# Postulates, Formulas Theorems, Properties, Corollaries, and Congruence Statements

## 72 terms · Super important things to know from the Holt McDougal Geometry book.

### 1.1.1

Through any two points there is exactly one line.

### 1.1.2

Through any three noncollinear points there is exactly one plane containing them.

### 1.1.3

If two points lie in a plane, then the line containing those points lies in the plane.

### 1.1.4

If two lines intersect, then they intersect in exactly one point.

### 1.1.5

If two planes intersect, then they intersect in exactly one line.

### Ruler postulate

The points on a line can be put into a one-to-one correspondence with the real numbers.

### Segment addition postulate

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

### Protractor postulate

Given line AB and a point O on 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 angle PQR, then the measure of PQS + the measure of SQR = the measure of PQR.

### Circumference of a circle

The circumference C of a circle is given by the formula C = 𝜋d or C = 2𝜋r.

### Area of a circle

The area A of a circle is given by the formula A = 𝜋r2.

### Midpoint formula

The midpoint M of segment AB with endpoints A and B is found by M = (x1 + x2 / 2, y1 + y2 / 2).

### Distance formula

In a coordinate plane, the distance d between two points (x1, y1) and (x2, y2) is d = √(x2 - x1)squared + (x2 - y2)squared.

### 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 (a2 + b2 = c2).

### Law of detachment

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.

### Addition property of equality

If a = b, then a + c = b + c.

### Subtraction property of equality

If a = b, then a - c = b - c.

### Multiplication property of equality

If a = b, then ac = bc.

### Division property of equality

If a =b and c ≠ 0, then a/c = b/c.

a = a

### Symmetric property of equality

If a = b, then b +a.

### Transitive property of equality

If a = b and b = c, then a = c.

### Substitution property of equality

If a = b, then b can be substituted for a in any expression.

### Reflexive property of congruence

figure A is congruent to figure A

### Symmetric property of congruence

If figure A is congruent to figure B, then figure B is congruent to figure A.

### Transitive property of congruence

If figure A is congruent to figure B and figure B is congruent to figure C, then figure A is congruent to figure C.

### 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, then the two angles are congruent.

### Right angle congruence theorem

All right angles are congruent.

### Congruent complements theorem

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

### Common segments theorem

Given collinear points A, B, C, and D arranged in alphabetical order, if segment AB is congruent to segment CD, then segment AC is congruent to segment BD.

### Vertical angles theorem

Vertical angles are congruent.

### 2.7.3

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 interior 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 theorem

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

### Converse of the corresponding 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 exterior angles 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.

### 3.4.1

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.

### 3.4.3

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 is and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.

### Parallel lines

same slope, different y-intercept

different slopes

### Coinciding lines

same slope, same y-intercept

### Triangle sum theorem

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

### 4.2.2

The acute angles of a right triangle are complementary.

### 4.2.3

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

### 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 angles theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third pair angles are congruent.

### SSS

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

### 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 triangles are congruent.

### 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 triangles are congruent.

### AAS

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.

### HL

If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and 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 isosceles triangle theorem

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

### Equilateral triangle

If a triangle is equilateral, then it is equiangular.

### Equiangular triangle

If a triangle is equiangular, then it is equilateral.

### Perpendicular bisector theorem

If a point 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 it 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 it is on the bisector of the angle.

### Circumcentre theorem

The circumcentre of a triangle is equidistant from the vertices of the trangle (PA=PB =PC).

### Incentre theorem

The incentre of a triangle is equidistant from the sides of the triangle (PX = PY = PZ).

### Centroid theorem

The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side (AP = 2/3AY; BP = 2/3BZ; CP = 2/3 CX).