Download 1.4-1.5 Angle Relations PPT

Geometry R/H
1.4 – Angle Measures
1.5 – Angle Relationships
Line Segments
A
B
• A line segment is part of a line containing two
endpoints and all points between them.
• Unlike lines, which extend forever in both
directions, line segments have a definite
beginning and end
• A line segment is named with the endpoints
AB or BA
Rays
B
A
• A ray is part of a line that consists of one endpoint and all
points of the line extending in one direction
• Name a ray using the endpoint first and then another point on
the ray
– When naming, make sure the arrow points away from the endpoint.
AB
, not BA
Rays, continued
B
AB
A
B
A
• Are these two rays the same?
No
• Different endpoints
• Extend in different directions
BA
Opposite Rays
J
K
L
• Are two collinear rays with the same endpoint
• Always form a line
KJ and KL are opposite rays
Parallel Lines
• Lines that
– Never intersect
– Extend in the same directions
– Coplanar
– Have the same slope
Skew lines
Lines that:
• Never intersect
• Are noncoplanar
• Extend in different directions
Parallel Planes
• Parallel planes are planes that never intersect
• A line and plane that never intersect are also
parallel
Learning Check and Summary
• Which of the following has no endpoints?
– Ray, Line Segment, Line
• Which of the following has two endpoints?
– Ray, Line Segment, Line
• Which of the following has one endpoint and
extends in one direction?
– Ray, Line Segment, Line
• Which of the following extend in the same direction?
– Parallel lines, skew lines
• Which of the following extend in different directions?
– Parallel lines, skew lines
Types of Angles
Vertex
Side
Naming Angles
• Angles are measured in degrees. The measure
of ÐA is written as mÐA .
• Angles with the same measure are congruent.
ÐABC
ÐB
Ð1
1
Angle Addition
• If point B is the interior
of ÐAOC , then
B
A
mAOB  mBOC  mAOC
• If ÐAOC is a straight
angle, then
O
C
B
mÐAOB + mÐBOC =180.
A
O
C
(on Word doc)
PROTRACTOR EXERCISE
What it is?
• Angle addition is adding (or subtracting) two
(or more) Angles
• If ∠ABC is a right angle
• m∠ABD is 50o
• find m∠DBC.
Variables (not scary!)
•
•
•
•
If ∠ABC is a right angle
∠ABD is (2x + 3)o
m∠DBC is (x + 6)o
solve for x
Example #1
• m∠1 = x
• m∠2 = 2x – 10
• m∠3 = 2x + 10
Find x and find m 3
Solve for x
B
A
C
D
1
O
mÐAOC = 7x - 2,mÐAOB = 2x + 8,mÐBOC = 3x +14
2
mÐAOB = 4x - 2,mÐBOC = 5x +10,mÐCOD = 2x +14
3
mÐAOB = 28,mÐBOC = 3x - 2,mÐAOD = 6x
4
mÐAOB = 4x + 3,mÐBOC = 7x,mÐAOD =16x -1
Bisecting an Angle
• An angle bisector is a ray that divides an angle
into two adjacent angles that are congruent.
• Ray FH Bisects Angle GFI because it divides the
angle into two congruent angles.
•
In the book, matching congruence arcs identify congruent angles in
diagrams.
G
F
H
I
Which angles are adjacent?
*Think: What does adjacent mean?
<1 & <2, <2 & <3, <3 & <4,
<4 & <1
Then what do we call <1 & <3?
2
1
3
4
Vertical Angles – 2 angles
that share a common vertex &
whose sides form 2 pairs of
opposite rays. Vertical Angles
are congruent.
<1 & <3,
<2 & <4
Linear Pair (of angles)
• 2 adjacent angles whose non-common sides
are opposite rays.
• The sum of a linear pair = 180 degrees
Ex 1:
• Vertical angles?
• Adjacent angles?
• Linear pair?
2
(2 adjacent angles=180 degrees)
1
• Adjacent angles not a linear pair?
5
3
4
Ex 2:
• If m<5=130o, find
m<3
m<6
m<4
4
5
3
6
Ex 3:
• Find the
values of
x&y
&
m<ABE
m<ABD
m<DBC
m<EBC
A
E
B
D
C
Complementary Angles
• 2 angles whose sum is 90o
35o
1
2
<1 & <2 are complementary
<A & <B are complementary
55o
A
B
Supplementary Angles
• 2 angles whose sum is 180o
<1 & <2 are
supplementary.
<X & <Y are
supplementary.
130o
X
50o
Y
Ex 4: <A & <B are supplementary. m<A is 5
times m<B. Find m<A & m<B.
m<A + m<B = 180o
m<A = 5(m<B)
Now substitute!
5(m<B) + m<B = 180o
6(m<B)=180o
m<B=30o
m<A=150o