Proofs, Theorems, and Postulates

Ruler Postulate

The points on any line can be matched one-to-one with real numbers (coordinates), allowing us to measure the exact distance between points.

Segment Addition Postulate

If point $B$ lies between points $A$ and $C$ on a line, then the parts equal the whole: $AB + BC = AC$.

Angle Addition Postulate

If point $D$ lies in the interior of $\angle ABC$, then the two smaller adjacent angles add up to the larger angle: $m\angle ABD + m\angle DBC = m\angle ABC$.

Supplement Theorem

If two angles form a linear pair, then they are automatically supplementary ($m\angle 1 + m\angle 2 = 180^\circ$).

Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are automatically complementary ($m\angle 1 + m\angle 2 = 90^\circ$ ).

Reflexive Property of Congruence

Any geometric figure is congruent to itself (e.g., $\overline{AB} \cong \overline{AB}$ or $\angle A \cong \angle A$).

Symmetric Property of Congruence

If one figure is congruent to a second, then the second is congruent to the first (e.g., If $\angle A \cong \angle B$, then $\angle B \cong \angle A$).

Transitive Property of Congruence

If figure 1 is congruent to figure 2, and figure 2 is congruent to figure 3, then figure 1 is congruent to figure 3 (e.g., If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$).

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then those two angles are congruent to each other (e.g., If $\angle 1 + \angle 2 = 180^\circ$ and $\angle 3 + \angle 2 = 180^\circ$, then $\angle 1 \cong \angle 3$).

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then those two angles are congruent to each other (e.g., If $\angle 1 + \angle 2 = 90^\circ$ and $\angle 3 + \angle 2 = 90^\circ$, then $\angle 1 \cong \angle 3$).

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then each pair of corresponding angles is exactly congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary (they add up to 180°).

Consecutive Exterior Angles Theorem

If two parallel lines are cut by a transversal, then each pair of consecutive exterior angles is supplementary (they add up to 180°).

Converse of the Corresponding Angles Postulate

If two lines are cut by a transversal so that corresponding angles are congruent, then the two lines must be parallel.

Parallel Postulate

Given a line and a point not on that line, there exists exactly one line through that point that is parallel to the given line.

Converse of the Alternate Interior Angles Theorem

If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines must be parallel.

Converse of the Alternate Exterior Angles Theorem

If two lines are cut by a transversal so that alternate exterior angles are congruent, then the two lines must be parallel.

Converse of the Consecutive Interior Angles Theorem

If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the two lines must be parallel.

Converse of the Consecutive Exterior Angles Theorem

If two lines are cut by a transversal so that consecutive exterior angles are supplementary, then the two lines must be parallel.

Perpendicular Transversal Converse

In a flat plane, if two separate lines are both perpendicular to the exact same transversal line, then those two lines are parallel to each other.

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.

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.