Transcript Slide 1

Segments and Angles
Chapter 2
Chapter Objectives
• Analyze segment bisectors and angle
bisectors
• Identify complementary angles,
supplementary angles, vertical angles, and
linear pairs.
• Use the properties of equality and
congruence to justify mathematical
statements.
Chapter Goals
• Introduction to special angles and their
properties.
Chapter Content
• 2.1 Segment Bisectors
• 2.2 Angle Bisectors
• 2.3 Complementary and Supplementary
Angles
• 2.4 Vertical Angles
• 2.6 Properties of Equality and Congruence
2.1 Segment Bisectors
Objectives:
• Bisect a segment.
• Find the coordinates of the
midpoint of a segment.
Key Vocabulary
• Midpoint
• Segment bisector
• Bisect
Midpoint of a Segment
• The midpoint of a segment is the
point halfway between the
endpoints of the segment. If X is
the midpoint of AB, then AX = XB.
• The midpoint divides a segment
into two congruent segments.
Midpoint
• The point that bisects a segment.
• Bisects?
Divides the segment into 2 congruent parts.
A
M
12x+3
12x+3=10x+5
2x=2
x=1
B
10x+5
AM=12x+3=12(1)+3=15
MB=10x+5=10(1)+5=15
If you know the length of AB, multiply
AB by ½ to find AM and MB.
Example 1
M is the midpoint of AB. Find AM and MB.
SOLUTION
M is the midpoint of AB, so AM and MB are each half the
length of AB.
1
1
·
AB
AM = MB =
= · 26 =13
2
2
ANSWER
AM = 13 and MB = 13.
Example 2
P is the midpoint of RS. Find PS and
RS.
SOLUTION
P is the midpoint of RS, so PS = RP. Therefore, PS = 7.
You know that RS is twice the length of RP.
RS = 2 · RP =
ANSWER
2 · 7 = 14
PS = 7 and RS = 14.
Your Turn:
1. Find DE and EF.
ANSWER
DE = 9; EF = 9
2. Find NP and MP.
ANSWER
NP = 11; MP = 22
Segment Bisector
• Definition: Any segment, line, or plane that
intersects a segment at its midpoint is
called a segment bisector.
Example - Segment Bisector
• A segment, ray, line, or plane that intersects a
segment at its midpoint.
• Line 𝓀 is a segment bisector of AB.
k
A
M
B
Example 3
Line l is a segment bisector of AB. Find the value of x.
SOLUTION
Line l bisects AB at point M.
AM = MB
5x = 35
5x
5
=
Substitute 5x for AM and 35 for MB.
35
Divide each side by 5.
5
Simplify.
x=7
CHECK
Check your solution by substituting 7 for x.
5x = 5(7) = 35
Example:
Finish these
Statements:
T
S
E
E is the Mid. Pt. of SO.
If TV bisects SO, then ___________________
SE  EO
If E is the Mid. Pt. of SO, then ____________
O
V
Midpoint of a Segment
• If the segment is on a coordinate plane,
we must use the midpoint formula for
coordinate planes which states given a
segment with endpoints (x1, y1) and
(x2, y2) the midpoint is…
 x1  x2 y1  y2 
M 
,

2 
 2
The Midpoint Formula
• The coordinates of the midpoint of a
segment are the averages of the
x-coordinates and the y-coordinates of
the endpoints.
• The midpoint of the segment joining
A(x1, y1) and B(x2, y2) is
 x1  x2 y1  y2 
M
,

2 
 2
Example 4
Find the coordinates of the midpoint of AB.
a. A(1, 2), B(7, 4)
b. A(–2, 3), B(5, –1)
SOLUTION
First make a sketch. Then use the Midpoint Formula.
a.
M=
=
Let (x1, y1) = (1, 2)
and (x2, y2) = (7, 4).
x1 + x2
2
1+7
2
= (4, 3)
,
,
y1 + y2
2
2+4
2
b.
x1 + x2
M=
2
–2 + 5
=
Let (x1, y1) = (–2, 3)
and (x2, y2) = (5, –1).
=
2
3
2
,1
,
,
y1 + y2
2
3 +(– 1)
2
Your Turn:
Sketch PQ. Then find the coordinates of its midpoint.
1. P(2, 5), Q(4, 3)
ANSWER
2. P(0, –2), Q(4, 0)
ANSWER
3. P(–1, 2), Q(–4, 1)
ANSWER
(3, 4)
(2, –1)
–
5
2
,
3
2
More About Midpoints
• You can also find the coordinates of an
endpoint of a segment if you know the
coordinates of the other endpoint and its
midpoint.
Example 5: The midpoint of AB is M(2,4). One
endpoint is A(-1,7). Find the coordinates of B.
 x1  x2 y1  y2 
,

  (midpoint)
2 
 2
y1  y2
4
2
x1  x2
2
2
1  x2
2
2
7  y2
4
2
1  x2  4
7  y2  8
y2  1
x2  5
 5,1
Example 6:
Find the coordinates of D if E(–6, 4) is the midpoint
of
and F has coordinates (–5, –3).
Let F be
in the Midpoint Formula.
Write two equations to find the coordinates of D.
Example 6:
Solve each equation.
Multiply each side by 2.
Add 5 to each side.
Multiply each side by 2.
Add 3 to each side.
Answer: The coordinates of D are (–7, 11).
Your Turn:
Find the coordinates of R if N(8, –3) is the midpoint
of
and S has coordinates (–1, 5).
Answer: (17, –11)
Example 7:
What is the measure of
if Q is the midpoint of
?
Solution:
Definition of midpoint
Distributive Property
Subtract 1 from each side.
Add 3x to each side.
Divide each side by 10.
Solution:
Now substitute
for x in the expression for PR.
Original measure
Simplify.
Your Turn:
What is the measure of
Answer: 3
if B is the midpoint of
?
Joke Time
• What lies at the bottom of the ocean
and twitches?
• A nervous wreck.
• What kind of coffee was served on
the Titanic?
• Sanka.
• And what kind of lettuce?
• Iceberg.
Assignment:
• Section 2.1, pg. 56-59: 1-35 odd, 42, 4551 all