Geometry Proofs: Definitions, Properties, and Postulates for Congruence

Given

The facts provided in the problem.

Midpoint

If a point is the midpoint of a segment, then it divides the segment into two congruent parts.

Angle Bisector

If a ray/segment bisects an angle, then it divides the angle into two congruent angles.

Perpendicular Lines

If two lines are perpendicular, then they form right angles.

Congruent Right Angles

Any two right angles are equal in measure.

Vertical Angles Theorem

Vertical angles (opposite angles when two lines cross) are congruent.

Reflexive Property

Any segment or angle is congruent to itself.

Triangle Congruence Postulates - SSS

Three sides of one triangle ≅ three sides of another.

Triangle Congruence Postulates - SAS

Two sides and the included angle ≅.

Triangle Congruence Postulates - ASA

Two angles and the included side ≅.

Triangle Congruence Postulates - AAS

Two angles and a non-included side ≅.

Triangle Congruence Postulates - HL

Hypotenuse and one leg of a right triangle ≅.

CPCTC

Corresponding Parts of Congruent Triangles are Congruent.

Angle Addition Postulate

If point B is between A and C, then AB + BC = AC.

Segment Addition Postulate

If point D lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Proof Strategy

Start with Given.

Shared Sides

Look for shared sides (Reflexive Property).

Vertical Angles

Look for vertical angles if lines intersect.

Midpoint Usage

If you see 'midpoint,' use Definition of Midpoint.

Bisects Usage

If you see 'bisects,' use Definition of Angle Bisector.

Perpendicular Lines Usage

If you see perpendicular lines (⟂), write 'they form right angles' (Definition of Perpendicular), then 'All right angles are congruent.'

Triangle Congruence Rule Selection

Pick the correct triangle congruence rule (SSS, SAS, ASA, AAS, HL).

Proving Parts Congruent

If you need to prove parts congruent after triangles are congruent, use CPCTC.