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