Lake Zurich Middle School South - Gifted Education

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Theorems are mathematical statements that can be proved. They are NOT always reversible. For example, the theorem "If two angles are right, then they are congruent" CANNOT be reversed to "If two angles are congruent, then they are right angles."

Theorem 1: If two angles are right, then they are congruent.

Theorem 2: If two angles are straight angles, then they are congruent.

Theorem 3: If a conditional statement is true, then the contrapositive of the statement is also true.

Theorem 4: If angles are supplementary to the same angle, then they are congruent.

Theorem 5: If angles are supplementary to congruent angles, then they are congruent.

Theorem 6: If angles are complementary to the same angle, then they are congruent.

Theorem 7: If angles are complementary to congruent angles, then they are congruent.

Theorem 8: If a segment is added to two congruent segments, then the sums are congruent. (Addition Property)

Theorem 9: If an angle is added to two congruent angles, then the sums are congruent. (Addition Property)

Theorem 10: If congruent segments are added to congruent segments, then the sums are congruent. (Addition Property)

Theorem 11: If congruent angles are added to congruent angles, then the sums are congruent. (Addition Property)

Theorem 12: If a segment (or angle) is subtracted from congruent segments (or angles), then the differences are congruent. (Subtraction Property)

Theorem 13: If congruent segments (or angles) are subtracted from congruent segments (or angles), then the differences are congruent. (Subtraction Property)

Theorem 14: If segments (or angles) are congruent, then their “like multiples” are congruent. (Multiplication Property)

Theorem 15: If segments (or angles) are congruent, then their “like divisions” are congruent. (Division Property)

Theorem 16: If angles (or segments) are congruent to the same angle (or segment), then they are congruent to each other. (Transitive Property)

Theorem 17: If angles (or segments) are congruent to congruent angles (or segments), then they are congruent to each other. (Transitive Property)

Theorem 18: Vertical Angles are congruent.

Theorem 19: All radii of a circle are congruent.

Theorem 20: If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

Theorem 21: If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

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Theorem 23: If two angles are both supplementary and congruent, then they are right angles.

Theorem 24: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of the segment.

Theorem 25: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.

Theorem 26: If two nonvertical lines are parallel, then their slopes are equal

Theorem 27: If the slopes of two nonvertical lines are equal, then the lines are parallel.

Theorem 28: If two lines are perpendicular and neither is vertical, then each line's slope is the opposite reciprocal of the others.

Theorem 29: If a line's slope is the opposite reciprocal of another line's slope, then the two lines are perpendicular.

Theorem 30: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

Theorem 31: If two lines are cut by a transversal such that two alternate interior angles are congruent, then the lines are parallel.

Theorem 32: If two lines are cut by a transversal such that two alternate exterior angles are congruent, then the lines are parallel.

Theorem 33: If two lines are cut by a transversal such that two corresponding angles are congruent, then the lines are parallel.

Theorem 34: If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, then the lines are parallel.

Theorem 35: If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, then the lines are parallel.

Theorem 36: If the two coplanar lines are perpendicular to the third line, then they are parallel.

Theorem 37: If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.

Theorem 38: If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.

Theorem 39: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles are congruent.

Theorem 40: If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.

Theorem 41: If two parallel lines are cut by a transversal, then each pair of interior angles on the same side of the transversal are congruent.

Theorem 42: If two parallel lines are cut by a transversal, then each pair of exterior angles on the same side of the transversal are congruent.

Theorem 43: In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Theorem 44: If two lines are parallel to a third line, then, they are parallel to each other, (Transitive Property of Parallel Lines)

Theorem 45: A line and a point not on a line determine a plane.

Theorem 46: Two intersecting lines determine a plane.

Theorem 47: Two parallel lines determine a plane.

Theorem 48: If a line is perpendicular to two distinct lines, that lie in a plane and that pass through its foot, then it is perpendicular to the plane.

Theorem 49: If a plane intersects two parallel planes, then the lines of intersection are parallel.

Theorem 50: The sum of the measures of the three angles of a triangle is 180.

Theorem 51: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Theorem 52: A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side.

Theorem 53: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No Choice Theorem)

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Lake Zurich Middle School South

Geometry
Page By: Matt B.
Created: October, 2004
Background From: Adrian Bruce

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