Geometry R/H 1.4 – Angle Measures 1.5 – Angle Relationships.
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Transcript of Geometry R/H 1.4 – Angle Measures 1.5 – Angle Relationships.
Geometry R/H1.4 – Angle Measures
1.5 – Angle Relationships
Line Segments
• 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
A B
AB or BA
Rays
• 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.
A
B
AB , not BA
Rays, continued
• Are these two rays the same?
A
B
A
B
No• Different endpoints• Extend in different directions
BA
AB
Opposite Rays
• Are two collinear rays with the same endpoint• Always form a line
JK
L
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
Side
Naming Angles
• Angles are measured in degrees. The measure of is written as .
• Angles with the same measure are congruent.
1
Angle Addition
• If point B is the interior of , then
• If is a straight angle, then
AOCmBOCmAOBm
A
A
B
B
O
OC
C
PROTRACTOR EXERCISE(on Word doc)
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 Find x and find m 3• m∠2 = 2x – 10• m∠3 = 2x + 10
Solve for x
A
B
C
D
O1
2
3
4
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.
F
G
H
I
Which angles are adjacent? *Think: What does adjacent mean?
1 32
4
<1 & <2, <2 & <3, <3 & <4, <4 & <1
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
Then what do we call <1 & <3?
Linear Pair (of angles)• 2 adjacent angles whose non-common sides
are opposite rays.• The sum of a linear pair = 180 degrees
1 2
Ex 1: • Vertical angles?• Adjacent angles?• Linear pair? (2 adjacent angles=180 degrees)
• Adjacent angles not a linear pair?
1 3
2
5 4
Ex 2:
• If m<5=130o, findm<3m<6m<4
5 3
4
6
Ex 3:• Find the
values of x & y
&m<ABEm<ABDm<DBCm<EBC
3x+5o y+20ox+15o 4y-15o
A
B
C
D
E
Complementary Angles
• 2 angles whose sum is 90o
1
2
35o
A
55o
B<1 & <2 are complementary
<A & <B are complementary
Supplementary Angles
• 2 angles whose sum is 180o1
2
130o 50o
X Y
<1 & <2 are supplementary.
<X & <Y are supplementary.
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