Chapter 1-2 Points, Lines, and Planes

Undefined Terms:

A point has no size but is often represented

by a dot and usually named by a capital letter.

. A

A line extends in two directions without ending.

Lines can be named by using two points on the

line or a single lowercase letter.

A B m

A plane is a flat surface with no edges but is

often represented by a four sided figure. A plane

is named by a capital letter.

R

Definitions:

Collinear points are points all in one line.

Coplanar points are points all in one plane.

.

. .

The intersection of two figures is the set of

points that are in both figures.

Chapter 1–3 Segments

Definitions:

Segment AC consists of points A and C and all

the points between A and C.

A C

The length of AC (written AC) is the distance

between A and C.

Ray ED consists of segment ED and all other

points P such that D is between E and P.

E D

Segments that have the same length are called

congruent segments, written as AB CD.

A B C D

The midpoint of a segment is the point that

divides the segment into 2 congruent segments.

A M B

A bisector of a segment is a line, ray, or plane

that intersects the segment at its midpoint.

A B

A statement that is accepted without proof is

called a Postulate.

Segment Addition Postulate:

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

A B C

Example: If B is between A and C, with

AB = 2x, BC = x + 10, and AC = 37, write and

solve an equation for x. Is B the midpoint of

AC?

Chapter 1–4 Angles

An angle is a figure formed by two rays that

have the same endpoint. A

B

C

The two rays are called sides and the common

endpoint is called the vertex.

Angles can be named, ABC, CBA, or B.

Angles will be measured in degrees, written as

m ABC = 40°

Angle Names:

Acute angle has measure between 0° & 90°

Right angle has measure 90°

Obtuse angle has measure between 90° & 180°

Straight angle has measure 180°

Congruent angles have equal measure.

Adjacent angles have a common endpoint and a

common side but no common interior points.

The bisector of an angle is a ray that divides the

angle into two congruent adjacent angles.

Angle Addition Postulate:

If point B is interior of AOC, then

m AOB + m BOC = m AOC

A

O B

C

Example: Find x, if m AOB = 2x + 16,

m BOC = 3x – 1, and m AOC = 60°. Does

OB bisect AOC ?

Chapter 2–4 Special Angle Pairs

Complementary angles are two angles whose

measures sum to 90.

Supplementary angles are two angles whose

measures sum to 180.

Vertical angles are two angles such that the

sides of one angle are opposites rays to the sides

of the other angle.

1 2

Vertical Angle Theorem:

Vertical angles are congruent.

1 2

Example: Write and solve an equation for x.

2x

3x + 5

Example: Write and solve an equation for x.

2x – 15 3x – 5

Example: Write and solve an equation for x.

2(x + 4) 4(x – 5)

Chapter 3–1 Parallel Lines

Parallel lines are coplanar lines that do not

intersect, m // n.

m

n

Perpendicular lines are coplanar lines that

intersect to form four right angles, m n.

m

n

Skew lines are non-coplanar lines.

n

m

A transversal, t , is a line that intersects two or

more coplanar lines in different points.

m

t

n

m n t

Alternate interior angles are nonadjacent

interior angles on opposite sides of the

transversal.

m

t

n

m n t

Same-side interior angles are two interior

angles on the same side of the transversal.

m

t

n

m n t

Corresponding angles are two angles in

corresponding positions relative to the two lines.

m

t

n

m n t

Chapter 3–2 Properties of Parallel Lines

Corresponding Angle Postulate

If two parallel lines are cut by a transversal, then

corresponding angles are congruent.

1

2

1 2

Alternate Interior Angle Theorem

If two parallel lines are cut by a transversal, then

alternate interior angles are congruent.

1

2

1 2

Same Side Interior Angle Theorem

If two parallel lines are cut by a transversal, then

same side interior angles are supplementary.

1

2

m 1 + m 2 = 180°

Example: Solve for x and y. Explain your

reasoning.

120 2x

3y + 6

Chapter 3–4 Angles of a Triangle

A triangle is the figure formed by three

segments joining three noncollinear points. Each

point is called a vertex. The segments are the

sides of the triangle, ∆ABC.

A

B C

Names of Triangles:

Scalene triangle has no congruent sides.

Isosceles triangle has two congruent sides.

Equilateral triangle has three congruent sides.

Acute triangle has three acute angles.

Obtuse triangle has one obtuse angle.

Right triangle has one right angle.

Equiangular triangle has all angles congruent.

Triangle Sum Theorem:

The sum of the measures of the interior angles

of a triangle is 180º.

x

y z

x + y + z = 180°

Exterior Angle Theorem:

The measure of an exterior angle of a triangle

equals the sum of the measures of its remote

interior angles.

x

y z

z = x + y

Chapter 3–5 Angles of a Polygon

A polygon is a figure formed by n coplanar

segments such that each segment intersects

exactly two other segments at the endpoints.

quadrilateral pentagon hexagon octagon

n = 4 n = 5 n = 6 n = 8

Find the sum of the interior angles of each

polygon.

number of 4 5 6 8

sides

interior angle

sum

Polygon Interior Sum Theorem:

The sum of the measures of the interior angles

of a polygon with n sides is (n – 2)180°.

Interior Sum = (n – 2)180°

Polygon Exterior Sum Theorem:

The sum of the measures of the exterior angles

of a polygon, one at each vertex, is 360°.

Exterior Sum = 360°

A regular polygon has all interior angles

congruent and all sides congruent.

Interior Angle Measure:

An interior angle of a regular polygon with n

sides is (n – 2)180° divided by n.

Interior Angle = –

Exterior Angle Measure:

An exterior angle of a regular polygon with n

sides is 360° divided by n

Exterior Angle =

Example: A polygon has 10 sides, find its

interior angle sum and its exterior angle sum.

Example: Find the measure of each interior and

each exterior angle of a regular polygon with 9

sides.

Chapter 10-1 Constructions

A Straightedge is used to construct lines or

parts of a line. A Compass is used to construct

circles or parts of a circle.

Construction 1: Congruent Segments

Given segment AB construct congruent segment

CD:

Construction Steps

A B

Construction 2: Congruent Angles

Given angle ABC construct congruent angle

XYZ:

Construction Steps

A

B C

Construction 3: Angle Bisector

Given angle ABC construct angle bisector BZ:

Construction Steps

A

B C

Construction 4: Segment Bisector

Given segment AB construct segment bisector

CD:

Construction Steps

A B