Conjectures, Postulates, and Theorems

 

C-15  Vertical Angles Theorem – If two angles are vertical angles, then they are congruent.

 

C-16  Linear Pair Postulate – If two angles are a linear pair of angles, then they are supplementary.

 

P-7  Segment Addition Postulate – If point B is on  and between points A and C, then AB + BC = AC.  (please copy the picture on p. 726 in your textbook)

 

P-8 Angle Addition Postulate – If point D lies in the interior of ÐABC, then

mÐABD + mÐDBC = mÐABC.  (please copy the picture on p. 726 in your textbook)

 

Properties of Algebra and Properties of Equality – refer to pp720-721 in your textbook. You must know these properties!

 

P-1  Line Postulate – You can construct exactly one line through any two points.

 

P-2  Line Intersection Postulate – The intersection of two lines is exactly one point.

 

P-3  Midpoint Postulate – You can construct exactly one midpoint on any line segment.

 

P-4  Angle Bisector Postulate – You can construct exactly one angle bisector in any angle.

 

P-5  Parallel Postulate -  Through a point not on a given line, you can construct exactly one line parallel to the given line.

 

P-6  Perpendicular Postulate – Through a point not on a given line, you can construct exactly one line perpendicular to the given line.

 

Linear Pair Theorem – If two angles form a linear pair, then the sum of their measures equals 180°.

 

Congruent Supplements Theorem – If two angles are supplementary to two congruent angles (or to the same angle), then they are congruent.

 

Congruent Complements Theorem – If two angles are complementary to two congruent angles (or to the same angle), then they are congruent.

 

Common Segment Theorem –

1)      If , then .

2)      If , then .

 

Common Angle Theorem –

1)      If ÐABC @ ÐDBE, then ÐABD @ ÐCBE.

2)      If ÐABD @ ÐCBE, then ÐABC @ ÐDBE.

 

 

C-1  Perpendicular Bisector Conjecture

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

 

C-2  Converse of the Perpendicular Bisector Conjecture

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

 

C-3  Shortest Distance Conjecture

The shortest distance from a point to a line is measured along the perpendicular from the point to the line.

 

C-4  Angle Bisector Conjecture

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

 

C-5  The measure of each angle of an equilateral triangle is 60°.

 

C-6 The three angle bisectors of a triangle are concurrent. (Their point of concurrency is the incenter)

 

C-7   The three perpendicular bisectors of a triangle are concurrent.  (Their point of concurrency is the circumcenter)

 

C-8  The three altitudes (or the lines through the altitudes) of a triangle are concurrent. (Their point of concurrency is the orthocenter)

 

C-9  The circumcenter of a triangle is equidistant from the vertices of the triangle.

 

C-10  The incenter of a triangle is equidistant from the sides of the triangle.

 

C-11  The three medians of a triangle are concurrent. (Their point of concurrency is the centroid)

 

C-17  Parallel Lines Conjecture

If two parallel lines are cut by a transversal, then corresponding angles are congruenrt, alternate interior angles are congruent, and alternate exterior angles are congruent.

 

 

C-17a  Corresponding Angle Theorem (CA Theorem)

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

 

C-17b  Alternate Interior Angle Postulate (AIA Postulate)

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

 

C-17c  Alternate Exterior Angle Theorem (AEA Theorem)

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

 

C-18  Converse of the Parallel Lines Conjecture

If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior angles, and congruent alternate exterior angles, then the lines are parallel.

 

Converse of the AIA Postulate

If two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel.

 

Converse of the CA Theorem

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

 

Converse of the AEA Theorem

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

 

Formulas from Algebra I

 

Distance Formula                                  Midpoint Formula                                Slope Formula

 

                                         

 

 

In a coordinate plane, two distinct lines are parallel if and only if their slopes are equal.

 

In a coordinate plane, two nonvertical lines are perpendicular if and only if their slopes are opposite reciprocals.

 

Equations of Lines

 

Slope-intercept form:  y = mx + b

 

Point-slope form:  (y – y1) = m(x – x1)

 

Standard Form:  Ax + By = C

C-25 Triangle Sum Conjecture

The sum of the measures of the three angles in a triangle is 180°.

 

C-32  Triangle Exterior Angle Conjecture

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

 

C-27  Isosceles Triangle Conjecture

If a triangle is isosceles, then its base angles are congruent.

 

C-28  Converse of the Isosceles Triangle Conjecture

If a triangle has two congruent angles, then it is an isosceles triangle.

 

C-29  Equilateral Triangle Conjecture

An equilateral triangle is equiangular, and conversely, an equiangular triangle is equilateral.

 

C-30  Triangle Inequality Conjecture

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

 

C-31  Side-Angle Inequality Conjecture

In a triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

 

 

P-11  SSS Congruence Postulate

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

 

P-12  SAS Congruence Postulate

If two sides and the angle between them in one triangle are congruent to two sides and the angle between them in another triangle, then the two triangles are congruent.

 

P-13  ASA Congruence Postulate

It two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle, then the two triangles are congruent.

 

SAA Theorem

If two angles and a side that is not between them in one triangle are congruent to the corresponding two angles and side in another triangle, then the triangles are congruent.