# Geometry Theorems Chapter 1-5

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### Ruler Postulate

The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.

If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

### Protractor Postulate

Let Ray OA and Ray OB be opposite rays in a plane. Ray OA, Ray OB, and all the rays with the endpoint O that can be drawn on one side of Line AB can be paired with the real numbers from 0 to 180 so that a. Ray OA is paired with 0 and Ray OB is paired with 180. b. If Ray OC is paired with x and Ray OD is paired with y, then the measurement of Angle COD = |x-y|.

If point B is in the interior of Angle AOC, then the measurement of Angle AOB + the measurement of Angle BOC = the measurement of Angle AOC. If Angle AOC is a straight angle, then the measurement of Angle AOC + the measurement of Angle BOC = 180.

### Law of Detachment

If a conditional is true and its hypotheses is true, then its conclusion is true. In symbolic form: If p-->q is a true statement and p is true, then q is true.

### Law of Syllogism

If p-->q and q-->r are true statements, then p-->r is a true statement.

### Reflexive Property

Line AB is congruent to Line AB and Angle A is congruent to Angle A.

### Symmetric Property

If Line AB is congruent to Line CD, then Line CD is congruent to Line AB. If Angle A is congruent to Angle B, then Angle B is congruent to Angle A.

### Transitive Property

If Line AB is congruent to Line CD, and Line CD is congruent to Line EF, then Line AB is congruent to Line EF. If Angle A is congruent to Angle B and Angle B is congruent to Angle C, then Angle A is congruent to Angle C.

### Vertical Angles Theorem

Vertical angles are congruent.

### Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

### Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

### Corresponding Angles Postulate

If a transversal intersects two parallel lines, then the corresponding angles are congruent.

### Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

### Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

### Alternate Exterior Angles Theorem

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

### Same-Side Exterior Angles Theorem

If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.

### Converse of the Corresponding Angles Postulate

If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.

### Converse of the Alternate Interior Angles Theorem

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.

### Converse of the Same-Side Interior Angles Theorem

If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

### Converse of the Alternate Exterior Angles Theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.

### Converse of the Same-Side Exterior Angles Theorem

If two lines and a transversal form same-side exterior angles that are supplementary, then the lines are parallel.

### Triangle Angle-Sum Theorem

The sum of the measures of the angles of a triangle is 180.

### Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

### Triangle Exterior Angle Theorem Corollary

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

### Parallel Postulate

Through a point not on a line, there is one and only one line parallel to the given line.

### Polygon Angle-Sum Theorem

The sum of the measures of the angles of an n-gon is (n-2)180.

### Polygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

### Slopes of Parallel Lines

If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertcal lines are equal, the lines are parallel. Any two vertical lines are parallel.

### Slopes of Perpendicular Lines

If two nonvertical lines are perpendicular, the product of their slopes is -1. If the slopes of two lines have a product of -1, the lines are perpendicular. Any horizontal line and vertical line are perpendicular.

### SSS Postulate

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

### SAS Postulate

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

### ASA 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 two triangles are congruent.

### AAS Theorem

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

### Isosceles Triangle Theorem Corollary

If a triangle is equilateral, then the triangle is equiangular.

### Converse of the Isosceles Triangle Theorem

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

### Converse of the Isosceles Triangle Theorem Corollary

If a triangle is equiangular, then the triangle is equilateral.

### HL 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.

### Triangle Midsegment Theorem

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.

### Perpendicular Bisector Theorem

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

### Converse of the Perpendicular Bisector Theorem

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

### Angle Bisector Theorem

If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

### Converse of the Angle Bisector Theorem

If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

### Comparison Property of Inequality

If a=b+c and c>0, then a>b.

### Triangle Inequality Theorem

The sum of the length of any two sides of a triangle is greater than the length of the third side.

Example: