182 terms

Space

the set of all points

Collinear points

points all in one line

Coplanar points

points all in one plane

Intersection

the set of points that are in both figures

Postulates

Statements that are accepted without proof

Congruent

Two objects that have the same size and shape

Congruent segments

segments that have equal lengths

midpoint of a segment

the point that divides the segment into two congruent segments

bisector of a segment

a line, segment, ray, or plane that intersects the segment at its midpoint

angle

the figure formed by two rays that have the same endpoint

sides (angles)

The two rays of an angle

vertex

the common endpoint of an angle

Congruent angles

angles that have equal measures

Adjacent angles

two angles in a plane that have a common vertex and a common side but no common interior points.

bisector of an angle

is the ray that divides the angle into two congruent adjacent angles

theorems

Important statements that are proved

Segment Addition Postulate

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

Angle Addition Postulate

If point 𝐵 lies in the interior of ∠𝐴𝑂𝐶, then 𝑚∠𝐴𝑂𝐵 + 𝑚∠𝐵𝑂𝐶 = 𝑚∠𝐴𝑂𝐶

line

contains at least two points

plane

contains at least three points not all in one line

space

contains at least four points not all in one plane

Through any two points

there is exactly one line

Through any three points

there is at least one plane

through any three noncollinear points

there is exactly one plane

If two points are in a plane

then the line that contains the points is in that plane

If two planes intersect

then their intersection is a line

If two lines intersect (point)

then they intersect in exactly one point

Through a line and a point not in the line

there is exactly one plane

If two lines intersect (plane)

then exactly one plane contains the lines

Three undefinable terms

point, line, plane

Conditional Statements

If p, then q (q if p) (p implies q)

Converse

If q, then p

counterexample

An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false

biconditional

If a conditional and its converse are both true, they can be combined into a single statement by using the words "if and only if." A statement that contains the words "if and only if"

Deductive Reasoning

proving statements by reasoning from accepted postulates, definitions, theorems, and given information

Complementary angles

two angles whose measures have the sum 90°.

Supplementary angles

two angles whose measures have the sum 180°.

Vertical angles

two angles such that the sides of one angle are opposite rays to the sides of the other angle

Perpendicular lines

two lines that intersect to form right angles

APOE

If 𝑎 = 𝑏 and 𝑐 = 𝑑, then 𝑎 + 𝑐 = 𝑏 + 𝑑.

SPOE

If 𝑎 = 𝑏 and 𝑐 = 𝑑, then 𝑎 − 𝑐 = 𝑏 − 𝑑.

MPOE

If 𝑎 = 𝑏, then 𝑐𝑎 = 𝑐𝑏.

DPOE

If 𝑎 = 𝑏 and 𝑐 ≠ 0, then a/c = b/c

Subst

If 𝑎 = 𝑏, then either 𝑎 or 𝑏 may be substituted for the other in any equation (or inequality).

RPOE

𝑎 = 𝑎

SyPOE

If 𝑎 = 𝑏, then 𝑏 = 𝑎.

TPOE

If 𝑎 = 𝑏 and 𝑏 = 𝑐, then 𝑎 = 𝑐

Dist.

𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐

RPOC

𝐷𝐸 ≅ 𝐷𝐸

∠𝐷 ≅ ∠𝐷

∠𝐷 ≅ ∠𝐷

SyPOC

If 𝐷𝐸 ≅ 𝐹𝐺, then 𝐹𝐺 ≅ 𝐷𝐸

If ∠𝐷 ≅ ∠𝐸, then ∠𝐸 ≅ ∠𝐷

If ∠𝐷 ≅ ∠𝐸, then ∠𝐸 ≅ ∠𝐷

TPOC

If 𝐷𝐸 ≅ 𝐹𝐺 and 𝐹𝐺 ≅ 𝐽𝐾, then 𝐷𝐸 ≅ 𝐽𝐾.

If ∠𝐷 ≅ ∠𝐸 and ∠𝐸 ≅ ∠𝐹, then ∠𝐷 ≅ ∠𝐹

If ∠𝐷 ≅ ∠𝐸 and ∠𝐸 ≅ ∠𝐹, then ∠𝐷 ≅ ∠𝐹

Mdpt Thm

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

Angle Bisector Thm

If ray BX is the bisector of ∠𝐴𝐵𝐶, then 𝑚∠𝐴𝐵𝑋 = 1 /2 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝑋𝐵𝐶 = 1 /2

𝑚∠𝐴𝐵𝐶

𝑚∠𝐴𝐵𝐶

Vertical Angle Thm

Vertical angles are congruent

If two lines are perpendicular,

then they form congruent adjacent angles

If two lines form congruent adjacent angles,

then the lines are perpendicular

If the exterior sides of two adjacent acute angles are perpendicular,

then the angles are complementary

If two angles are supplements of congruent angles (or of the same angle),

then the two angles are congruent.

If two angles are complements of congruent angles (or of the same angle),

then the two angles are congruent.

Skew lines

noncoplanar lines that are neither parallel nor intersecting

Parallel lines

coplanar lines that do not intersect

transversal

a line that intersects two or more coplanar lines in different points

Alternate interior angles

two nonadjacent interior angles on opposite sides of the transversal

Same-side interior angles

two interior angles on the same side of the transversal

Corresponding angles

two angles in corresponding positions relative to the two lines

If two parallel lines are cut by a transversal, (corresponding angles)

then corresponding angles are congruent

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

then the lines are parallel

If two parallel planes are cut by a third plane,

then the lines of intersection are parallel.

If two parallel lines are cut by a transversal,

then alternate interior angles are congruent.

If two parallel lines are cut by a transversal,

then same-side interior angles are supplementary

If a transversal is perpendicular to one of two parallel lines,

then it is perpendicular to the other one also

If two lines are cut by a transversal and alternate interior angles are congruent,

then the lines are parallel

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

then the lines are parallel

In a plane, two lines perpendicular to the same line

are parallel.

Through a point outside a line, (parallel)

there is exactly one line parallel to the given line

Through a point outside a line (Perpendicular)

there is exactly one line perpendicular to the given line.

Two lines parallel to a third line

are parallel to each other

triangle

the figure formed by three segments joining three noncollinear points

Vertex

Each of the three points of the triangle

sides (triangle)

The segments of the triangle

Scalene triangle

no congruent sides of triangle

Isosceles triangle

at least two congruent sides of triangle

Equilateral triangle

all sides congruent of triangle

acute triangle

three acute angles in triangle

obtuse triangle

one obtuse angle in triangle

right triangle

one right angle in triangle

Equiangular triangle

triangle where all angles are congruent

auxiliary line

a line (or ray or segment) added to a diagram to help in a proof

corollary

a statement that can be proved easily by applying the theorem

remote interior angles

The two interior angles of the triangle that are nonadjacent to the exterior angle

exterior angle

Formed when one side of a triangle is extended

convex polygon

a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon

Quadrilateral

Polygon with 4 sides

Pentagon

Polygon with 5 sides

Hexagon

Polygon with 6 sides

Heptagon

Polygon with 7 sides

Octagon

Polygon with 8 sides

Nonagon

Polygon with 9 sides

Decagon

Polygon with 10 sides

Undecagon

Polygon with 11 sides

Dodecagon

Polygon with 12 sides

n-gon

Polygon with n sides

diagonal

A segment joining two nonconsecutive vertices

regular polygons

Polygons that are both equiangular and equilateral

(n-2)180

Sum of the interior angles of a convex 𝑛-gon

(n-2)180/n

Measure of one interior angle of a regular 𝑛-gon

360

Sum of the exterior angles of a convex 𝑛-gon

360/n

Measure of one exterior angle of a regular 𝑛-gon

n-3

Number of diagonals you can draw from one vertex of a convex 𝑛-gon

n(n-3)/2

Total number of diagonals you can draw in a convex 𝑛-gon

The sum of the measures of the angles of a triangle

180°

If two angles of one triangle are congruent to two angles of another triangle,

then the third angles are congruent.

Each angle of an equiangular triangle has measure

60°

In a triangle, (type of angles and how many)

there can be at most one right angle or obtuse angle

The acute angles of a right triangle

are complementary

The measure of an exterior angle of a triangle equals

the sum of the measures of the two remote interior angles

The sum of the measures of the angles of a convex polygon with 𝑛 sides

(𝑛 − 2)180°

The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex

360°

All right angles are

congruent

Since congruent triangles have the same shape, their corresponding angles

are congruent

Since congruent triangles have the same shape, their corresponding sides

are congruent

Two triangles are congruent if and only if their vertices can be matched up so that

the corresponding parts (angles and sides) of the triangles are congruent

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

isosceles triangle

triangle with at least two congruent sides

All equilateral triangles are

isosceles

Some isosceles triangles are

equilateral

hypotenuse

In a right triangle, the side opposite the right angle

legs (triangle)

other two sides of a triangle that isnt hypotenuse

SSS

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

SAS

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

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

Isosceles Triangle Thm

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

An equilateral triangle is also

equilangular

An equilateral triangle has

three 60° angles

The bisector of the vertex angle of an isosceles triangle is

perpendicular to the base at its midpoint

If two angles of a triangle are congruent,

then the sides opposite those angles are congruent.

AAS

If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent

HL

If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent

median (triangle)

a segment from a vertex to the midpoint of the opposite side. Every triangle has 3. Always lie inside the triangle

altitude

the perpendicular segment from vertex to the line that contains the opposite side. In an acute triangle, they are all inside the triangle. In a right triangle, two of them are parts of the triangle. They are the legs of the right triangle. The third one is inside the triangle. In an obtuse tringle, two of them are outside the triangle

perpendicular bisector

a line (or ray or segment) that is perpendicular to the segment at its midpoint. There are three of them that can be drawn in any triangle.

angle bisector

a segment, ray, or line which bisects an angle of the triangle

concurrent

When two or more lines interesct in one point they are said to be this

point of concurrency

the point in which concurrent lines intersect

centroid

the point where the medians of a triangle meet and always lies inside the triangle

If a point lies on the perpendicular bisector of a segment,

then the point is equidistant from the endpoints of the segment.

If a point is equidistant from the endpoints of a segment,

then point lies of the perpendicular sector of the segment

If a point lies on the bisector of an angle,

then the point is equidistant from the sides of the angle

If a point is equidistant from the sides of an angle,

then the point lies on the bisector of the angle

The centroid of a triangle is two-thirds of the distance from

each vertex to the midpoint of the opposite side

parallelogram

a quadrilateral with both pairs of opposite sides parallel

rectangle

quadrilateral with four right angles

rhombus

quadrilateral with four congruent sides

square

quadrilateral that is both a rectangle and a rhombus

trapezoid

quadrilateral with exactly one pair of parallel sides

bases

parallel sides of a trapezoid

legs (trapezoid)

sides of trapezoid that arent the bases

isosceles trapezoid

trapezoid with congruent legs

median (trapezoid)

the segment that joins the midpoints of the legs of a trapezoid

opposite sides of a parallelogram are

congruent

opposite angles of a parallelogram are

congruent

diagonals of a parallelogram

bisect each other

consecutive angles of a parallelogram are

supplementary

diagonal of a parallelogram forms

two congruent triangles

If both pairs of opposite sides of a quadrilateral are parallel

then the quadrilateral is a paralleogram

If both pairs of opposite sides of a quadrilateral are congruent

then the quadrilateral is a paralleogram

If one pair of opposite sides of a quadrilateral is both congruent and parallel

then the quadrilateral is a parallelogram

If both pairs of opposite angles of a quadrilateral are congruent

then the quadrilateral is a parallelogram

If the diagonals of a quadrilateral bisect each other

then the quadrilateral is a parallelogram

If two lines are parallel

then all points on one line are equidistant from the other line

If three parallel lines cut off congruent segments on one transversal

then they cut off congruent segments on every transversal

A line that contains the midpoint of one side of a triangle and is parallel to another side passes through

the midpoint of the third side

Midsegment

segment that joins the midpoints of two sides of a triangle

Midsegment is parallel to and half as long as

the third side

The diagonals of a rectangle are

congruent

The diagonals of a rhombus are

perpendicular

Each diagonal of a rhombus bisects

two angles of the rhombus

The midpoint of the hypotenuse of a right triangle

is equidistant from the three vertices

If an angle of a parallelogram is a right angle

then the parallelogram is a rectangle

If two consecutive sides of a parallelogram are congruent

then the parallelogram is a rhombus

Base angles of an isosceles trapezoid are

congruent

Median of trapezoid is

parallel to the bases and has a length equal to the average of the base lengths