Proof Reasons

The names shown in green are okay to use directly as proof reasons. For the other

definitions, postulates, properties, and theorems, you should use the descriptions shown.

 

Miscellaneous Reasons

Given

Simplification

 

Basic Properties of Numbers (Info Sheet #13)

Commutative

Associative

Distributive

 

Algebraic Properties of Equality (Info Sheet #13)

Addition

Subtraction

Multiplication

Division

Substitution

 

Equivalence Properties of Equality and Congruence (Info Sheet #13)

Reflexive

Symmetric

Transitive

 

Definition of Perpendicular Lines (Info Sheet #3)

Two lines intersecting at a right (90°) are .

Two lines intersect at a right (90°) .

 

Definition of Segment Midpoint (Info Sheet #3)

The midpoint of a segment is the point that divides the segment into two segments.

 

Definition of Segment Bisector (Info Sheet #3)

A segment bisector (a point, ray, line, segment, or plane) intersects a segment at its midpoint (dividing it into two segments).

 

Segment Congruence Postulate (Info Sheet #3)

Two segments of the same length are .

Two segments are the same length.

 

Segment Addition Postulate (Info Sheet #3)

If point R is between points P and Q on a line, then PR + RQ = PQ.

 

Definition of Right Angle (Info Sheet #4)

A right is an with a measure of exactly 90°.

 

Definition of Angle Bisector (Info Sheet #4)

An bisector (a ray or line) divides an into two ≅ ∠s.

 

Angle Congruence Postulate (Info Sheet #4)

Two s with = measures are .

Two s have = measures.

 

Angle Addition Postulate (Info Sheet #4)

If point S is in the interior of PQR, then mPQS + mSQR = mPQR.

 

Definition of Supplementary Angles (Info Sheet #4)

Two s adding to 180° are supplementary.

Supplementary s add up to 180°.

 

Definition of Complementary Angles (Info Sheet #4)

Two s adding to 90° are complementary.

Complementary s add up to 90°.

 

Linear Pair Property (Info Sheet #4)

Two s forming a linear pair are supplementary.

 

Definition of Right Triangle (Info Sheet #5)

A right ∆ is a ∆ that contains a right (90°) interior .

 

Vertical Angles Theorem (Info Sheet #5)

Vertical s are .

 

Triangle Sum Theorem (Info Sheet #5)

Sum of 3 interior s = 180°.

 

Overlapping Segments Theorem (Info Sheet #9)

Given points A, B, C, D on a line (in that order): If AB = CD, then AC = BD.

Given points A, B, C, D on a line (in that order): If AC = BD, then AB = CD.

 

Overlapping Angles Theorem (Info Sheet #9)

Given AOD with points B and C in its interior: If mAOB = mCOD, then mAOC = mBOD.

Given AOD with points B and C in its interior: If mAOC = mBOD, then mAOB = mCOD.

 

Corresponding Angles Postulate and Converse (Info Sheet #11 & Info Sheet #12)

Transversal with lines → corresponding s.

Transversal with corresponding s → lines.

 

Alternate Exterior Angles Theorem and Converse (Info Sheet #11 & Info Sheet #12)

Transversal with lines → alternate exterior s.

Transversal with alternate exterior s → lines.

 

Alternate Interior Angles Theorem and Converse (Info Sheet #11 & Info Sheet #12)

Transversal with lines → alternate interior s.

Transversal with alternate interior s → lines.

 

Same-Side Interior Angles Theorem and Converse (Info Sheet #11 & Info Sheet #12)

Transversal with lines → supplementary same-side interior s.

Transversal with supplementary same-side interior s → lines.

 

Perpendicular and Parallel Lines (Info Sheet #12)

Two coplanar lines to same line → lines.

Two coplanar lines to same line → lines.

 

Miscellaneous Theorem

Multiple adjacent s forming a line → sum of ∠s = 180°. (Info Sheet #12)

 

Parallel Postulate (Info Sheet #15)

Given a line and a point not on the line, there is exactly one line through the given point that is parallel to the given line.

 

Perpendicular Postulate (Info Sheet #15)

Given a line and a point not on the line, there is exactly one line through the given point that is perpendicular to the given line.

 

Miscellaneous Theorem

Two and supplementary s are right (90°) s. (Info Sheet #15)

 

Right Angle Congruence Theorem (Info Sheet #15)

All right (90°) s are .

 

Congruent Supplements/Complements Theorems (Info Sheet #16)

Two s are if they are supplements of the same or s.

Two s are if they are complements of the same or s.