Download Answer - Math with ms. Taylor

Geometry
Points, Lines, Planes, &
Angles
Table of Contents
Points, Lines, & Planes
Line Segments
Pythagorean Theorem
Distance between points
Area of Figures in Coordinate
Plane
Midpoint formula
Locus & Constructions
Angles & Angle Addition Postulate
Angle Pair Relationships
Angle Bisectors & Constructions
Table of Contents for Videos
Demonstrating Constructions
Congruent Segments
Midpoints
Circle
Equilateral Triangles
Congruent Angles
Angle Bisectors
Points, Lines, & Planes
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 two- dimensions. 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.
However, in geometry, a line is
composed of an unlimited
(infinite) number of points. There
are no spaces between the points
that make a line.
....................................
A
B
You can always find a point between any two other
The line above would be called line
or line
Points & Lines
Line
Points are labeled with
letters.
,
, or
all refer to the same line
Line a
Lines are named by using any two points OR by using a single lowercase letter. Arrowheads show that the line continues without end in
opposite directions.
Collinear Points
Line
,
, or
all refer to the same line
Points A, B, and C are NOT
collinear points since they do
not lie on the same (one) line.
Line a
Points D, E, and F are called collinear points, meaning they all lie on the same
line.
Postulate: Any two points are always collinear.
Answer
Example:
Give six different names for the line
that contains points U, V, and W.
Intersecting Lines
If two non-parallel lines intersect in a plane they do so at only one point.
and
intersect at K.
Postulate: two lines intersect at exactly one point.
Example
Answer
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?
Rays
or
Rays are also portions of a line.
Rays start at an initial point, here endpoint A, and continues infinitely in one
direction.
is read "ray AB"
Naming Rays
or
Rays
and
are NOT the same. They have different initial points and extend in
different directions.
Opposite Rays
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.
Collinear Rays
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
Answer
False
Hint
Read the notation carefully. Are they asking about
lines, line segments, or rays?
click
2
is the same as
.
True
Answer
False
3
Line p contains just three points.
True
Answer
False
Hint
click to reveal
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.
Points D, H, and E are collinear.
True
False
Answer
4
5
Points G, D, and H are collinear.
True
Answer
False
6
Ray LJ and ray JL are opposite rays.
Explain your answer.
Yes
Answer
No
Which of the following are opposite rays?
A
B
and
and
C
and
D
and
Answer
7
A
J
B
K
C
L
Answer
Name the initial point of
8
A
J
B
K
C
L
Answer
Name the initial point of
9
Example
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
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.
(These 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 above.
However, you could draw a plane to contain any three points
Collinear points are points that are on the same line.
J,G, and K are three collinear points.
F,G, and H are three collinear points.
J,G, and H are three non-collinear points.
Any three non-collinear points can name a plane.
Coplanar points are points that lie on the same plane.
F, G, H, and I are coplanar.
F, G, H, and J are also coplanar, but the plane is not drawn.
F,G, and H are coplanar in addition to being collinear.
G, I, and K are non-coplanar and non-collinear.
Intersecting Planes
The intersection of these two
planes is shown by line
A
B
If two planes intersect, they intersect along exactly one 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.
Through any three non-collinear 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?
No
Answer
Yes
11
Plane LMN does not contain point P. Are points P, M, and N
coplanar?
Yes
Answer
No
Hint:
click to reveal
What do we know about any three points?
12
Plane QRS contains
a picture)
. Are points Q, R, S, and V coplanar? (Draw
Yes
Answer
No
13
Plane JKL does not contain
. Are points J, K, L, and N coplanar?
Yes
Answer
No
and
intersect at
A
Point A
B
Point B
C
Point C
D
Point D
Answer
14
Which group of points are noncoplanar with points A, B, and F on
the cube below.
A
E, F, B, A
B
A, C, G, E
C
D, H, G, C
D
F, E, G, H
Answer
15
16
Are lines
and
coplanar on the cube below?
Yes
Answer
No
Plane ABC and plane DCG intersect at _____?
A
C
B
line DC
C
Line CG
D
they don't intersect
Answer
17
Planes ABC, GCD, and EGC intersect at _____?
A
line
B
point C
C
point A
D
line
Answer
18
19
A
B
B
C
C
D
D
F
Answer
Name another point that is in the same plane as
points E, G, and H
20
A
H
B
B
C
D
D
A
Answer
Name a point that is coplanar with points E, F, and C
A
Always
B
Sometimes
C
Never
Answer
Intersecting lines are __________ coplanar.
21
Two planes ____________ intersect at
exactly one point.
A
Always
B
Sometimes
C
Never
Answer
22
A plane can __________ be drawn so that
any three points are coplaner
A
Always
B
Sometimes
C
Never
Answer
23
A plane containing two points of a line
__________ contains the entire line.
A
Always
B
Sometimes
C
Never
Answer
24
Four points are ____________ noncoplanar.
A
Always
B
Sometimes
C
Never
Answer
25
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.
click to reveal
HINT
Answer
26
Line Segments
Line Segments
or
Line segments are portions of a line.
or
endpoint
endpoint
is read "segment AB"
Line Segment
or
are different names for the same segment. 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
-10
-9
-8
B
-7
-6
C
-5
-4
D
-3
-2
-1
E
0
1
F
2
3
4
5
6
7
A
F
coordinate
coordinate
Distance
AF = |-8 - 6| = 14
8
9
10
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.
A
-10
-9
-8
B
-7
-6
C
-5
-4
D
-3
-2
-1
E
0
1
F
2
3
4
5
6
7
A
F
coordinate
coordinate
8
9
10
Distance
AF = |-8 - 6| = 14
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.
Congruent Segments
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."
Constructing Congruent Segments
Given:
Given: AB
1. Draw a reference line w/ your
straight edge. Place a reference point
to indicate where your new segment
start on the line.
2. Place your compass point on point A.
3. Stretch the compass out so that
the pencil tip is on point B.
Constructing Congruent Segments
(cont'd)
4. Without stretching out your
compass, place the compass point on
the reference point & swing the pencil
so that it crosses the reference line.
Constructing Congruent Segments
(cont'd)
5. Make a point where the pencil
crossed your line. Label your new
segment.
Constructing Congruent Segments
Try This!
Construct a Congruent Segment on the horizontal line.
Teacher Notes
1)
Constructing Congruent Segments
Try This!
Construct a Congruent Segment on the slanted line.
2)
Video Demonstrating Constructing Congruent Segments
using Dynamic Geometric Software
Click here to see video
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.
= 8click- 2 = 6 cm
b.
=
1.5 cm
click
27
Find a segment that is 4 cm long
B
C
D
cm
Answer
A
28
Find a segment that is 6.5 cm long
A
C
D
cm
Answer
B
29
Find a segment that is 3.5 cm long
B
C
D
cm
Answer
A
30
Find a segment that is 2 cm long
B
C
D
cm
Answer
A
31
Find a segment that is 5.5 cm long
B
C
D
cm
Answer
A
If point F was placed at 3.5 cm on the ruler, how
from point E would it be?
A
5 cm
B
4 cm
C
3.5 cm
D
4.5 cm
cm
Answer
32
Segment Addition Postulate
If B is between A and C, Then AB + BC = AC.
Or, 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
||
x
Can you finish the rest?
6
||
x
5
2x+11 = 27
2x = 16
click
x = 8 = CD
click
click
=click19
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
2) From the segment addition postulate, we know that
KP + PM = MK (the parts equal the whole)
3) Solve for x
(x + 17) + (2x + 4) = 14x - 56
Answer
1) Draw a diagram and insert the information
given into the diagram
Example
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
Answer
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
For next six questions we are given the following information about the
collinear points:
Hint: always start these problems by placing the information you have into the
diagram.
We are given the following information about the
collinear points:
What is
,
, and
?
Answer
33
We are given the following information about
the collinear points:
What is
?
Answer
34
We are given the following information about the
collinear points:
What is
?
Answer
35
We are given the following information about
the collinear points:
What is
?
Answer
36
We are given the following information about the
collinear points:
What is
?
Answer
37
We are given the following information about
the collinear points:
What is
?
Answer
38
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
Answer
39
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
Answer
40
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
Answer
41
The Pythagorean Theorem
Pythagorean Theorem
c
b
c2 = a2 + b2
a
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 theorem 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.
Pythagorean Theory Visual Proof 1
Pythagorean Theory Visual Proof 2
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.
25 = a2 + 9
5
c2= a2 + b2
-9
a=?
-9
16 =
a2
= a
4 =
3
You will use the Pythagorean Theorem often.
a
Answer
Example
What is the length of side c?
Answer
42
The longest side of a triangle is called the ___________
What is the length of side a?
Answer
43
Hint:
click to reveal
Always determine which side is the hypotenuse first
What is the length of c?
B
Answer
44
What is the length of the missing side?
Answer
45
What is the length of side b?
Answer
46
What is the measure of x?
x
8
17
Answer
47
Pythagorean Triples
are three positive integers for side lengths that satisfy
a2 + b2 = c2
( 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
Answer
No
49
A triangle has sides 9, 12 , and 15, is it a right triangle?
Yes
Answer
No
A triangle has sides √3, 2 , and √5, is it a right triangle?
50
Yes
Answer
No
Distance Between Points
Computing distance
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.
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.
Relationship between the
Pythagorean Theorem & Distance Formula
The Pythagorean Theorem states a relationship
among the sides of a right triangle.
The distance formula
calculates the distance
using the points' coordinates.
(x2, y2)
c2= a2 + b2
c
c
a
b
(x1, y1)
(x2, y1)
The Distance Formula
The distance 'd' between any two points with coordinates
is given
(x2,
y2) by the formula:
(x1, y1)and
d=
The distance between two points, whether on a line or in a coordinate plane, is
computed using the distance formula.
Note: recall that all coordinates are (x-coordinate, y-coordinate).
Example
Calculate the distance
from Point K to Point I
(x1, y1)
(x2, y2)
d=
Plug the coordinates into the distance formula
KI =
Answer
Label the points - it does not matter
which one you label point 1 and point 2.
Your answer will be the same.
51
Calculate the distance from Point J to Point K
A
B
C
Answer
D
52
Calculate the distance from H to K
A
B
D
Answer
C
Calculate the distance from Point G to Point K
53
A
B
C
Answer
D
54
Calculate the distance from Point I to Point H
A
B
C
Answer
D
55
Calculate the distance from Point G to Point H
A
B
C
Answer
D
Area of Figures in the
Coordinate Plane
Calculating Area of Figures in the
Coordinate Plane
Area Formulas:
Area of a Triangle: A =
bh
1
2
Area of a Rectangle: A = lw
Steps to calculate the area:
1) Calculate the desired distances using the distance formula
> Ex: base & height in a triangle
> Ex: length & width in a rectangle
2) Calculate the area of the shape
Example: Calculate the area of the rectangle.
Steps to calculate the area:
1) Calculate the desired distances using
the distance formula
> = FG & w = EF
=
2) Calculate the area of the shape
=w
Steps to calculate the area:
1) Calculate the desired distances using the distance formula
> b = AC & h = BD
Answer
Example: Calculate the area of the triangle.
Answer
Example: Calculate the area of the triangle.
2) Calculate the area of the shape
56
Calculate the area of the rectangle.
A
C
10
D
20
Answer
B
Calculate the area of the triangle.
A
75
B
144
C
84.5
D
169
Answer
57
Calculate the area of the rectangle.
D
A
10,000
B
100
C
50
Answer
58
59
Calculate the area of the triangle.
A
C
22.5
D
45
Answer
B
Midpoint Formula
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
the point
(x2,with
y2) coordinates
and
(x1,
is y1)
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 xcoordinate first. (x, y)
The coordinates of M, the midpoint of PQ, are (6, 5)
(
,
)
Find the midpoint coordinates (x,y) of the segment
and B(5,6)
A
(4, 3)
B
(3, 4)
C
(6, 8)
D
(2.5, 3)
Hint:
Always label the points coordinates first
click
connecting points A(1,2)
Answer
60
61
A
(-1, -1)
B
(-3, -8)
C
(-8, -3)
D
(1, 1)
Answer
Find the midpoint coordinates (x,y) of the segment connecting the points A(2,5) and B(4, -3)
62
A
(-1, 1)
B
(1, 1)
C
(-1, -1)
D
(1, -1)
Answer
Find the coordinates of the midpoint (x, y) of the segment with endpoints R(-4,
6) and Q(2, -8)
63
A
(-3, 3)
B
(6, -4)
C
(-4, 6)
D
(4, 6)
Answer
Find the coordinates (x, y) of the midpoint of the segment with endpoints B(-1,
3) and C(-7, 9)
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)
Answer
64
Example: Finding the coordinates of an
endpoint of an segment
Use the midpoint formula to
write equations using x and y.
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)
Answer
65
Find the other endpoint of the segment with
the endpoint (1, 4) and midpoint (5, -2)
A
(11, -8)
B
(9, 0)
C
(9, -8)
D
(3, 1)
Answer
66
Locus
&
Constructions
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.
Bisect a Line Segment
We can find the midpoint of a line segment by constructing the locus between two
points.
Given
Bisect a Line Segment
We can find the midpoint of a line segment by constructing the locus between two
points.
Given
1. Place the point of the compass on A
and open it so the pencil is more than
halfway, it does not matter how far, to
point B
Bisect a Line Segment
We can find the midpoint of a line segment by constructing the locus between two
points.
Given
2. Keeping the compass locked,
place the point of the compass on
B and draw an arc so that it
intersects the one you just drew
Bisect a Line Segment
We can find the midpoint of a line segment by constructing the locus between two
points.
Given
M
3. Draw a line through
the arc intersections.
M marks the midpoint
of
Bisect a Line Segment
We can find the midpoint of a line segment by constructing the locus between two
points.
2. Keeping the compass locked,
place the point of the compass on
B and draw an arc so that it
intersects the one you just drew
Given
M
1. Place the point of the compass on A
and open it so the pencil is more than
halfway, it does not matter how far, to
point B
3. Draw a line through
the arc intersections.
M marks the midpoint
of
Bisect a Line Segment
Try this!
Construct the midpoint of the
segment below.
3)
Bisect a Line Segment
Try this!
Construct the midpoint of the
segment below.
4)
Bisect a Line Segment w/ a string & rod
We can also construct the locus between two points with string, a rod (e.g. marker,
closed pen, etc) & a pencil.
Given
1. Place the rod on A and extend your
piece of string out. The pencil should
be more than halfway across the
segment. It does not matter how far, to
point B.
Bisect a Line Segment w/ a string & a rod
We can also construct the locus between two points with string, a rod (e.g. marker,
closed pen, etc) & a pencil.
Given
2. Place the rod on B and extend
your piece of string out. The pencil
should be more than half way
across the segment. It does not
matter how
far, to
point A.
Bisect a Line Segment w/ a string & a rod
We can also construct the locus between two points with string, a rod (e.g. marker,
closed pen, etc) & a pencil.
Given
M
3. Draw a line through
the arc intersections.
M marks the midpoint
of
Bisect a Line Segment w/ a string & a rod
We can find the midpoint of a line segment by constructing the locus between two
points.
2. Place the rod on B and extend
your piece of string out. The pencil
should be more than half way
across the segment. It does not
matter how
far, to
point A.
Given
M
1. Place the rod on A and extend your
piece of string out. The pencil should
be more than halfway across the
segment. It does not matter how far, to
point B.
3. Draw a line through
the arc intersections.
M marks the midpoint
of
Bisect a Line Segment w/ a string & a rod
Try this!
Construct the midpoint of the
segment below.
5)
Bisect a Line Segment w/ a string & a rod
Try this!
Construct the midpoint of the
segment below.
Bisect a Line Segment w/ Paper Folding
3. Unfold your patty paper. Draw a line
along the fold.
Other paper edge
2. Fold your patty paper so that point
D lines up with point E. Crease the
fold.
FOLD
1. On patty paper, draw a segment. Label
the endpoints D & E.
4. Draw & label your
midpoint M.
Bisect a Line Segment w/ Paper Folding
Try this!
Construct the midpoint of the
segment below.
7)
Bisect a Line Segment w/ Paper Folding
Try this!
Construct the midpoint of the
segment below.
8)
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.
C
D
6. Draw a line connecting B with D
7. Set the compass width back to AC and step along DB
making 3 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.
Example: Construction
Divide the line segment into 3 congruent segments.
67
Point C is on the locus between point A and point B
True
Answer
False
68
Point C is on the locus between point A and point B
True
Answer
False
How many points are equidistant from
the endpoints of
?
A
2
B
1
C
0
D
infinite
Answer
69
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
Answer
70
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
Answer
71
Videos Demonstrating Constructing Midpoints using Dynamic
Geometric Software
Click here to see video using
compass and segment tool
Click here to see video using
menu options
Construction of a Circle
Answer
What is a circle?
Extension: Construction of a Circle
1. Find the midpoint of a line segment 1st, using any of the methods that we
discussed.
Teacher Notes
M
Extension: Construction of a Circle
2. Take your compass (or rod, string & pencil) and place the tip (or rod) on your
Midpoint. Extend out your pencil as much as you would like. Turn your compass (or
string & pencil) 3600 to make your circle.
M
Construction of a Circle
Try this!
Create a circle using the
segment below.
9)
Construction of a Circle
Try this!
Create a circle using the
segment below.
10)
Video Demonstrating Constructing a Circle using Dynamic
Geometric Software
Click here to see video
Construction of an Equilateral Triangle
Answer
What is an equilateral triangle?
Extension: Construction of an Equilateral Triangle
1. Construct a segment of any
length.
2. Take your compass and place
the tip on one endpoint & make
an arc.
3. Take your compass and place the tip
on the other endpoint & make an arc
that intersects your 1st arc.
4. Make a point where your two arcs
intersect. Connect your 3 points.
C
Construction of an Equilateral Triangle
Try this!
Create an equilateral triangle
using the segment below.
11)
Construction of an Equilateral Triangle
Try this!
Create an equilateral triangle
using the segment below.
12)
Extension: Construction of an Equilateral Triangle w/ rod,
string & pencil
1. Construct a segment of any
length.
3. Switch the places of your rod &
pencil tip. Make an arc that intersects
your 1st arc.
2. Place your rod on one
endpoint & the pencil on the
other. Extend your pencil to
make an
arc.
4. Make a point where your two arcs
intersect. Connect your 3 points.
C
Construction of an Equilateral Triangle w/ rod, string &
pencil
Try this!
Create an equilateral triangle
using the segment below.
13)
Construction of an Equilateral Triangle w/ rod, string &
pencil
Try this!
Create an equilateral triangle
using the segment below.
14)
Video Demonstrating Constructing Equilateral Triangles
using Dynamic Geometric Software
Click here to see video
Angles &
Angle Addition Postulate
Identifying Angles
An angle is formed by two rays with a common endpoint (vertex)
The angle shown can be called
,
, or
.
C
When there is no chance
of confusion, the angle
may also be identified
by its vertex B.
(Side)
32°
The sides of
are rays BC and BA
B
(Vertex)
(Side)
The measure of the angle is 32 degrees.
"The measure of
is equal to the measure of
..."
A
Congruent angles
Two angles that have the same measure are congruent angles.
The single mark through the arc
shows that the angle
measures are equal
Exterior
Interior
We read this as
is congruent to
The area between the rays that form an angle is called the interior. The exterior is
the area outside the angle.
Constructing Congruent angles
Given: <FGH
1. Draw a reference line w/ your straight edge. Place a reference point to indicate
where your new segment starts on the line.
2. Place your compass
point on the vertex (point
G).
4. Swing an arc with the pencil so that it
crosses both sides of <FGH.
3. Stretch the compass to any length so
long as it stays ON the angle.
Constructing Congruent angles (cont'd)
5. Without changing the span of the compass,
place the compass tip on your reference point
& swing an arc that goes through the line &
extends above the line.
6. Go back to <FGH & measure the width of the arc from where it crosses one side
of the angle to where it crosses the other side of the angle.
The width of the arc is 2.2 cm
Constructing Congruent angles (cont'd)
7. With this width, place the compass point on the reference line where your new
arc crosses the reference line. Mark off this width on your new arc.
The width of the arc is 2.2 cm
8. Connect this new intersection point to the
starting point on the reference line.
Constructing Congruent angles
Try this!
Construct a congruent angle on the given line.
15)
Constructing Congruent angles
Try this!
Construct a congruent angle on the given line.
16)
Video Demonstrating Constructing Congruent Angles
using Dynamic Geometric Software
Click here to see video
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 same.
D
A
C
B
is a 23° degree angle
The measure of
is 23° degrees
is a 119° degree angle
The measure of
is 119° degrees
In
and
, notice that the vertex is written in
between the sides
M
L
Example
K
N
O
P
J
90o
clic
32o
clic
180o
clic
k
k
63o
clic
k
k
135o
clic
k
L
M
Example
K
N
O
P
J
Challenge Questions
clic
k
148
-117
31o
clic
k
117
- 90
27o
clic
k
90
-45
45o
clic
k
45o
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.
Click here for a Math
Is Fun website
activity.
Adjacent Angles
O
K
Adjacent angles are angles that have a
common ray coming out of the vertex
going between two other rays.
P
In other words, they are angles that are side by side, or adjacent.
J
Angle Addition Postulate
if a point S lies in the interior of
then PQS + SQR = PQR.
PQR,
58°
32°
26°
m
PQS = 32°
+ m
SQR = 26°
m
PQR = 58°
Just as from the Segment Addition Postulate,
"The whole is the sum of the parts"
Example
72
Given m∠ABC = 22° and m∠DBC = 46°.
Answer
Find m∠ABD
Hint:
Always label your diagram with the
information given
click
Given m∠OLM = 64° and m∠OLN = 53°. Find m∠NLM
A
28
B
15
C
11
D
117
64°
53°
Answer
73
Given m∠ABD = 95° and m∠CBA = 48°.
Find m∠DBC
Answer
74
Given m∠KLJ = 145° and m∠KLH = 61°.
Find m∠HLJ
Answer
75
Given m∠TRQ = 61° and m∠SRQ = 153°.
Find m∠SRT
Answer
76
C is in the interior of ∠TUV.
If m∠TUV = (10x + 72)⁰,
m∠TUC = (14x + 18)⁰ and
m∠CUV = (9x + 2)⁰
solve for x.
Answer
77
Hint:
Draw a diagram and label it with the
given information
click
D is in the interior of ∠ABC.
If m∠CBA = (11x + 66)⁰,
m∠DBA = (5x + 3)⁰ and
m∠CBD= (13x + 7)⁰
solve for x.
Answer
78
Hint:
Draw a diagram and label it with the
given information
click
F is in the interior of ∠DQP.
If m∠DQP = (3x + 44)⁰,
m∠FQP = (8x + 3)⁰ and
m∠DQF= (5x + 1)⁰
solve for x
Answer
79
Angle Pair Relationships
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.
K
136 degrees
44 degrees
P
O
J
Example
Solution:
Choose a variable for the angle - I'll choose "x"
Example
Two angles are complementary. The larger angle is twice the
size of the smaller angle. What is the measure of both
angles?
Let x = the angle
Since the angles are complementary we know their sum
must equal 90 degrees.
90 = 2x + x
90 = 3x
30 = x
80
An angle is 34° more than its complement.
Answer
What is its measure?
Hint:
Choose a variable for the angle.
What is a complement?
click
81
An angle is 14° less than its complement.
Answer
What is the angle's measure?
Hint:
Choose a variable for the angle.
What is a complement?
click
82
An angle is 98 more than its supplement.
Answer
What is the measure of the angle?
Hint:
Choose a variable for the angle
What is a supplement?
click
An angle is 74° less than its supplement.
What is the angle?
Answer
83
An angle is 26° more than its supplement.
What is the angle?
Answer
84
Linear Pair of Angles
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 non- common sides on the
same line. A line could also be called a straight angle with 180°
Vertical Angles
Given: m∠ABC = 55o
Find the measures of the remaining angles.
A
55o
B
E
D
C
Answer
Vertical Angles: Two angles whose sides form two pairs of opposite rays
- In diagram below, ∠ABC & ∠DBE are vertical angles, and ∠ABE & ∠CBD are
vertical angles.
Vertical Angles
Given: m∠ABC = 55o
What do you notice about your vertical angles?
A
125o
E
55o
55o
B
125o
C
D
click to reveal theorem
Vertical Angles Theorem: All vertical angles are congruent.
Example
Find the m < 1, m < 2 & m < 3. Explain your answer.
1
36o
3
2
36 + m < 1 = 180
-36
m < 1 = 144o
Linear pair angles are
supplementary
m < 2 = 36o; Vertical angles are congruent (original angle & < 2)
m < 3 = 144o; Vertical angles are congruent (< 1 & < 3)
-36
What is the m < 1?
A
77o
1
2
B
103o
C
113o
D
none of the above
77o
3
Answer
85
What is the m < 2?
A
77o
1
2
B
103o
C
113o
D
none of the above
77o
3
Answer
86
What is the m < 3?
A
77o
1
2
B
103o
C
113o
D
none of the above
77o
3
Answer
87
What is the m < 4?
A
112o
4
112o
6
B
78o
C
102o
D
none of the above
5
Answer
88
What is the m < 5?
A
112o
4
112o
6
B
68o
C
102o
D
none of the above
5
Answer
89
What is the m < 6?
A
102o
4
112o
6
B
78o
C
112o
D
none of the above
5
Answer
90
Example
Find the value of x.
The angles shown are vertical
angles, so they are congruent
(13x + 16)o
(14x + 7)o
13x + 16 = 14x + 7
-13x
-13x
16 = x + 7
-7
-7
9=x
Example
Find the value of x.
The angles shown are a linear
pair, so they are supplementary
(2x + 8)o
(3x + 17)o
2x + 8 + 3x + 17 = 180
5x + 25 = 180
- 25
5x = 155
5
x = 31
- 25
5
Find the value of x.
A
95
B
50
C
45
D
85o
(2x - 5)o
40
Answer
91
92
Find the value of x.
A
75
75o
(6x + 3)o
B
17
C
13
12
Answer
D
93
Find the value of x.
A
122o
13.1
(9x - 4)o
B
14
C
15
122
Answer
D
Find the value of x.
A
12
(7x + 54)o
B
13
C
42
D
42o
138
Answer
94
Angle Bisectors &
Constructions
Angle Bisector
bisects
An angle bisector is 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.
Constructing Angle Bisectors
1. With the compass point on
the vertex, draw an arc across
each side
3. With a straightedge, connect the
vertex to the arc intersections
2. Without changing the
compass setting, place the
compass point on the arc
intersections of the sides
4. Label your point
X
m∠UVX = m∠WVX
∠UVX
∠WVX
Try this!
Bisect the angle
17)
Try this!
Bisect the angle
18)
Constructing Angle Bisectors w/ string,
rod, pencil & straightedge
1. With the rod on the vertex, draw an arc
across each side.
3. With a straightedge, connect the
vertex to the arc intersections
2. Place the rod on the arc
intersections of the sides & draw 2
arcs, one from each side showing an
intersection point.
4. Label your point
X
m∠UVX = m∠WVX
∠UVX
∠WVX
Try this!
Bisect the angle with string, rod,
pencil & straightedge.
19)
Try this!
Bisect the angle with string, rod,
pencil & straightedge.
20)
Constructing Angle Bisectors by Folding
1. On patty paper, create any angle of your 2. Fold your patty paper so that BA
choice. Make it appear large on your patty lines up with BC. Crease the fold.
paper. Label the points A, B & C
3. Unfold your patty paper. Draw a ray
along the fold, starting at point B.
4. Draw & label your point.
Try this!
21)
Bisect the angle with
folding.
Try this!
22)
Bisect the angle with
folding.
Videos Demonstrating Constructing Angle Bisectors
using Dynamic Geometric Software
Click here to see video using
a compass and segment tool
Click here to see video using
the menu options
Finding the missing measurement.
Answer
Example: ABC is bisected by BD. Find the measures
of the missing angles.
A
D
52o
B
C
Example: EFG is bisected by FH. The
m EFG = 56o. Find the measures of the missing
angles.
Answer
E
H
56o
F
G
Example: MO bisects
LMN. Find the value of x.
L
(x + 10)o
(3x - 20)o
N
O
Answer
M
JK bisects
HJL. Given that m
HJL = 46o, what is m
HJK?
Answer
95
Hint:
click to does
reveal bisect mean?
What
Draw & label a picture.
NP bisects
MNO Given that m
MNP = 57o, what is m
MNO?
Answer
96
Hint:
What
does bisect mean?
click to reveal
Draw & label a picture.
RT bisects
QRS Given that m
QRT = 78o, what is m
QRS?
Answer
97
VY bisects
UVW. Given that m
UVW = 165o, what is m
UVY?
Answer
98
99
BD bisects
ABC. Find the value of x.
A
D
Answer
(7x + 3)o
(11x - 25)o
B
C
100
FH bisects
EFG. Find the value of x.
H
E
Answer
(9x - 17)o
(3x + 49)o
F
G
JL bisects
IJK. Find the value of x.
I
L
(7x + 1)o
(12x - 19)o
J
K
Answer
101