2.1: Triangles

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2.1.1 Basic Triangle Theorems

Note all theorems in this section can and should be proved without using the parallel postulate.

Definition: Vertical Angles

The opposing angles formed by the intersection of two lines are called vertical angles.

Definition: Congruent Angles

Two angles are congruent (∠ABC ≌ ∠DEF) if and only if their measures are equal (m∠ABC ≌ m∠DEF).

Theorem: Vertical Angle Congruence

Vertical angles are congruent.

A-B-C means that the points A, B, and C are colinear and B is between A and C.

Theorem: Pasch's Axiom

If a line ℓ intersects a triangle △ABC at a point D such that A-D-B then ℓ must intersect AC or BC.

Theorem: Crossbar

If X is a point in the interior of △ABC then ray AX intersects BC at a point D such that B-D-C.

Definition: Congruent Line Segments

Two line segments are congruent (AB ≌ CD) if and only if their measures (length) are equal (|AB| = |CD|).

Definition: Isosceles

A triangle is isosceles if and only if two sides are congruent.

Theorem: Isosceles Triangle

In an isosceles triangle the angles opposite the equal sides are congruent.

Theorem: Perpendicular Bisector

A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints.

Definition: Exterior Angle

The supplementary angle formed by extending one side of a triangle is called an exterior angle.

Theorem: Exterior Angle

The measure of an exterior angle of a triangle is greater than the measure of either of the opposing angles of the triangle.

Definition: Congruent Triangles

Two triangles are congruent if and only if all their sides and angles are congruent (△ABC ≌ △DEF).

2.1.2 Triangle Congruence Theorems

Determine if two triangles with two congruent sides and a congruent angle not between the two sides are congruent.

Theorem: Angle-Side-Angle

Two triangles are congruent if and only if two corresponding angles and the side between them are congruent.

Theorem: Angle-Angle-Side

Two triangles are congruent if and only if two corresponding angles and a side not between them are congruent.

Theorem: Side-Side-Side

Two triangles are congruent if and only if all three corresponding sides are congruent.

Theorem: Right Angle-Side-Side

Two right triangles are congruent if and only if two corresponding sides and a right angle not between those sides are congruent.

Theorem: Converse of Isosceles Triangle

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

Theorem: Extended Inverse of Isosceles Triangle

If two sides of a triangle are not congruent then the angles opposite those sides are not congruent. Further the larger angle is opposite the longer side.

Theorem: Triangle Inequality

The sum of the lengths of any two sides of a triangle is larger than the length of the other side.