G Points Lines Planes Angles

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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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Points, Lines, Planes,
&
Angles
October 4, 2011
www.njctl.org
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Table of Contents
Points, Lines, & Planes
Line Segments
Pythagorean Theorem
Distance between points
Midpoint formula
Locus & Constructions
Angles & Angle Relationships
Angle Addition Postulate
Angle Bisectors & Constructions
Click on the topic to
go to that section
Points, Lines, & Planes
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of Contents
Definitions
An "undefined term" is a word or term that does not require
further explanation. There are three undefined terms in
geometry:
Points - A point has no dimensions (length, width,
height), it is usually represent by a capital letter and a dot on a
page. It shows position only.
Lines - composed of an unlimited number of points
along a straight path. A line has no width or height and
extends infinitely in opposite directions.
Planes - a flat surface that extends indefinitely in twodimensions. A plane has no thickness.
Points & Lines
A television picture is composed of many dots placed closely
together. But if you look very closely, you will see the spaces.
A
B
However, in geometry, a line is composed of an
unlimited/infinite number of points. There are no spaces
between the point that make a line. You can always find a point
between any two other points.
The line above would b called line
or line
Line
,
, or
all refer to the same line
Line a
Points are labeled with letters. (Points A, B, or C)
Lines are named by using any two points OR by using a single
lower-cased letter. Arrowheads show the line continues without
end in opposite directions.
Line
, , or ,
all refer to the same line
Line a
Collinear Points - Points D, E, and F above are called collinear
points, meaning they all lie on the same line.
Points A, B, and C are NOT collinear point since they do not lie
on the same (one) line.
Postulate: Any two points are always collinear.
Example
Give six different names for the line that contains points U, V,
and W.
Answer
(click)
Postulate: two lines intersect at exactly one point.
If two non-parallel intersect they do so at only one point.
and
intersect at K.
Example
a. Name three points that are collinear
b. Name three sets of points that are noncollinear
c. What is the intersection of the two lines?
Answer
a. A, D, C
b. A,B,D / A,C,B / C,D,B (others)
Rays are also portions of a line.
or
is read ray AB.
Rays start at an initial point, here endpoint A, and
continues infinitely in one direction.
Ray
has a different initial point, endpoint B, and
continues infinitely in the direction marked.
Rays
and
are NOT the same. They have different
initial points and extend in different directions.
Suppose point C is between points A and B
Rays
and
are opposite rays.
Opposite rays are two rays with a common endpoint that point
in opposite directions and form a straight line.
Recall: Since A, B, and C all lie on the same line, we know they
are collinear points.
Similarly, segments and rays are called collinear, if they lie on the
same line. Segments, rays, and lines are also called coplanar if
they all lie on the same plane.
Example
Name a point that is
collinear
with the given points.
a. R and P
b. M and Q
c. S and N
d. O and P
Example
Name two opposite rays on the given line
e.
f.
g.
h
1
is the same as
.
True
False
Hint
Read the notation carefully. Are they asking about lines, line
segments, or rays?
2
is the same as
True
False
3
Line p contains just three points
True
False
Answer
Remember that even though only three points are marked, a
line is composed of an infinite number of points. You can
always find another point in between two other points.
4
Points D, H, and E are collinear.
True
False
5
Points G, D, and H are collinear.
True
False
6
Ray LJ and ray JL are opposite rays.
Yes
No
Explain your answer?
7
Which of the following are opposite rays?
A
and
B
and
C
and
D
and
8
Name the initial point of
A
J
B
K
C
L
9
Name the initial point of
A
J
B
K
C
L
Are the three points collinear? If they are, name the line
they lie on.
a
L, K, J
b
N, I, M
c
M, N, K
d
P, M, I
Planes
Collinear points are points that are on the same line.
F,G, and H are three collinear points.
J,G, and K are three collinear points.
J,G, and H are three non-collinear
points.
F, G, H, and I are coplanar.
F, G, H, and J are also coplanar, but
the plane is not drawn.
Coplanar points are points that lie on the same plane.
F,G, and H are coplanar in addition to being collinear.
G, I, and K are non-coplanar and non-collinear.
Any three noncollinear points can name a plane.
Planes can be named by any three noncollinear points:
- plane KMN, plane LKM, or plane KNL
- or, by a single letter such as Plane R
(all name the same plane)
Coplanar points are points that lie on the same plane:
- Points K, M, and L are coplanar
- Points O, K, and L are non-coplanar in the diagram
However, you could draw a plane to contain any three points
A
B
Postulate:
If two planes intersect,
they intersect along
exactly one line.
The intersection of the two planes above is shown by line
As another example, picture the intersections of the four walls in
a room with the ceiling or the floor. You can imagine a line laying
along the intersections of these planes.
Postulate: Through any three noncollinear points there is
exactly one plane.
Example
Name the following points:
A point not in plane HIE
A point not in plane GIE
Two points in both planes
Two points not on
10
Line BC does not contain point R. Are points R,
B, and C collinear?
Yes
No
11
Plane LMN does not contain point P. Are points
P, M, and N coplanar?
Yes
No
Hint: what do we know about any three points?
12
Plane QRS contains
. Are points Q, R, S, and
V coplanar? (Draw a picture)
Yes
No
13
Plane JKL does not contain
L, and N coplanar?
Yes
No
. Are points J, K,
14
and
A
D
B
C
C
A
D
B
intersect at
15
Which group of points are noncoplanar with
points A,B, and F.
A
E, F, B, A
B
A, C, G, E
C
D, H, G, C
D
F, E, G, H
16
Are lines
and
coplanar?
Yes
No
Answer
17
Plane ABC and plane DCG intersect at ?
A
C
B
line DC
C
Line CG
D
they don't intersect
Answer
18
Planes ABC, GCD, and EGC intersect at ?
A
line
B
point C
C
point A
D
line
Answer
19
Name another point that is in the same plane as
points E, G, and H
A
B
B
C
C
D
D
F
Answer
20
Name a point that is coplanar with points E, F,
and C
A
H
B
B
C
D
D
A
Answer
21
Intersecting lines are __________ coplanar.
A
Always
B
Sometimes
C
Never
22
Two planes ____________ intersect at exactly one
point.
A
Always
B
Sometimes
C
Never
23
A plane can __________ be drawn so that any
three points are coplaner
A
Always
B
Sometimes
C
Never
24
A plane containing two points of a line
__________ contains the entire line.
A
Always
B
Sometimes
C
Never
25
Four points are ____________ noncoplanar.
A
Always
B
Sometimes
C
Never
26
Two lines ________________ meet at more
than one point.
A
Always
B
Sometimes
C
Never
Look what happens if I place line
y directly on top of line x.
Hint
Line Segments
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of Contents
Line Segments
or
Line segments are portions of a line.
or
endpoint
endpoint
is read segment AB..
Line Segment
segment.
or
are different names for the same
It consists of the endpoints A and B and all the points on the line
between them.
Ruler Postulate
On a number line, every point can be paired with a
number and every number can be paired with a point.
Coordinates indicate the point's position on the number line.
The symbol AF stands for the length of
. This distance
from A to F can be found by subtracting the two coordinates
and taking the absolute value.
A
B
C
D
E
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
2 3 4
A
coordinate
Distance
AF = |-8 - 6| = 14
F
5 6 7 8 9 10
F
coordinate
X
Why did we take the Absolute Value when calculating
distance?
In our previous slide, we were seeking the distance between
two points.
Distance is a physical quantity that can be measured distances cannot be negative.
When you take the absolute value between two numbers, the
order in which you subtract the two numbers
does not matter
Definition: Congruence
Equal in size and shape. Two objects are congruent
if they have the same dimensions and shape.
Roughly, 'congruent' means 'equal', but it has a precise
meaning that you should understand completely when you
consider complex shapes.
Line Segments are congruent if they have the same
length. Congruent lines can be at any angle or
orientation on the plane; they do not need to be parallel.
Read as:
"The line segment DE is
congruent to line segment HI."
Definition: Parallel Lines
Lines are parallel if they lie in the same plane, and are the
same distance apart over their entire length. That is, they
do not intersect.
Example
Find the measure of each segment in centimeters.
cm
a.
CE =
b.
AB =
27
Find a segment that is 4 cm long
cm
A
DA
B
BD
C
DE
D
CD
28
Find a segment that is 6.5 cm long
cm
A
DA
B
BE
C
DE
D
CD
29
Find a segment that is 3.5 cm long
cm
A
AC
B
BE
C
BD
D
DC
30
Find a segment that is 2 cm long
cm
A
DE
B
CA
C
BD
D
DC
31
Find a segment that is 5.5 cm long
cm
A
CE
B
EB
C
DB
D
DA
32
If point F was placed at 3.5 cm on the ruler,
how from point E would it be?
cm
A
5 cm
B
4 cm
C
3.5 cm
D
4.5 cm
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
Or, said another way,
If AB + BC = AC, then B is between A and C.
AB
BC
AC
Simply said, if you take one part of a segment (AB),
and add it to another part of the segment (BC), you
get the entire segment.
The whole is equal to the sum of its parts.
Example
The segment addition postulate works for
three or more segments if all the segments lie
on the same line (i.e. all the points are
collinear).
In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6
Find CD and BE
Start by filling in the information you are given
AE
AB
BC
CD DE
In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6
27
||
6
Can you finish the rest?
||
5
CD =
BE =
Example
K, M, and P are collinear with P between K and M
PM = 2x + 4, MK = 14x - 56, and PK = x + 17
Solve for x
1) Draw a diagram and insert the information
given into the diagram
2) From the segment addition postulate, we know that
KP + PM = MK (the parts equal the whole)
3) (x + 17) + (2x + 4) = 14x - 56
3x + 21 = 14x - 56
+ 56
+ 56
3x + 77 = 14x
-3x
- 3x
77 = 11x
7=x
Example
P, B, L, and M are collinear and are in the following order:
a) P is between B and M
b) L is between M and P
Draw a diagram and solve for x, given:
ML = 3x +16, PL = 2x +11, BM = 3x +140, and PB = 3x + 13
1) First, arrange the points in order and draw a diagram
a) BPM
b) BPLM
2) Segment addition postulate gives
3x+13 + 2x+11 + 3x+16 = 3x+140
3) Combine like terms and isolate/solve for the variable x
8x + 40 = 3x + 140
5x + 40 = 140
5x = 100
x = 20
For next group of questions (#31-36), we are given the
following information about the collinear points:
AE = 20
BD = 6
AB = BC = CD
Hint: always start these problems by placing the information
you have into the diagram.
33
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is AB, BC, and CD?
34
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is DE?
35
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is CA?
36
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is CE?
37
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is DA?
38
We are given the following information about
the collinear points:
AE = 20
BD = 6
AB = BC = CD
What is BE?
39
X, B, and Y are collinear points, with Y between
B and X. Draw a diagram and solve for x,
given:
BX = 6x + 151
XY = 15x - 7
BY = x - 12
40
Q, X, and R are collinear points, with X
between R and Q. Draw a diagram and solve
for x, given:
XQ = 15x + 10
RQ = 2x + 131
XR = 7x +1
41
B, K, and V are collinear points, with K
between V and B. Draw a diagram and solve
for x, given:
KB = 5x
BV = 15x + 125
KV = 4x +149
The Pythagorean Theorem
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Pythagorean Theorem
Pythagoras was a philosopher, theologian,
scientist and mathematician born on the
island of Samos in ancient Greece and
lived from c. 570–c. 495 BC.
The Pythagorean Theorem
2
2
c
b
2
c =a +b
a
states that in a right triangle the area of the
square on the hypotenuse (the side
opposite the right angle) is equal to the
sum of the areas of the squares of the
other two sides.
Proof
Proof
Using the Pythagorean Theorem
In the Pythagorean Theorem, c always stands for the longest
side. In a right triangle, the longest side is called the
hypotenuse. The hypotenuse is the side opposite the right
angle.
2
2
2
2
c =a +b
5
a=?
25 = a + 9
-9
-9
2
16 = a
= a
4 = a
3
You will use the Pythagorean Theorem often.
Example
Answer
42
What is the length of side c?
(The longest side of a triangle is called the ?)
c2= a2 + b2
Answer
43
What is the length of side a?
Hint: Always determine which side is the hypotenuse first
44
What is the length of c?
B
45
What is the length of the missing side?
46
What is the length of side b?
47
What is the measure of x?
x
8
17
Pythagorean Triples
are three positive integers for side lengths that satisfy
2
2
2
a +b =c
( 3 , 4 , 5 ) ( 5, 12, 13) (6, 8, 10)
( 7, 24, 25)
( 8, 15, 17) ( 9, 40, 41) (10, 24, 26) (11, 60, 61)
(12, 35, 37) (13, 84, 85) etc.
There are many more.
Remembering some of these combinations may save you
some time
48
A triangle has sides 30, 40 , and 50, is it a right
triangle?
Yes
No
49
A triangle has sides 9, 15 , and 12, is it a right
triangle?
Yes
No
50
A triangle has sides √3, 2 , and √5, is it a right
triangle?
Yes
No
Distance between Points
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Computing the distance between two points in the plane is an
application of the Pythagorean Theorem for right triangles.
Computing distances between points in the plane is
equivalent to finding the length of the hypotenuse of a right
triangle.
Relationship between the
Pythagorean Theorem & Distance Formula
The Pythagorean Theorem states a
relationship among the sides of a
right triangle.
2
2
The distance formula
calculates the distance
using point's coordinates.
(x2, y2)
2
c =a +b
c
c
a
b
(x1, y1)
(x2, y1)
The Pythagorean Theorem is true for all right triangles. If we
know the lengths of two sides of a right triangle then we
know the length of the third side.
Distance
The distance between two points, whether on a line or in a
coordinate plane, is computed using the distance formula.
The Distance Formula
The distance 'd' between any two points with coordinates
(x1 ,y1) and (x2 ,y2) is given by the formula:
d=
Note: recall that all coordinates are (x-coordinate, y-coordinate).
Example
Calculate the distance
from Point K to Point I
(x2, y2)
(x1, y1)
Label the points - it does not matter
which one you label point 1 and point 2.
Your answer will be the same.
d=
Plug the coordinates into the
distance formula
KI =
KI =
=
=
51
Calculate the distance from Point J to Point K
A
B
C
D
52
Calculate the distance from H to K
A
B
C
D
53
Calculate the distance from Point G to Point K
A
B
C
D
54
Calculate the distance from Point I to Point H
A
B
C
D
55
Calculate the distance from Point G to Point H
A
B
C
D
Midpoint Formula
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Midpoint of a line segment
A number line can help you find the midpoint of a segment.
The midpoint of GH, marked by point M, is -1.
Here's how you calculate it using the endpoint coordinates.
Take the coordinates of the endpoint G and H, add them
together, and divide by two.
=
= -1
Midpoint Formula Theorem
The midpoint of a segment joining points with coordinates (x1, y1)
and (x2, y2) is the point with coordinates
Calculating Midpoints in a Cartesian Plane
Segment PQ contain the
points (2, 4) and (10, 6).
The midpoint M of
is
the point halfway between
P and Q.
Just as before, we find the
average of the coordinates.
Remember that points are written
with the x-coordinate first. (x, y)
(
,
)
The coordinates of M, the midpoint
of PQ, are (6, 5)
56
Find the midpoint coordinates (x,y) of the
segment connecting points A(1,2) and B(5,6)
A
(4, 3)
B
(3, 4)
C
(6, 8)
D
(2.5, 3)
Hint: Always label the points coordinates first
57
Find the midpoint coordinates (x,y) of the segment
connecting the points A(-2,5) and B(4, -3)
A
(-1, -1)
B
(-3, -8)
C
(-8, -3)
D
(1, 1)
58
Find the coordinates of the midpoint (x, y) of the
segment with endpoints R(-4, 6) and Q(2, -8)
A
(-1, 1)
B
(1, 1)
C
(-1, -1)
D
(1, -1)
59
Find the coordinates (x, y) of the midpoint of the
segment with endpoints B(-1, 3) and C(-7, 9)
A
(-3, 3)
B
(6, -4)
C
(-4, 6)
D
(4, 6)
60
Find the midpoint (x, y) of the line segment
between A(-1, 3) and B(2,2)
A
(3/2, 5/2)
B
(1/2, 5/2)
C
(1/2, 3)
D
(3, 1/2)
Example: Finding the coordinates of an
endpoint of an segment
Use the midpoint formula to
write equations using x and y.
61
Find the other endpoint of the segment with
the endpoint (7,2) and midpoint (3,0)
A
(-1, -2)
B
(-2, -1)
C
(4, 2)
D
(2, 4)
62
Find the other endpoint of the segment with
the endpoint (1, 4) and (5, -2)
A
(1, 3)
B
(3, 1)
C
(3, -3)
D
(-3, 3)
Locus
&
Constructions
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Introduction to Locus
Definition:
Locus is a set of points that all satisfy a certain condition, in
this course, it is the set of points that are the same distance
from something else.
Theorem: locus between two points
All points on the perpendicular bisector of a line segment
connecting two points are equidistant from the two points.
Theorem: locus from a given line
All points equidistant from a given line is a parallel line.
Theorem: locus between two lines
The locus of points equidistant from two given parallel lines
is a parallel line midway between them.
locus: equidistant from two points
The locus of points equidistant from two points, A and B, is the
perpendicular bisector of the line segment determined by the
two points.
X
Y
The distance (d) from point A to the locus is equal to the
distance (d') from Point B to the locus. The set of all these
points forms the red line and is named the locus.
Point M is the midpoint of
. X is equidistant (d=d') from A
and B. Y lies on the locus; it is also equidistant from A and B.
Constructions
Dividing a line segment into x congruent segments.
Let us divide AB into 3 equal segments - we
could choose any number of segments.
1. From point A, draw
a line segment at an
angle to the given line,
and about the same
length. The exact
length is not
important.
2. Set the compass on A, and set its
width to a bit less than 1/3 of the
length of the new line.
Step the compass along the line,
marking off 3 arcs. Label the last C.
3. Set the compass width to CB.
4. Using the compass set to CB, draw an arc below A
5. With the compass width set to AC, draw an arc from B
intersecting the arc you just drew in step 4. Label this D.
6. Draw a line connecting B with D
7. Set the compass width back to AC and step along DB
making 4 new arcs across the line
8. Draw lines connecting the arc along AC and BD. These lines
intersect AB and divide it into 3 congruent segments.
63
Point C is on the locus between point A and
point B
True
False
64
Point C is on the locus between point A and
point B
True
False
65
How many points are equidistant from
the endpoints of
?
A
2
B
1
C
0
D
infinite
66
You can find the midpoint of a line segment by
A
measuring with a ruler
B
constructing the midpoint
C
finding the intersection of the locus and line segment
D
all of the above
67
The definition of locus
A
a straight line between two points
B
the midpoint of a segment
C
the set of all points equidistant from two other points
D
a set of points
Example: Construction
Divide the line segment into 3 congruent segments.
Angles
&
Angle Relationships
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Identifying Angles
An angle is formed by two rays with a common endpoint (vertex)
The angle shown can be called
When there is no chance
of confusion, the angle
may also be identified
by its vertex B.
The sides of
are CB and AB
,
, or
.
C
(Side)
32°
B
(Vertex)
(Side)
A
The measure of the angle is 32 degrees.
"The measure of
is equal to the measure of
..."
Two angles that have the same measure are congruent angles.
Exterior
Interior
We read this as
The single mark
through the arc
shows that the
angle measures
are equal
is congruent to
The area between the rays that form an angle is called the
interior. The exterior is the area outside the angle.
Angle Measures
Angles are measured in degrees, using a protractor.
Every angle has a measure from 0 to 180 degrees.
Angles can be drawn any size, the measure would still be the
D
same.
A
B
is a 23° degree angle
The measure of
is 23° degrees
C
is a 119° degree angle
The measure of
is 119° degrees
In
and
, notice that the vertex is
written in between the sides
L
M
Example
K
P
N
J
O
Challenge Questions
X
Angle Relationships
Once we know the measurements of angles, we can categorize
them into several groups of angles:
0° < acute < 90°
right = 90°
90° < obtuse < 180°
straight = 180°
180°
180° < reflex angle < 360°
Two lines or line segments that meet
at a right angle are said to be perpendicular.
Link
Complementary Angles
A pair of angles are called complementary angles if the sum
of their degree measurements equals 90 degrees. One of the
angles is said to be the complement of the other.
These two angles are complementary (58° + 32° = 90°)
+
=
We can rearrange the angles so they are adjacent, i.e. share a
common side and a vertex. Complementary angles do not have
to be adjacent.
If two adjacent angles are complementary, they form a right
angle
Supplementary Angles
Supplementary angles are pairs of angles whose
measurements sum to 180 degrees. Supplementary angles do
not have to be adjacent or on the same line; they can be
separated in space. One angle is said to be the supplement of
the other.
Definition: Adjacent Angles
are angles that have a common ray coming out of the vertex
going between two other rays. In other words, they are
angles that are side by side, or adjacent.
+
=
If the two supplementary angles are adjacent, having a
common vertex and sharing one side, their non-shared sides
form a line.
A linear pair of angles are two adjacent angles whose noncommon sides on the same line. A line could also be called a
straight angle with 180°
Example
Solution:
Choose a variable for the angle - I'll choose "x"
Example
Let x = the angle
Since the angles are complementary we know their sum
must equal 90 degrees.
90 = 2x + x
90 = 3x
30 = x
68
An angle is 34° more than its complement.
What is its measure?
Hint: Choose a variable for the angle
What is a complement?
angle = complement + 34
angle = (90 - x) + 34
Answer
x = 90 - x +34
2x = 124
x = 62
69
An angle is 14° less than its complement.
What is the angle's measure?
Hint: What is a complement?
Choose a variable for the angle
angle = complement - 14
angle = (90 - x) - 14
x = 90 - x - 14
Answer
2x = 90 - 14
2x = 76
x = 38
70
An angle is 98 o more than its supplement.
What is the measure of the angle?
Hint: Choose a variable for the angle
What is a complement?
angle = (180 - x) + 98
x Answer
= 180 - x + 98
2x = 278
x = 139
71
An angle is 74° less than its supplement.
What is the angle?
angle = supplement - 74
x = (180 - x) - 74
Answer
2x = 180 - 74
2x = 106
x = 53
72
An angle is 26° more than its supplement.
What is the angle?
angle = supplement + 26
x = (180 - x) + 26
Answer
2x = 180 + 26
2x = 206
x = 106
Angle Addition
Postulate
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of Contents
Angle Addition Postulate
if a point S lies in the interior of ∠PQR,
then ∠PQS + ∠SQR = ∠PQR.
32°
26°
m∠PQS = 32° + m∠SQR = 26°
58°
m∠PQR = 58°
Just as from the Segment Addition Postulate,
"The whole is the sum of the parts"
Example
73
Given m∠ABC = 23° and m∠DBC = 46°.
Find m∠ABD
Hint: Always label your diagram with the information given
74
Given m∠OLM = 64° and m∠OLN = 53°. Find
m∠NLM
A
28
B
15
C
11
D
117
75
Given m∠ABD = 95° and m∠CBA = 48°.
Find m∠DBC
76
Given m∠KLJ = 145° and m∠KLH = 61°.
Find m∠HLJ
77
Given m∠TRQ = 61° and m∠SRQ = 153°.
Find m∠SRT
78
Hint: Draw a diagram and label it with the given information
79
Hint: Draw a diagram and label it with the given information
80
Hint: Draw a diagram and label it with the given information
Angle Bisectors
& Constructions
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of Contents
Definition: Angle Bisector
bisects
Angle Bisector:
A ray or line which
starts at the vertex and
cuts an angle into two equal
halves
Bisect means to cut it into two equal parts. The 'bisector' is the
thing doing the cutting.
The angle bisector is equidistant from the sides of the angle
when measured along a segment perpendicular to the sides of
the angle.
Try this!
Bisect the angle
Try this!
Bisect the angle

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