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Points, Lines, Planes, and Angles
Points, Lines, Planes, & Angles
Table of Contents
Points, Lines, & Planes
Line Segments
Simplifying Perfect Square Radical Expressions
Rational & Irrational Numbers
Simplifying Non­Perfect Square Radicands
Pythagorean Theorem
Distance between points
Midpoint formula
Angles & Angle Relationships
Angle Addition Postulate
www.njctl.org
Definitions
An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry:
Points, Lines, & Planes
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. Return to Table of Contents
Planes ­ a flat surface that extends indefinitely in two­
dimensions. A plane has no thickness.
1
Points, Lines, Planes, and Angles
Points are labeled with letters. (Points A, B, or C)
Points & Lines
A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces.
Lines are named by using any two points OR by using a single lower­cased letter. Arrowheads show the line continues without end in opposite directions.
Line ,
, or
all refer to the same line
....................................
B
A
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.
Line a
The line above would b called line or line Collinear Points ­ Points D, E, and F above are called collinear points, meaning they all lie on the same line.
Example
Points A, B, and C are NOT collinear point since they do not lie on the same (one) line. Give six different names for the line that contains points U, V, and W.
Postulate: Any two points are always collinear. Line ,
, or
all refer to the same line
Answer
(click) Line a
2
Points, Lines, Planes, and Angles
Postulate: two lines intersect at exactly one point.
If two non­parallel lines intersect in a plane they do so at only one point.
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
and intersect at K.
Rays are also portions of a line. a. A, D, C
b. A,B,D / A,C,B / C,D,B (others) Move
Suppose point C is between points A and B
or Rays and are opposite rays.
is read ray AB.
Rays start at an initial point, here endpoint A, and continues infinitely in one direction.
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.
Ray has a different initial point, endpoint B, and continues infinitely in the direction marked.
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.
Rays and are NOT the same. They have different initial points and extend in different directions.
3
Points, Lines, Planes, and Angles
Example
Example
Name a point that is collinear
with the given points.
Name two opposite rays on
the given line
e. b. M and Q
f. c. S and N
g. d. O and P
h. is the same as .
True True False False Answer
2
is the same as . Answer
1
a. R and P
Hint
Read the notation carefully. Are they asking about lines, line Move
segments, or rays?
4
Points, Lines, Planes, and Angles
Line p contains just three points.
4
Points D, H, and E are collinear.
True True False False Answer
Answer
3
Hint
Remember that even though only three points are marked, a line is composed of an infinite number of points. You can Move
always find another point in between two other points. 5Ray LJ and ray JL are opposite rays.
6Which of the following are opposite rays?
Explain your answer.
Yes A and No B and C and No, Opposite Rays have same endpoint but point in opposite directions
Answer
Answer
D and 5
Points, Lines, Planes, and Angles
Are the three points collinear? If they are,
name the line they lie on.
7Name the initial point of A J
a. L, K, J
b. N, I, M
c. M, N, K
d. P, M, I
B K
Answer
C L
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 non­collinear 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 non­coplanar and non­collinear.
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 non­coplanar in the diagram above
However, you could draw a plane to contain any three points
6
Points, Lines, Planes, and Angles
Postulate: Through any three noncollinear
points there is exactly one plane. 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. Example
8Line BC does not contain point R. Are points R, B, and C collinear?
Name the following points:
Yes A point not in plane HIE
No Two points in both planes
Answer
A point not in plane GIE
Two points not on 7
Points, Lines, Planes, and Angles
9Plane LMN does not contain point P. Are points P, M, and N coplanar?
10Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture)
Yes No No Answer
Answer
Yes Hint: Move
11Plane JKL does not contain . Are points J, K, L, and N coplanar?
12 and intersect at A
Yes Answer
No Point A
B
Point B
C
Point C
D
Point D
Answer
What do we know about any three points?
8
Points, Lines, Planes, and Angles
13Which group of points are noncoplanar with points A, B, and F on the cube below.
14Are lines and coplanar on the cube below?
A E, F, B, A
B A, C, G, E
Yes C D, H, G, C
No Answer
Answer
D F, E, G, H
15Plane ABC and plane DCG intersect at _____?
A C
16Planes ABC, GCD, and EGC intersect at _____?
A line B line DC
B point C
C Line CG
C point A D they don't intersect
B
Answer
Answer
D line B
9
Points, Lines, Planes, and Angles
17Name another point that is in the same plane as 18Name a point that is coplanar with points E, F, and C
points E, G, and H
A H
A B
B B
B C
C D
C D
D A
19Intersecting lines are __________ coplanar.
Answer
Answer
D F
C
D
20Two planes ____________ intersect at exactly one point.
A Always
A Always B Sometimes B Sometimes
C Never
Answer
Answer
C Never
10
Points, Lines, Planes, and Angles
21A plane can __________ be drawn so that any three points are coplaner
22A plane containing two points of a line
__________ contains the entire line.
A Always
A Always
B Sometimes
B Sometimes
C Never
Answer
Answer
C Never
A Always
B Sometimes
B Sometimes
C Never
C Never
Answer
A Always Answer
24Two lines ________________ meet at more than one point.
23Four points are ____________ noncoplanar.
Look what happens if I place line
y directly on top of line x.
Hint
11
Points, Lines, Planes, and Angles
Line Segments or Line segments are portions of a line.
or
Line Segments
endpoint
endpoint
is read segment AB.
Return to Table of Contents
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
Why did we take the Absolute Value when calculating distance? 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 F
­10
­9
­8
­7
­6
­5
­4
­3
­2
­1
0
1
2
3
4
5
6
7
8
F
coordinate
A
coordinate
9
10
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
Distance
AF = |­8 ­ 6| = 14
12
Points, Lines, Planes, and Angles
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
25Find a segment that is 4 cm long
A B C Answer
Find the measure of each segment in centimeters.
D cm
a. b.
= cm
= 13
Points, Lines, Planes, and Angles
27Find a segment that is 2 cm long
26Find a segment that is 3.5 cm long
C B Answer
B Answer
A A C D D cm
cm
28If point F was placed at 3.5 cm on the ruler, how far from point E would it be?
Segment Addition Postulate
B 4 cm
C 3.5 cm
Answer
A 5 cm
D 4.5 cm
AB
BC
AC
cm
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. 14
Points, Lines, Planes, and Angles
Example
Start by filling in the information you are given
AE
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). AB
CD
BC
DE
In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6 In the diagram, AE = 27, AB = CD, DE = 5, and BC = 6
27
Find CD and BE
||
Can you finish the rest?
K, M, and P are collinear with P between K and M. PM = 2x+4, MK = 14x­56, and PK = x+17
Solve for x. ||
6
5
CD = BE = 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
15
Points, Lines, Planes, and Angles
29We are given the following information about the collinear points: 30We are given the following information about the collinear points: What is ?
Answer
Answer
What is , , and ?
32We are given the following information about the collinear points: 31We are given the following information about the collinear points: Answer
Answer
What is ?
What is ?
16
Points, Lines, Planes, and Angles
33We are given the following information about the collinear points: 34We are given the following information about the collinear points: What is ?
35X, 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
Answer
What is ?
36Q, 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
Answer
7x + 1 + 15x + 1
22x + 11 = 2
20x = 1
x = 6
17
Points, Lines, Planes, and Angles
37B, 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
Simplifying Perfect Square Radical Expressions
Return to Table of Contents
Answer
4x + 149 + 5x = 15x + 125
9x + 149 = 15x + 125
6x = 24
x = 4
Can you recall the perfect squares from 1 to 169?
12 = 82 = 2 2 = 9 = 2
3 =
10 = 2 4 =
11 = 5 =
122 = 62 =
132 = 202 =
2 2 2
Square Root Of A Number
Recall: If b 2 = a, then b is a square root of a.
Example: If 42 = 16, then 4 is a square root of 16
What is a square root of 25? 64? 100?
2
72 = 18
Points, Lines, Planes, and Angles
Is there a difference between
Square Root Of A Number
Square roots are written with a radical symbol
&
?
Positive square root: = 4
Which expression has no real roots?
Negative square root: ­ = ­ 4
Positive & negative square roots: = 4
Evaluate the expression
Negative numbers have no real square roots
no real roots because there is no real number that, when squared, would equal ­16. Evaluate the expression
38 ?
is not real 19
Points, Lines, Planes, and Angles
39 = ?
40 41 = ?
A
3
B
­3
C
No real roots
Rational & Irrational
Numbers
Return to Table of Contents
20
Points, Lines, Planes, and Angles
Rational & Irrational Numbers
is rational because the radicand (number under the radical) is a perfect square 42Rational or Irrational?
A
Rational B
Irrational B
Irrational If a radicand is not a perfect square, the root is said to be irrational. Ex: 43Rational or Irrational?
A
Rational 44Rational or Irrational?
B
Irrational A
Rational 21
Points, Lines, Planes, and Angles
What happens when the radicand is not a perfect square?
Rewrite the radicand as a product of its largest perfect square factor.
Simplifying Non­Perfect Square Radicands
Return to Table of Contents
Try These.
Simplify the square root of the perfect square.
When simplified form still contains a radical, it is said to be irrational.
Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work.
Ex: Not simplified! Keep going!
Finding the largest perfect square factor results in less work:
Note that the answers are the same for both solution processes
22
Points, Lines, Planes, and Angles
45Simplify
46Simplify
A
A
B
B
C
C
D already in simplified form
47Simplify
D already in simplified form
48Simplify
A
A
B
B
C
C
D already in simplified form
D already in simplified form
23
Points, Lines, Planes, and Angles
49Simplify
50Simplify
A
A
B
B
C
C
D already in simplified form
51Which of the following does not have an irrational simplified form?
A
B
C
D
D already in simplified form
2
24
Points, Lines, Planes, and Angles
52Simplify 53Simplify A
A
B
B
C
D
C
D
54Simplify 55Simplify A
B
B
C
C
D
D
A
25
Points, Lines, Planes, and Angles
56Simplify A
B
C
D
The Pythagorean Theorem
Return to Table of Contents
Pythagorean Theorem
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. 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 c2 = a2 + b2 c
b
c2 = a2 + b2
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.
Click to see a Visual Proof
Proof
5
3
a = ?
25 = a2 + 9
­9
­9
2 16 = a
=a
4 =a
Proof
You will use the Pythagorean Theorem often.
26
Points, Lines, Planes, and Angles
57What is the length of side c?
Answer
Example
Answer
Hint:
The longest side of a triangle is called the?
Move
59What is the length of c?
58What is the length of side a?
Hint: Answer
Answer
B
Always determine which side is the hypotenuse first
Move
27
Points, Lines, Planes, and Angles
60What is the length of the missing side?
Answer
Answer
61What is the length of side b?
63 Calculate the value of the missing side. Leave your answer in simplest radical form.
62What is the measure of x?
x
10
8
Answer
17
Answer
8
28
Points, Lines, Planes, and Angles
64 Calculate the value of the missing side. Leave your answer in simplest radical form.
65 Calculate the value of x. Leave your answer in simplest radical form.
18
Pythagorean Triples
are three positive integers for side lengths that satisfy a2 + b 2 = 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
12
66A triangle has sides 30, 40 , and 50, is it a right triangle?
Yes No Answer
6
Answer
Answer
18
29
Points, Lines, Planes, and Angles
67 A triangle has sides 9, 12 , and 15, is it a right triangle?
68 A triangle has sides √3, 2 , and √5, is it a right triangle?
Yes No No Answer
Answer
Yes Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles.
Distance Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle.
Return to Table of Contents
30
Points, Lines, Planes, and Angles
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. Distance
The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula.
(x2, y2)
The Distance Formula
c = a + b
2
2
2
The distance 'd' between any two points with coordinates
(x2, y2)
and is given by the formula:
(x1, y1)
c
c
a
(x1, y1)
b
d = (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.
Note: recall that all coordinates are (x­coordinate, y­coordinate).
69Calculate the distance from Point J to Point K
Example
A B Calculate the distance from Point K to Point I
(x1, y1)
C (x2, y2)
D Answer
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 = D
=
=
31
Points, Lines, Planes, and Angles
71Calculate the distance from Point G to Point K
70Calculate the distance from H to K
A B B C C D D D
Answer
Answer
A 72Calculate the distance from Point I to Point H
A 73Calculate the distance from Point G to Point H
B A C B D C Answer
Answer
D 32
Points, Lines, Planes, and Angles
74 Calculate the distance between A(­3, 7) and B(8, 2).
75 Calculate the distance between A(3, ­7) and B(1, ­3).
76 Calculate the distance between A(3, 0) and B(­10, 6).
Midpoints
Return to Table of Contents
33
Points, Lines, Planes, and Angles
Midpoint of a line segment
A number line can help you find the midpoint of a segment.
Midpoint Formula Theorem
The midpoint of GH, marked by point M, is ­1.
(x1, y1)
The midpoint of a segment joining points with coordinates and is the point with coordinates
(x2, y2)
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
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.
A (4, 3) B (3, 4) Remember that points are written with the x­coordinate first. (x, y)
The coordinates of M, the midpoint of PQ, are (6, 5)
D (2.5, 3) Answer
C (6, 8) Just as before, we find the average of the coordinates.
( , )
77Find the midpoint coordinates (x,y) of the segment connecting points A(1,2) and B(5,6)
Hint: Always label the points coordinates first
34
Points, Lines, Planes, and Angles
79Find the coordinates of the midpoint (x, y) of the segment with endpoints R(­4, 6) and Q(2, ­8)
A (­1, 1) B (­3, ­8) B (1, 1) C (­8, ­3) C (­1, ­1) D (1, 1) D (1, ­1) Answer
A (­1, ­1) 80Find the coordinates (x, y) of the midpoint of the segment with endpoints B(­1, 3) and C(­7, 9)
B (1/2, 5/2) C (­4, 6) C (1/2, 3) D (3, 1/2) Answer
A (3/2, 5/2) B (6, ­4) Answer
81Find the midpoint (x, y) of the line segment between A(­1, 3) and B(2,2)
A (­3, 3) D (4, 6) Answer
78Find the midpoint coordinates (x,y) of the segment connecting the points A(­2,5) and B(4, ­3)
35
Points, Lines, Planes, and Angles
82Find the other endpoint of the segment with the endpoint (7,2) and midpoint (3,0)
Example: Finding the coordinates of an
endpoint of an segment
A (­1, ­2) B (­2, ­1)
C (4, 2) Answer
D (2, 4)
Use the midpoint formula to
write equations using x and y.
A
83Find the other endpoint of the segment with the endpoint (1, 4) and midpoint (5, ­2)
A (11, ­8) B (9, 0) Angles
&
Angle Relationships
C (9, ­8) Answer
D (3, 1)
C
Return to Table of Contents
36
Points, Lines, Planes, and Angles
Identifying Angles
Two angles that have the same measure are congruent angles.
An angle is formed by two rays with a common endpoint (vertex)
The angle shown can be called , , or . C
The sides of are CB and AB
The single mark through the arc shows that the angle measures are equal
Exterior
When there is no chance
of confusion, the angle may also be identified by its vertex B.
(Side)
Interior
32°
B
(Vertex)
A
(Side)
We read this as is congruent to The measure of the angle is 32 degrees.
The area between the rays that form an angle is called the interior. The exterior is the area outside the angle. "The measure of is equal to the measure of ..."
Angle Measures
L
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.
M
Example
K
N
A
P
B
is a 23° degree angle
The measure of
is 23° degrees
J
O
C
is a 119° degree angle
The measure of
is 119° degrees
Challenge Questions
In and , notice that the vertex is written in between the sides
37
Points, Lines, Planes, and Angles
Angle Relationships
Once we know the measurements of angles, we can categorize them into several groups of angles:
0° < acute < 90°
90° < obtuse < 180°
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°)
right = 90°
straight = 180°
180°
180° < reflex angle < 360°
Two lines or line segments that meet
at a right angle are said to be perpendicular.
Link
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 non­shared 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°
38
Points, Lines, Planes, and Angles
Example
Example
Two angles are complementary. The larger angle is twice the size of the smaller angle. What is the measure of both angles?
Solution:
Choose a variable for the angle ­ I'll choose "x"
Let x = the angle
Since the angles are complementary we know their sum
must equal 90 degrees.
90 = 2x + x
90 = 3x
30 = x
85An angle is 14° less than its complement.
84An angle is 34° more than its complement. Choose a variable for the angle. What is a complement? Move
Answer
Hint: angle = complement + 34
angle = (90 ­ x) + 34
Answer
What is the angle's measure?
What is its measure?
Hint: What is a complement?
Choose a variable for the angle
Move
39
Points, Lines, Planes, and Angles
87An angle is 74° less than its supplement.
86An angle is 98 more than its supplement.
What is the angle?
What is the measure of the angle?
Answer
Answer
angle = supp
Hint: Choose a variable for the angle
Move
What is a supplement?
89
and
are a linear pair. What is the value
of x if and .
88An angle is 26° more than its supplement.
What is the angle?
Answer
angle = supplement + 26
40
Points, Lines, Planes, and Angles
Angle Addition Postulate
if a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR. Angle Addition Postulate
32°
58°
26°
Return to Table of Contents
m∠PQS = 32° + m∠SQR = 26°
m∠PQR = 58°
Just as from the Segment Addition Postulate,
"The whole is the sum of the parts"
90 Given m∠ABC = 22° and m∠DBC = 46°. Find m∠ABD
Answer
Example
Hint: Always label your diagram with the information given
Move
41
Points, Lines, Planes, and Angles
91 Given m∠OLM = 64° and m∠OLN = 53°. Find m∠NLM
92 Given m∠ABD = 95° and m∠CBA = 48°. Find m∠DBC
A 28
B 15
C 11
64°
Answer
53°
Answer
D 117
94 Given m∠TRQ = 61° and m∠SRQ = 153°. 93 Given m∠KLJ = 145° and m∠KLH = 61°. Answer
Answer
Find m∠SRT
Find m∠HLJ 42
Points, Lines, Planes, and Angles
95 C is in the interior of ∠TUV.
96 D is in the interior of ∠ABC.
m∠DBA = (5x + 3)⁰ and
m∠CUV = (9x + 2)⁰
m∠CBD= (13x + 7)⁰
solve for x.
solve for x.
11x + 66 = 5x +
Answer
If m∠CBA = (11x + 66)⁰,
m∠TUC = (14x + 18)⁰ and
Answer
If m∠TUV = (10x + 72)⁰,
11x + 66 = 7x = x = Hint: Hint: Draw a diagram and label it with the given information
Move
Draw a diagram and label it with the given information
Move
97 F is in the interior of ∠DQP.
If m∠DQP = (3x + 44)⁰,
m∠FQP = (8x + 3)⁰ and
m∠DQF= (5x + 1)⁰
Answer
solve for x
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