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Chapter 1
Foundations of Geometry: Points, Lines, and Planes
Objectives(Goals)
 Identify and model points, lines, and planes.
 Identify collinear and coplanar points and
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intersecting lines and planes in space.
Use algebra to compute segment measures
Find the distance between two points
Find the midpoint of a segment.
Measure and classify angles.
Identify and use congruent angles and the
bisector of an angle.
Identify and use special pairs of angles.
Find the measures of angles.
Undefined Terms: most basic figures in
geometry that can’t be defined using other
figures
Term
Point
Line
Plane
Facts
Names a location
and has no shape
or size.
A straight path that
has no thickness
and extends
forever.
A flat surface that
has no thickness
and extends
forever.
Diagram
Name
Collinear: points that lie on the same line
Collinear:
Noncollinear:
Coplanar: points that lie in the same plane
Coplanar:
Noncoplanar:
Postulate (axiom): a statement that is accepted as true without proof.
P-1-1: Through any two points there is exactly one line.
P-1-2: Through any three noncollinear points there is exactly one plane
containing them.
P-1-3: If two points lie in a plane, then the line containing those points lie in the
plane.
P-1-4: If two lines intersect, then they intersect in exactly one point.
P-1-5: If two planes intersect, then they intersect in exactly one line.
Name a line that contains point T.
Name three collinear points.
Name three noncollinear points.
Name the intersection of line n and line m.
What is another name for line n ?
Name the intersection of plane R and plane P.
Name three points that are nonplanar.
What is another name for plane P ?
Line Segment: part of a line consisting of two
endpoints and all the points between
them.
Diagram:
Name:
Fact: line segments can be measured.
Ruler Postulate: The points on a line can be put into a one-one to
correspondence with the real numbers.
Betweenness of Points
Point D is between points C and E if and only if C, D, and E
are collinear and CD + DE = CE.
Segment Addition Postulate: If B is between A and C, then
AB + BC = AC
Example 1 of Using Segment Addition Postulate
B is between A and C, AC =14 and BC =11.4. Find AB.
Example 2 of Using the Segment Addition Postulate
S is between R and T. Find the value of x and RT if RS = 2x+7,
ST = 28, and RT = 4x
Congruent Segments: segments that have the same
length(measure).
Symbol:
 “is congruent to”
Tick marks (slashes) on figures also indicate congruence.
Distance Between Two Points
 Number Line
The distance between any two
points is the absolute value of the
distance of the coordinates.
 Coordinate Plane
The distance d between two
points with coordinates (x1, y1)
and (x2, y2) is
d  ( x2  x1 ) 2  ( y2  y1 ) 2
Example:
CD =
AC =
CF =
Example: (5, 1) and (-3, -3)
Midpoint of Segment
The point halfway between two endpoints. A midpoint is the point that bisects
(divides) into two congruent segments.
If K is the midpoint of AB , then AK = KB.
 Number Line
The coordinate of the midpoint
of a segment is the mean of the
endpoints.
.
 Coordinate Plane
The coordinates of the midpoint
of a segment whose endpoints
(x1, y1) and (x2, y2) are
 x1  x2 y1  y2 
,


2 
 2
Find the midpoint of (-12, 9)
and (7, 4).
Finding the coordinates of a missing endpoint given
the coordinates of one endpoint and the midpoint.
Given that S is the midpoint of RT . T(-4, 3) and
S(-1, 5).
Using Algebra to Find Measures
Example 1: E is the midpoint of DF, DE = 2x + 4,
and EF = 3x – 1. Find DE, EF, and DF.
Example 2: Q bisects PR , PQ = 3y, and PR = 42.
Find y and QR.
Rays and Angles
Ray: part of a line that starts at an endpoint and
extends forever in one direction.
Diagram:
Name:
Opposite Rays: two rays that have a common
endpoint and form a line.
Angle: figure formed by two rays that have a common
endpoint called the vertex.
Diagram:
Name:
4 Types of Angles
1. Acute
2. Right
3. Obtuse
4. Straight
Congruent Angles: angles that have the same measure.
Diagram:
Name:
Arc Marks are used to indicate that two angles are congruent.
Angle Addition Postulate
If S is in the interior of PQR, then the
mPQS  mSQR  mPQR.
Example:
Find the mCAB
Angle Bisector: a ray that divides an angle into two
congruent angles.
Diagram:
Finding the measure of an Angle
BD bisects ABC , mABD  6 x  3, and mDBC  8x  7. Find mABD.
Angle Relationships
5 types of special angle pairs
1.
Adjacent Angles: two angles that have a common vertex, and
a common side, but no common interior
points.
2.
Vertical Angles: two nonadjacent angles formed by two
intersecting lines. Vertical Angles are
congruent.
3. Linear Pair: a pair of adjacent angles whose noncommon sides are
opposite angles. In others, adjacent angles that form a
line.
4. Complementary Angles: two angles whose measures have a sum
of 90 degrees.
5. Supplementary Angles: two angles who measures have a sum of
180 degrees.
Angle Problems
1. ABD and BDE are supplementary. mABD  3x  12
and mBDE  7 x  32. Find the measure of each angle.
2. ABD and BDC are complementary. mABD  5 y  1 and
mBDC  3 y  7.
Find the measures of both angles.
3. Solve for x. Then find the measure of each angle.
4. Find the measure of each angle.
5. Solve for x and y. Find the measure of each angle.
6. mBGC  35. Find the measure of each
remaining angle.