruence postulate, prove triangles congruent step-by-step, then use CPCTC.

Congruence in Proofs

A triangle with one angle measuring 90°. The side opposite the right angle is the hypotenuse; the other two sides are legs.

Right Triangle

A triangle where all three interior angles are acute (measure less than 90°).

Acute Triangle

A triangle with one obtuse interior angle (measure greater than 90°).

Obtuse Triangle

A triangle where all three sides have different lengths, and all three angles have different measures.

Scalene Triangle

A segment from a vertex perpendicular to the opposite side or to the line containing the opposite side.

Altitude of a Triangle

A segment from a vertex to the midpoint of the opposite side.

Median of a Triangle

A segment from a vertex that bisects the angle at that vertex and extends to the opposite side.

Angle Bisector of a Triangle

A line or segment that passes through the midpoint of a side and is perpendicular to that side.

Perpendicular Bisector of a Triangle