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Definition of Perpendicular lines

Two lines are perpendicular if and only if the two lines meet to form congruent adjacent angles

Definition of a midpoint

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

Definition of segment bisector

The two sides of the line are congruent to each other when cut by a segment bisector.

Definition of angle bisector

The two sides of the angle are congruent to each other when cut by an angle bisector.

Definition of perpendicular bisector

a line, ray, segment, or plane that passes through the midpoint of the segment and is perpendicular to the segment

Definition of Supplementary angles

Two angles whose measures have a sum of 180 degrees

Definition of Complementary Angles

Two angles are complementary if and only if their sum= 90 degrees

Segment Congruence Postulate

Two segments are congruent if and only if they have the same length.

Segment Addition Postulate

If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.

Angle Addition Postulate

D is in the interior of angle ABC and only if m<ABD + m<DBC = m<ABC.

Angle congruence Postulate

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

Linear pair property

if two angles form a linear pair, then they are supplementary

Addition Property of Equality

For all real numbers x, y, and z, if x = y, then x + z = y + z.

Subtraction Property of Equality

For all real numbers x, y, and z, if x = y, then x - z = y - z.

Multiplication Property of Equality

For all real numbers x, y, and z, if x = y, then xz = yz.

Division Property of Equality

For all real numbers x, y, and z, if x = y, and z ≠ 0, then x/z = y/z. You can divide each side of an equation by the same non-zero number and not change its truth value.

Substitution Property of Equality

For all real numbers x and y, if x = y , then y can be substituted for x in any expression and vice versa.

Distribution Property of Equality

a(b+c) = ab + bc

Reflexive Property of Congruence

Line AB is congruent to line AB.

Symmetric Property of Congruence

If line AB is congruent to line CD, then line CD is congruent to line AB.

transitive property of congruence

If line AB is congruent to line CD and line CD is congruent to line EF, then line AB is congruent to line EF.

Overlapping Segments Theorem

given a segment with points A,B, C, and D arranged as shown, the following statements are true, if AB=CD, then AC=BD, if AC=BD, then AB=CD

Overlapping Angles Theorem

given <AOD with points B and C in its interior, the following statements are true: 1. If m<AOB=m<COD, then m<AOC=m<BOD. 2.If m<AOC=m<BOD, then m<AOB=m<COD.

Vertical angles theorem

If two angles are vertical angles, then they are congruent.

Congruent supplements theorem

if two angles are supplementary to the same angle (or to congruent angles), then they are congruent

midpoint theorem

if M is the midpoint of AB, then AM is congruent to MB

angle bisector theorem

if a point is on the bisector of an angle, then it is equidistant from the other two sides of the angle

Definition of a Transversal

a line, ray, or segment that intersects two or more coplanar lines, rays, or segments, each at a different point.

Corresponding angles postulate

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

Alternate Interior Angles Theorem

If two lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.

Alternate Exterior Angles Theorem

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

Same-side interior angles theorem

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

Converse of the corresponding angles postulate

if two lines cut by a transversal in such a way that the corresponding angles are congruent, then the two lines are parallel.

Converse of the same-side interior angles theorem

if two lines cut by a transversal in such a way that the same-side interior angles are supplementary, the angles are parellel.

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 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.

Parallel Perpendicular Theorem

if two lines are perpendicular to the same line, then the two lines are parallel

Parallel Parallel Theorem

if two lines are parallel to the same then, then the two lines are parallel to each other

Triangle Sum Theorem

angles add up to 180°

Polygon Sum Theorem

the sum of the measures of the interior angles of a triangle is n-2(180) where n=sides

Equiangular Polygon

polygon with all angles equal in measure

Equilateral Polygon

polygon with all sides equal in length

regular polygon

a convex polygon in which all sides and angles are congruent

Triangle Midsegment

segment formed by connecting the midpoints of 2 sides of a triangle

Triangle Midsegment Theorem

A midsegment of a triangle is parallel to the side of a triangle, and its length is half the length of that side.

Trapezoid Midsegment

endpoints are the midpoints of the 2 non-parallel sides of a trapezoid

Trapezoid Midsegment Theorem

parallel to the 2 bases of trapezoid length is equal to the average of the measures of the 2 bases

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

Reflectional Symmetry

A figure that can be reflected over a line so its image matches the original figure.

Rotational Symmetry

If you can rotate (or turn) a figure around a center point by fewer than 360° and the figure appears unchanged, then the figure has rotation symmetry.

Incenter

The point where three angle bisectors of a triangle intersect

Inscribed Circle

Inside circle

circumcenter

The point where three perpendicular bisectors of a triangle intersect

Circumscribed Circle

Outside circle

Median

A segment or Ray that joins a vertex to the midpoint of the opposite side

altitude

What is the apothem called In a triangle?

Isosceles Triangle Theorem

If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent

Converse of the Isosceles Triangle Theorem

If 2 angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollary of the Isosceles triangle theorem

The bisector of the vertex of an isosceles triangle is the perpendicular of the base