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1.2 Points,
Lines, &
Planes
1
• Point – indicates a location and has no size.
–Name it with a capital letter, such as A.
• Line –a straight path that extends in opposite
directions without end and has no thickness.
–Contains infinitely many points.
–Name a line by any two points on the line, or
by a single lowercase letter.
• Plane –a flat surface that extends without end
and has no thickness.
–Contains infinitely many lines.
–Name a plane by capital letter or by at least
three points in the plane that aren’t on same
line.
•Collinear points – points
that lie on the same line.
•Coplanar points – points
that lie in the same plane.
–All the points of a line are coplanar.
Naming Points, Lines, and Planes
1) What are two other
ways to name 𝑨𝑪?
2) What are two ways to
name Plane S?
3) What are the names
of three collinear
points?
4) What are the names
of four coplanar points?
• Segment – part of line that consists of two
endpoints and all points between them.
–Name a segment by its two endpoints.
• Ray – segment extended infinitely in one
direction from single endpoint.
–Name a ray by its endpoint and another
point on the ray.
• Opposite rays – two rays that share the
same endpoint and form a line.
–Can name opposite rays by their shared
endpoint and any other point on each
ray.
Name Segments and Rays 5) What are
the names of
the segments
in the figure?
6) What are the
names of the rays in
the figure?
7) Which of the rays in the previous
question are opposite rays?
8) What is the
intersection of
plane RST and
plane STW?
• Postulate 1-4
• Through any three
noncollinear points, there
is exactly one plane.
9) The intersection of two lines is a
.
10) The intersection of two planes is a
.
11) Any three points can define a
.
12) Any two points can define a
.
Warm Up Problems
1. Name the
intersection of plane
AEH and plane GHE.
2. What plane contains
points B, F, and C?
3. What plane contains
points E, F, and D?
1.3
Measuring
Segments
11
• Postulate 1.5 – Ruler Postulate
–Every point on a line can be paired
with a real number.
–The real number that corresponds
to a point is called the coordinate of
the point.
–Allows you to measure lengths of
segments and will allow you to find
the distances between points.
Measuring Segment Lengths
1) What is 𝐴𝐵?
2) What is 𝐴𝐶?
3) What is 𝐷𝐸?
4) What is 𝐵𝐷?
Postulate 1.6 Segment Addition
Postulate
• If three points A, B, and C are
collinear and B is between A
and C, then 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶.
Using the Segment Addition Postulate
5) If 𝐴𝐶 ,38 = what are 𝐴𝐵 and 𝐵𝐶?
6) Find 𝑅𝑇
and 𝑅𝑆.
Congruent Segments
If two segments have the same
length, then the segments are
congruent segments.
Comparing Segment Lengths
7)
8)
• The midpoint of a segment is a point
that divides the segment into two
congruent segments.
• A point, line, ray, or other segment
that intersects a segment at its
midpoint is said to bisect the segment.
–The point, line, ray, or segment is
called a segment bisector.
Using the Midpoint
9) In the figure, B is the midpoint
of 𝑨𝑪. Find the length of 𝑨𝑪, 𝑨𝑩,
& 𝑩𝑪.
1.7 Midpoint and Distance in the
Coordinate Plane
20
Formulas for Midpoint
Number Line
Coordinate
Plane
Find the midpoint with
the given endpoints.
1)-9 and 6
2) -7 and -14
Find the midpoint of 𝑨𝑩.
3) A (-1, 3) and B (-9, 2)
4) A (-4, -5) and B (-1, 1)
5) Given the midpoint of 𝑿𝒀 is
(-3, 4) and point Y is located at
(-12, 0), find point X.
Distance Formula
The distance between two
points A(x1 , y1) and B(x1 , y1) is
Find the distance between
each pair of points. If
necessary, round to the nearest
hundredth.
6) G (3, 0) and Y (0, 12)
7) M (-3, 5) and S (2, -9)
8) T (4, -3) and W (11, -3)
1.4 Measuring
Angles
27
An angle is formed by
two rays with the
same endpoint.
The rays are the sides
of the angle and the
endpoint is the vertex
of the angle.
Ways to name an
angle:
• The vertex
• A point on each
ray & the vertex
(the vertex
must go in the
middle)
• A number
1) Name angle 1 and
angle 2 in two other
ways.
The interior of an angle is the region
containing all of the points between the
two sides of the angle.
The exterior of an angle is the region
containing all of the points outside of
an angle.
Postulate 1.7 Protractor Postulate
• allows you to find the measure of an angle.
• The measure of the angle is the absolute value of the
difference of the real numbers paired with the sides
of the angle.
Types of Angles
2) Given the figure, find the measure
of angle APB, angle BPC, and angle
APC.
Congruent Angles
Angles are congruent if they have
the same angle measure.
iff reads
“if and
only if”
3) Name the congruent
angles in the figure.
4) If the measure
of angle QPW is
55, what is the
measure of angle
PWV?
Angle
Addition
Postulate
5) Assume angle ABC is a straight line,
find the measure of angle ABD & DBC.
6) Given that angle
ABD = 120, find the
measure of angle
ABC and angle CBD.
1.5 Exploring
Angle Pairs
38
Adjacent angles – two
coplanar angles with a
common side, a
common vertex, and
no common interior
points.
Vertical angles – two
angles whose sides are
opposite rays; vertical
angles are congruent to
one another.
Complementary angles –
two angles whose measures
have a sum of 90°; each
angle is called the
complement of the other.
Supplementary angles – two angles whose
measures have a sum of 180°; each angle is
called the supplement of the other.
2) Find the
value of x,
angle CEA,
and angle
DEA.
1) Name a pair of adjacent, vertical,
complementary, and supplementary
angles in the given figure.
Linear pair- a pair of adjacent angles whose
noncommon sides are opposite rays; angles
of a linear pair form a straight angle.
Postulate 1.9 Linear Pair Postulate
If two angles form a linear pair, then they
are supplementary.
Angle bisector- a ray that divides an
angle into two congruent angles; its
endpoint is at the angle vertex.
Within the ray, a segment
with the same endpoint
is also an angle bisector;
the ray or segment
bisects the angle.
3) Given the figure, find q
and then find the measure
of angle ABD, angle CBD.
4) BD bisects angle CBA.
Find the measure of
angle CBA