Download Ch. 9 GEOMETRY Remember the “Cartesian Coordinate System

Ch. 9 GEOMETRY
Remember the “Cartesian Coordinate System”?
That represented the “xy-plane in space”.
A ray is like “half a line”. It starts at
one point (called the endpoint) and
goes on to infinity in one direction.
The other point tells you which
direction the line goes. The ray
below is the ray DB.
A line is infinitely long and is
denoted by two points that lie on it.
The line below can be denoted
AB or BA.
y- axis
3
Point B
2
1
Point A
x- axis
0
-4
-3
-2
-1
0
-1
1
origin
(0,0)
2
3
4
-2
Point D
-3
Point C
-4
All the points on the line connecting points C and D (that is all the points in
between C and D and including C and D) are called a line segment. Points C and
D are the endpoints of the line segment.The line segment is denoted: CD or DC
Angles are formed by two rays, lines, or line segments with a common
endpoint. The common endpoint is called the vertex. In the angle
below, point A is the vertex. The angle can be denoted as
BAC or
CAB
Note that the vertex, point A, must be in the middle.
B
vertex
The unit of measurement of an angle is degrees.
A
C
Right angle
90°
Acute angle
Less than 90°
Straight angle
Obtuse angle
180°
Greater than 90°
B
C
Intersection Point
A
E
D
Intersecting lines
form angles at their
intersection point.
Angles that share a
common side are
called adjacent
angles. The angles
that are not
adjacent are called
vertical angles.
Example 4 on p.562
EAB and BAC are adjacent angles.
BAC and CAD are adjacent angles.
EAD and BAC are vertical angles.
(3x+15)°
(4x-20)°
The vertical angles property allows us to make an equation:
3x+15 = 4x -20 and solve for x.
Vertical angles have equal measures of degrees.
We call this being “congruent.”
Two angles are supplementary angles when the sum of their measures is 180°.
When two lines intersect, adjacent angles are supplementary because the sides that
are not in common form a straight angle.
EAD and
DAC are supplementary angles.
DAC = 180°
Two angles are complementary angles
when the sum of their measures is 90°.
When two adjacent angles form a right
angle with the sides that are not in common,
these angles are complementary because the
measures add up to 90°.
B
C
90°
A
EAD +
D
BAC and
CAD are complementary angles.
Parallel lines are lines in the same plane that never intersect
(that have the same slope). If two lines, l1 and l2 are parallel,
we say l1 || l2.
Transversal – a line that intersects two or more lines on
the same plane.
A
Alternate Interior Angle Property
A transversal intersects parallel lines at
congruent angles.
Because of this and also because of the property
of supplementary angles and the property of
vertical angles, we can rewrite the diagram on
the left as this:
B
C
D
E
F
G
H
A
The angles on the inside of the parallel
lines that are congruent are called
“Alternate Interior Angles.”
F
C=
F are alternate interior angles.
C and
D=
D and
E
E are alternate interior angles.
180° - A
180° - A
A
A
180° - A
180° - A
A
Example 5 p. 570
Given that l1|| l2, solve for x
l1
(3x+20)°
l2
(3x-80)°
(3x-80)°
l1
(3x+20)°
l2
These two angles are not equal, but
we can still solve for x by using
other properties.
Since these two lines are parallel,
the transveral intersects them at
congruent angles, so the angle
adjacent to (3x+20)° is (3x-80)°.
(3x-80)°
The supplementary angle property
says that if two adjacent angles
form a straight angle with
uncommon sides, they are
supplementary, so
(3x – 80) + (3x + 20) = 180.
Now that we have an equation, we
can solve for x.