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Math in Our World
Section 10.1
Points, Lines, Planes, and Angles
Learning Objectives
 Write names for angles.
 Use complementary and supplementary angles
to find angle measure.
 Use vertical angles to find angle measure.
 Find measures of angles formed by a
transversal.
Points, Lines, and Planes
The most basic geometric figures we will
study are points, lines, and planes.
A point is easiest thought of as a location,
like a particular spot on this page. We
represent points with dots, but in actuality
a point has no length, width, or thickness.
(We call it dimensionless.)
Points, Lines, and Planes
A line is a set of connected points that has
an infinite length, but no width. We draw
representations of lines, but again, in
actuality a line cannot be seen because it
has no thickness. We will assume that lines
are straight, meaning that they follow the
shortest path between any two points on the
line. This means that only two points are
needed to describe an entire line.
Points, Lines, and Planes
A plane is a two-dimensional flat surface
that is infinite in length and width, but has
no thickness. You might find it helpful to
think of a plane as an infinitely thin piece
of paper that extends infinitely far in each
direction.
Points, Lines, and Planes
Points and lines can be used to make
other geometric figures.
A line segment is a finite portion of a line
consisting of two distinct points, called
endpoints, and all of the points on a line
between them.
Points, Lines, and Planes
Any point on a line separates the line into
two halves, which we call half lines. A
half-line beginning at point A and
continuing through point B means that A
is the endpoint of the half-line, which is
not included.
When the endpoint of a half line is
included, we call the resulting figure a ray.
Points, Lines, and Planes
Below is a summary of the figures described on the
last few slides showing how each is symbolized.
Angles
An angle is a figure formed by two rays
with a common endpoint. The rays are
called the sides of the angle, and the
endpoint is called the vertex.
The symbol for angle is , and
there are a number of ways to
name angles. The angle shown
could be called ABC, CBA,
B, or 1.
Angles
You should only use a single letter to
denote an angle if there’s no question as to
the angle represented.
In this angle, B is
ambiguous, because
there are three different
angles with vertex at
point B.
EXAMPLE 1
Naming Angles
Name the angle shown here in four different ways.
SOLUTION
RST, TSR, S, and 3.
Measuring Angles
One way to measure an angle is in
degrees, symbolized by °. One degree is
defined to be 1/360 of a complete rotation.
Measuring Angles
The instrument that is used to measure an
angle is called a protractor.
The center of the base of the protractor is
placed at the vertex of the angle, and the
bottom of the protractor is placed on one
side of the angle.
The angle measure
m ABC  40
is marked where
the other side
falls on the scale.
Classifying Angles
Angles can be classified by their measures.
An acute angle has a
measure between 0 and 90.
A right angle has a
measure of 90.
Classifying Angles
Angles can be classified by their measures.
An obtuse angle has a
measure between 90 and 180.
A straight angle has a
measure of 180.
Pairs of Angles
Pairs of angles have various names
depending on how they are related.
Two angles are called adjacent angles if
they have a common vertex and a common
side.
Adjacent angles, ABC
and CBD are shown. The
common vertex is B and the
common side is the ray BC.
Pairs of Angles
Pairs of angles have various names
depending on how they are related.
Two angles are said to be complementary
if the sum of their measures is 90°.
Shown are two complementary
angles. The sum of the
measures of GHI and IHJ
is 90°.
Pairs of Angles
Pairs of angles have various names
depending on how they are related.
Two angles are said to be supplementary if
the sum of their measures is equal to
180°. are two supplementary
Shown
angles. The sum of the
measures of WXY and YXZ
is 180°.
EXAMPLE 2
Using Complementary Angles
If FEG and GED are complementary and
mFEG is 28°, find mGED.
SOLUTION
Since the two angles are
complementary, the sum of their
measures is 90°, so mGED +
mFEG = 90°, and solving we
get:
The complement of an angle with measure 28° has measure
62°.
EXAMPLE 3
Using Supplementary Angles
If RQS and SQP are supplementary and
mRQS = 135°, find mSQP.
EXAMPLE 3
Using Supplementary Angles
SOLUTION
Since the two angles are supplementary, the sum of their
measures is 180°, so mSQP + mRQS = 180°, and
solving we get:
The supplement of an angle with measure 135° has
measure 45°.
EXAMPLE 4
Using Supplementary Angles
If two adjacent angles are supplementary and one
angle is 3 times as large as the other, find the
measure of each.
SOLUTION
Let x = the measure of the smaller angle. The larger is
three times as big, so 3x = the measure of the larger. The
angles are supplementary, so their measures add to 180°.
This gives us an equation:
The smaller angle has
measure 45°, and the larger
has measure 3 x 45°, or
135°.
Pairs of Angles
Just as two intersecting streets have four
corners at which you can cross, when two
lines intersect, four angles are formed, as we
see in the figure. Angles 1 and 3 are called
vertical angles, as are angles 2 and 4.
It’s easy to believe
from the diagram that
two vertical angles have
the same measure.
EXAMPLE 5
Using Vertical Angles
Find m2, m3, and m4 when m1 = 40°.
SOLUTION
Since 1 and 3 are vertical angles
and m1 = 40, m3 = 40°.
Since 1 and 2 form a straight
angle (180°), m1 + m2 = 180°,
and
Since 2 and 4 are vertical angles, m4 = 140°.
Parallel Lines
Two lines in the same plane are called
parallel if they never intersect. You might
find it helpful to think of them as lines that
go in the exact same direction. If two lines
l1 and l2 are parallel, we write l1 || l2.
When two parallel lines are intersected by
a third line, we call the third line a
transversal.
Parallel Lines
Shown are two parallel lines cut by a transversal.
The angles between the parallel lines (angles 3–6)
are called interior angles, and the ones outside
the parallel lines (angles 1, 2, 7, and 8) are called
exterior angles.
Parallel Lines
Shown are two parallel lines cut by a transversal.
Alternate interior angles are the angles formed
between two parallel lines on the opposite sides of
the transversal that intersects the two lines.
Alternate interior angles have equal measures.
Parallel Lines
Shown are two parallel lines cut by a transversal.
Alternate exterior angles are the opposite
exterior angles formed by the transversal that
intersects two parallel lines. Alternate exterior
angles have equal measures.
Parallel Lines
Shown are two parallel lines cut by a transversal.
Corresponding angles consist of one exterior and
one interior angle with no common vertex on the same
side of the transversal that intersects two parallel lines.
Corresponding angles have equal measures.
EXAMPLE 6
Finding Angles Formed by a
Transversal
Find the measures of all the angles shown when
the measure of 2 is 50°.
EXAMPLE 6
Finding Angles Formed by a
Transversal
SOLUTION
First, let’s identify the angles that have the same measure
as 2. They are 3 (vertical angles), 6 (corresponding
angles), and 7 (alternate exterior angles). Mark all of these
as 50° on the diagram.
EXAMPLE 6
Finding Angles Formed by a
Transversal
SOLUTION
Since 1 & 2 are supplementary, m1 = 180° – 50° =
130°.
This allows us to find the remaining angles: 4 is a vertical
angle
1, so it has measure 130° as well.
Now 8with
is acorresponding
130
°
angle with 4, and 5 is an
alternate interior angle with
130
4, which means they both
°
130
°
have measure 130° as
well.
130
°