Download Chapter 1 - Catawba County Schools

Basics of Geometry
Chapter 1
1.2 Points, Lines, and Planes

Three undefined terms in Geometry:

Point: No size, no shape, only LOCATION.


Line: No thickness, only DIRECTION.


Named by a single Capital letter
Named by two points on the line.
Plane: No thickness, flat surface, infinate in ALL
DIRECTIONS.

Named by any three or more points on the plane.
Vocabulary

Collinear points – Three or more points that
lie on the same line.

Coplanar points – Four or more points that lie
on the same plane.

Segment – Part of a line that consists of two
endpoints and all points in between.

Name by the two endpoints in any order.
More Vocabulary

Ray – Part of a line that consists of one
endpoint and all points in one direction.

Named by the endpoint and a point on the ray (in that
order)

Opposite rays – two rays that share an endpoint and
form a straight line.
Examples

Describe what each of these symbols means:

PQ

PQ

PQ

QP
Examples

True or False?




Point A lies on line l.
Point B lies on line l.
A, B, and C are collinear.
A, B, and C are coplanar.
B
l
A
E
D
C
m
1.3 Segments and Their
Measures

The distance between points A and B, written
as AB, is the absolute value of the difference
between the coordinates of A and B.

AB is also called the length of AB.
A
AB
AB = l x2 – x1 l
B
Segments and Their Lengths

Another way to find the length of a segment is by
using the distance formula:
d  ( x 2  x1 )  ( y2  y1 )
2
2
Examples

Given A(3, 2) and B(-2, -1), find AB.
Terminology

Definitions: statements that are known facts
(do not have to be proven true)

Postulates (Axioms): Statements that are
accepted as true (do not have to be proven)

Theorems: Statements that MUST be proven
true.
Segment Addition Postulate

If B is between A and C, then AB + BC = AC.
Likewise, if AB + BC = AC, then B is between A
and C.
AC
A
B
AB
C
BC
Examples

If AB = 5, AC = 13, and BD = 15, what is AD?
Examples

Suppose M is between L and N, find x and
find the lengths of the segments LM and MN.



LM = 3x + 8
MN = 2x + 5
LN = 23
Examples

Use the distance formula to decide whether JK  KL



J (3, -5)
K (-1, 2)
L (-5, -5)
1.4 Angles and Their Measures



Angle – Consists of two different rays that
have the same endpoint.
Sides of the angle are the rays
Vertex of the angle is the common endpoint.

An angle that has sides AB and AC is denoted by
BAC, CAB, or A, if point A is the vertex.
C
A
B
Angle Addition Postulate

If P is in the interior of RST, then
mRSP  mPST  mRST
Angle Measures

MEASURES are EQUAL
mBAC  mDEF

ANGLES are CONGRUENT
BAC  DEF
Classifying Angles

Acute: 0  mA  90

Right: mA  90

Obtuse: 90  mA  180

Straight: mA  180
Examples

What is the measure of
ABC?
1.5 Segment and Angle
Bisectors


Midpoint of a segment is the point that
divides, or bisects, the segment into 2
congruent segments. (A midpoint divides it in
half.)
Segment bisector is a segment, line, ray, or
plane that intersects a segment at its
midpoint.
C
M
A
B
D
Midpoint Formula

If A(x1, y1) and B(x2, y2) are points in a
coordinate plane, then the midpoint of AB
has coordinates:
 x1  x 2 y1  y2 
,


2 
 2
Example

If AB has endpoints at (-2, 3) and (5, -2),
what is the midpoint?
Example

If the midpoint of RP is M(2, 4), and one
endpoint is R(-1, 7), what is the coordinate for
P?
Angle Bisector

An angle bisector is a ray that divides an
angle into 2 congruent, adjacent angles. (The
angle bisector divides the angle in half.)

Adjacent angles: share a common vertex and
side, but have no common interior points.
A
C
D
B
Example

The ray RQ bisects PRS. The measures of
the two congruent angles are (x + 40) and
(3x – 20) . Solve for x.
P
R
Q
S
1.6 Angle Pairs

Two angles are vertical angles if their sides
form two pairs of opposite rays.



1&3
2&4
1
4
2
3
Two adjacent angles are a linear pair if their
noncommon sides are opposite rays.

5&6
5
6
Example

Solve for x and y. Then find the angle
measures.
Vocabulary

Complementary Angles are two angles that
the sum of their measures is 90 .

Supplementary Angles are two angles that
the sum of their measures is 180 .
Examples






Are
Are
Are
Are
Are
Are
5 and
5 and
5 and
5 and
5 and
9 and
6 a linear pair?
9 a linear pair?
8 a linear pair?
8 vertical angles?
7 vertical angles?
6 vertical angles?
Always, Sometimes, or Never






If m 1 = 40 , then m 2 = 140 .
If m 4 = 130 , then m 2 = 50 .
1 and 4 are congruent.
m 2+m 3=m 1+m 4
~
2= 1
m 2 = 90 - m 3.