Holt Geometry Chapters 1-4 Postulates, Theorems, and Corollaries

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All Postulates, Theorems, and Corollaries can also be found on page S82 in the back of the Texas Holt Geometry Textbook.

Segment Addition Postulate

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

Protractor Postulate

Given line AB and a point O on line AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180.

Angle Addition Postulate

If S is in the interior of ∠PQR, then m∠ PQS + m∠ SQR = m∠PQR

Pythagorean Theorem

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Linear Pair Theorem

If two angles form a linear pair, then they are supplementary.

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. ex.: if angle 1 and 2 are supplementary, and angle 2 and 3 are supplementary, then angle 1 is congruent to angle 3

Right Angle Congruence Theorem

All right angles are congruent.

Congruent Complements Theorem

If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent.

Common Segments Theorem

Given collinear points, A, B, C, and D arranged as shown, if AB is congruent to CD, then AC is congruent to BD.

Vertical Angles Theorem

Vertical Angles are congruent.

Congruent Angles Supplementary Theorem

If two congruent angles are supplementary, then each angle is a right angle.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate initerior angles are congruent.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent.

Same-Side Interior Angles

If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary.

Converse of the Corrresponding Angles Postulate

If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.

Parallel Postulate

Through a point P not on line l, there is exactly one line parallel to l.

Converse of the Alternate Interior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.

Converse of the Alternate Extrerior Theorem

If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.

Converse of the Same-Side Interior Angles Theorem

If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel.

Intersecting Lines Form Congruent Linear Pair so Lines are Perpendicular

If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.

Perpendicular Transversal Theorem

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

Two Perpendiculars to Same Line so Parallel Theorem

If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other.

Parallel Lines Theorem

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

Perpendicular Lines Theorem

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their shapes is -1. Vertical and horizontal lines are perpendicular.

Triangle Sum Theorem

The sum of the angle measures of a triangle is 180 degrees.

Acute Angles of Right Triangle are Complementary

The acute angles of a right triangle are complementary.

Measure of Angles of Equiangular Triangle = 60

The measure of each angle of an equiangular triangle is 60 degrees.

Exterior Angle Theorem

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

Third Angel Theorem

If two angles of one triangle are conguent to two angles of another triangle, then the third pair of angles are 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 Postulate

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

Hypotenuse-Leg (HL) Congruence Theorem

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

Isosceles Triangle Theorem

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

Converse of the Isosceles Triangle Theorem

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

Equilateral Triangle is Equiangular Corollary

If a triangle is equilateral, then it is equiangular.

Equiangular Triangle is Equilateral Corollary

If a triangle is equiangular, then it is equilateral.

Definition of Perpendicular

If two lines are perpendicular, they create right angles.

Definition of Congruence

If the measures of two line segments are equal, then they are congruent.

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