Geometry proofs

Definition of a Congruent

Segments or angles that have the same measure (Used when you switch from an "=" sign to a "≅" sign or vice versa)

Definition of a Bisector

a line or segment that intersects a segment or angle and divides it into two congruent halves

Definition of a Midpoint

divides the segment into two congruent segments

Definition of Complementary Angles

two angles whose degree measures have a sum of 90º

Definition of Supplementary Angles

two angles whose degree measures have a sum of 180º

Vertical Angles

are always congruent and you can assume from a picture

Linear Pair

you can assume from a picture; they are supplementary

Reflexive Property of Equality

A line segment (or angle) is congruent to itself

Transitive Property of Equality

a=b, b=c, a=c [usually not = to a #]

Addition Property of Equality

The same thing can be added to both sides of an equation

Subtraction Property of Equality

The same thing can be subtracted from both sides of an equation

Multiplication Property of Equality

You can multiply both sides of an equation by the same thing

Substitution Property of Equality

Replace a variable or value with an equal variable or value

Distributive Property of Equality

a(b+c)=ab+ac

If Two Angles are Congruent and Supplementary

then each is a right angle

All Right Angles

are congruent to each other

Definition of Perpendicular Lines

Two intersecting lines that form four right angles

Definition of a Right Angle

measures 90º

Same Side Interior Angles Postulate

Angles inside two parallel lines on the same side of the transversal are supplementary

Alternate Interior Angles Theorem

Angles inside two parallel lines on opposite sides of the transversal are congruent

Corresponding Angles Theorem

The two angles are in the same position at each parallel line. They are congruent.

Alternate Exterior Angles Theorem

Angles outside two parallel lines on opposite sides of the transversal are congruent

Two lines parallel to the same line

are parallel to each other

Triangle Sum Theorem

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

Third Angles Theorem

if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent

SSS Postulate

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

SAS 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.

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 triangles are congruent.

AAS Theorem

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

CPCTC

If two triangles are congruent then all their corresponding parts are congruent. Used in a proof after showing triangles are congruent.

Definition of Isosceles Triangle

a triangle that has two congruent sides

Isosceles Triangle Theorem

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

A triangle is an equilateral

if and only if it is equiangular. Each angles of an equilateral triangle measures 60 degrees.

Hypotenuse Leg Theorem

if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent

Definition of a Parallelogram

If a quadrilateral is a parallelogram, then the quadrilateral's both pairs of opposite sides are parallel

(1/5 Characteristic of a Parallelogram)

If a quadrilateral is a parallelogram, then the opposite sides are _.

congruent (sides)

(2/5 Characteristic of a Parallelogram)

If a quadrilateral is a parallelogram, then the opposite angles are _.

congruent (angles)

(3/5 Characteristic of a Parallelogram)

If a quadrilateral is a parallelogram, then the consecutive angles are _.

supplementary

(4/5 Characteristic of a Parallelogram)

If a quadrilateral is a parallelogram, then the diagonals _.

bisect each other

(5/5 Characteristic of a Parallelogram)

If diagonals of a quadrilateral bisect each other, then the quadrilateral is a .

parallelogram

If diagonals of a quadrilateral bisect each other and are congruent, then the quadrilateral is a .

rectangle

If diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a .

rhombus

If diagonals of a quadrilateral bisect each other, are perpendicular, and are congruent, then the quadrilateral is a .

square

Median

A segment in a triangle that connects the vertex of an angle to the midpoint of the opposite side

AA~

Two triangles are similar if two pairs of angles are congruent

SSS~

Two triangles are similar if the corresponding sides are proportional

SAS~

Two triangles are similar if two corresponding sides are proportional and the included angle is congruent

Congruent Segments

Segments with equal lengths in geometry.

Midpoint

Point dividing a segment into two equal parts.

Median

Line segment from vertex to midpoint of opposite side.

Segment Bisector

Line that divides a segment into two equal parts.

Congruent Angles

Angles that have the same measure.

Angle Bisector

Ray that divides an angle into two equal angles.

Perpendicular Lines

Lines that intersect to form right angles.

Transversal

Line that intersects two or more lines.

Vertical Angles

Angles opposite each other when two lines intersect.

Reflexive Property

A property stating an element is congruent to itself.