Theorems, Postulates

Two Distinct points postulate

Through any two distinct points there exists one and only one line that can be drawn

Parallel Lines Postulate

Through a point not on a line, exactly one line is parallel to that line.

Side-Side-Side (SSS)

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

Rigid Motions

1. they map lines to lines, rays to rays, and segments to segments.

2. They preserve the distances between to mapped points

3. The preserve angle measures between tow mapped rays, lines, or segments.

*Rigid Motions preserve congruence

Isosceles Triangle Facts

1. two equal sides

2. base angles equal

3. The vertex is the perpendicular bisector of the base

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Line Reflections

When a point A is reflected across line m, we can find its image point by:

1. If A does not lie on line m then locate A' such that:

a. line m is perpendicular to line segment AA' and b AB = A'B

(m is the perpendicular bisector of AA')

2. If A does lie on line m, then A' in the same place as A.

90 degree clockwise rotation

(x,y) -> (y-,x)

90 degree counterclockwise rotation

(x,y) -> (-y,x)

Reflection across the origin

(x,y) -> (-x,-y)

Reflection across the x-axis

(x,y) -> (x,-y)

Reflection across x-axis

(x,y) -> (-x,y)

Side-Side-Side Theorem (SSS)

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

Side-Angle-Side Theorem (SAS)

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

Angle-Side-Angle Theorem (ASA)

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

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Parallel Lines Theorem

1. If two lines are crossed by a transversal and the alternate interior angles are congruent, then the lines are parallel

2. If two lines are crossed by a transversal and the corresponding angles are congruent, then the lines are parallel.

3. If two lines are crossed by a transversal and the alternate exterior angles are congruent, then the lines are parallel

Angle-Angle-Side 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.

Hypotenuse-Leg Theorem

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.