83 terms

congruent polygons

polygons in which all sides and all angles are congruent

corollary

A statement that follows immediately from a theorem

hypotenuse

The longest side of a right triangle, opposite the right angle

base angles of an isosceles triangle

The 2 non-vertex angles of an isosceles triangle, formed by the legs and base

base of an isosceles triangle

the side opposite the vertex angle, the non-congruent side

legs of an isosceles triangle

the 2 congruent sides of an isosceles triangle

legs of a right triangle

the 2 shortest legs of a right triangle, adjacent to the right angle

vertex angle of an isosceles triangle

the angle formed by the 2 congruent legs of an isosceles triangle

3rd Angles Theorem

If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are also congruent

SSS Postulate

If the 3 sides of 1 triangle are congruent to the 3 sides of another triangle, then the triangles are congruent (SSS)

SAS Postulate

If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent (SAS)

AAS Theorem

If 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle, then the triangles are congruent. (AAS)

ASA Postulate

If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent (ASA )

Isosceles triangle theorem

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

Converse of isosceles triangle theorem

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

Hypotenuse Leg Theorem

If the hypotenuse and a leg of 1 right triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent.

CPCTC

Corresponding Parts of Congruent Triangles are

bisector of isosceles vertex is Perpendicular bisector of base

If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector the base

If a triangle is equilateral... (Corollary 1)

If a triangle is equilateral, then it is also equiangular

Corollary 2 (An equilateral triangle has...)

An equilateral triangle has 3 60 degrees angles

Corollary 3 (The bisector of the vertex of an isosceles triangle...)

The bisector of the vertex of an isosceles triangle is perpendicular to the base at its midpoint.

An equiangular triangle is...

An equiangular triangle is also equilateral.

Median

A segment from a vertex to the midpoint of the opposite sides

Altitude

The perpendicular segment from a vertex to the line that contains the opposite side

Where do some altitudes fall in a right triangle?

2 of the 3 altitudes fall on the same line as the legs.

Where do altitudes fall in an obtuse triangle?

2 of the 3 altitudes fall outside the triangle

Perpendicular bisector

In a given plane, there is exactly one line perpendicular to a segment at its midpoint.

Theorem 4-5 (If a point lies on the perpendicular bisector...)

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

Theorem 4-6 (If a point t from the endpoints of a segment...)

If a point is equidistant from the endpoints of a segment, then the point lies in on the perpendicular bisector of a segment

Right angles theorem

All right angles are congruent

Theorem 4-7 (If a point lies on the bisector of an angle...)

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

Theorem 4-8 (If a point is equidistant from the sides of angle....)

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

Equilateral Triangle

A triangle with three congruent sides

Equiangular Triangle

A triangle with 3 congruent angles

Centriod

Point of concurrency formed by the medians of a triangle

Orthocenter

Point of concurrency formed by the altitudes of a triangle

The orthocenter of a right triangle

Falls on the vertex of the right angle

Circumcenter

point of concurrency of the perpendicular bisectors of the three sides of a triangle

The distance from a point to a line

The length of the perpendicular segment from the point to the line

incenter

Point of concurrency for the angle bisectors of a triangle

CA Angles

Perpendicular lines form congruent adjacent angles

shared lines...

shared lines are congruent

line Perpendicular to a plane

a line that intersects the plane in a point and is perpendicular to every line in the plane the passes through the line of intersection

A line and a plane are perpendicular

A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection

Measure of one interior angle of a polygon equation

(n-2)180/n

Corollary 4 (Acute angles of a right triangle)

The acute angles of a right triangle are complementary.

Sum of interior angles equations

(n-2)180

Sum of exterior angles of a polygon

360

Measure of one exterior angle equation

360/n

Number of diagonals equation

n(n-3)/2

Posutlate ten (corr, parralel)

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

third plane theorem

If two parallel planes are cut by a third plane, then the lines of intersection are parallel

S-S int theorem

If two parallel lines are cut by a transversal, then same-side interior angles are supplementary

2 lines perpendicular theorem

In a plane two lines perpendicular to the same line are parallel

Theorems 3.8 (outside ll)

Through a point outside a line there is exactly one line parallel to the given line

Theorem 3.9 (outside perpendicular)

Through a point outside a line, there is exactly one line perpendicular to the given line

Theorem 3.12 (Measure of exterior angle of a triangle)

The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

Remote interior angle

An interior angle that is not adjacent to the exterior angle

Exterior angle of a triangle

equals sum of opposite interior angles

Theorem 3.13 (angles of convex polygon)

The sum of the measures of the angles of a convex polygon with n sides is (n-2)180

Scalene triangle

Zero congruent sides

Isosceles triagnle

2 or more equal sides

Equilateral

All sides are equal

What polygon has 3 sides?

Triangle

A polygon with 4 sides

Quadrilateral

A polygon with 5 sides

Pentagon

A polygon with 6 sides

Hexagon

Polygon with 7 sides

Heptagon

A polygon with 8 sides

Octogon

polygon with 9 sides

nonagon

A polygon with 10 sides

Dodecagon

A polygon with 12 sides

dodecagon

Regular polygon is...

Equilateral and equiangular

Ruler Postulate

The points on a line can be put into a one-to-one correspondence with the real numbers.

Protractor Postulate

the measurement of an angle between two rays can be designated as a unique number, and this number would be between 0 and 180 degrees

Theorem 2.8 (complement of congruent angles)

If two angles are complements of congruent angles, then the two angles are congruent

Addition Postulate

If B is between A and C, then AB + BC is = AC

Angle Addition Postulate

the measure of the larger angle is the sum of the measures of the two smaller ones.

Postulate 6 (through any two points...)

Through any two points there is exactly one line

A line contains at least...

A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.

Midpoint theorem

If M is the midpoint of AB, then AM=1/2AB and MB=1/2AB

Angle bisector Theorem

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

Skew lines

If lines are skew, then they are not coplanar and they do not intersect