angle relationship worksheets

Name _______________________
WORKSHEET #3
Upward Bound Summer 2011: Geometry
OBJECTIVE:
I will be able to determine segment lengths using diagrams and the Segment Addition Postulate.
I will be able to find the distance between two points in the coordinate plane using the distance formula.
NEW CONCEPT:
New Vocabulary → Postulate, Segment Addition Postulate, midpoint
Say WHAT?
Geometry can be weird...
The Segment Addition Postulate:
“If AC = n, then AB + BC = n”
A
n
𝐴𝐶 means:
C
B
Midpoints:
̅̅̅̅
𝐴𝐶 means:
If point M is the midpoint of ̅̅̅̅ , then AM = MB
A
M
B
NOTES AND EXAMPLES:
Example 1: Point J is between points A and B on ̅̅̅̅
a)
If AJ = 8 and JB = 7, then AB = ____________
b)
If AB = 12 and JB = 4, then AJ = ____________
c)
If AJ = 8 and JB = 8, then point J is called the ____________ of ̅̅̅̅
Example 2: AC = 9. Solve for x and find the length of ̅̅̅̅ .
3x – 1
2x
A
C
B
Example 3: Point M is the midpoint of ̅̅̅̅. Solve for x and find the length of ̅̅̅̅ .
10x – 4
8x + 2
L
M
N
We can use the distance formula to find the distance between any two points in the coordinate
plane. In the formula below, d is distance and the two points are (x1 , y1) and (x2 , y2).
√
Example 4: Find the distance between the points (-2, 1) and (1, 5).
Example 5: Find the length of the line segment with endpoints (-2, 3) and (5, -3). Round to the
nearest tenth.
Example 6: Find how long each segment would be if it were drawn on the graph below. Round to the
nearest tenth if necessary.
a)
̅̅̅̅
A
b)
B
̅̅̅̅
C
c)
̅̅̅̅
Geometry Mathematics
Name_______________ Period
Unit 1 Segment Addition Worksheet
Segment Addition Postulate If B is between A and C, then AB BC AC .
If AB BC AC , then B is between A and C.
Write the Segment Addition Postulate for each problem. Also use Segment Addition
Postulate to solve the following problems.
1. If AB = 27 and BC = 13, then find the length of AC.
A
B
AC=
C
2. If TD = 32 and YD = 51, then find the length of YT.
Y
T
D
YT=
3. If RG = 7x + 3, GQ = 3x + 13, and RQ = 56, then find the value for x, RQ, and GQ.
R
G
Q
x=
RQ=
GQ=
4. If AB = x + 4, BC = 2x – 10, and AC =2x + 1, then find the value for x, AB, BC and AC.
A
B
C
x=
AB=
BC=
AC=
5. If AT = 6x – 2, TL = 4x – 12, and AL = 36, then find the value for x, AT, and TL.
A
T
L
x=
AT=
TL=
6. If RE = 4x + 7, ET = 2(3x – 4), and RT = 43, then find the value for x, RE, and ET.
R
E
T
x=
RE=
ET=
!
Geometry
WS 1.1 + (Congruence & Segment Addition)
!
Name___________________
Date:__________Period:____
Write the Segment Addition Postulate for the points described. Draw a picture to help.
1. S is between D and P
2. J is between S and H
!
!
!
3. C is between Q and R
!
!
!
!
4. T is between M and N
5. If AC = 24 in. and CE = 13 in., AE = _____.
6. If CE = 7in. and AE = 23 in., AC = _____.
C is between A and E. For each problem, draw a picture representing the three points and the
information given. Solve for indicated.
!
!
!
!
!
!
Find QR in the following problems. R is between Q and S.
7. If RS = 44.6 and SQ = 68.4, find QR.
8. If RS = 33.5 and RQ = 80, find SQ.
!
!
!
!
!
!
Refer to the figure and the given information to find each measure.
!
2x-8
x+17
9. Given :
AC = 39 m
C
AB = _______
BC = _______
4x- 3
!
x = ________
2x + 21
10. Given the
figure and DG =
60 ft.
!
!
!
x = _______
DO = ______
OG = ______
If U is between T and B, find the value of x and the lengths of the segments. (Hint: Draw a
picture for each problem with the given information and then write the equation to solve.)
11. TU = 2x, UB = 3x + 1, TB = 21
!
!
!
!
12. TU = 4x-1, UB = 2x -1, TB = 5x
x = _______
x = _______
TU = ______
TU = ______
UB = ______
UB = ______
TB = _______
Write an equation for the each:
13. Segment AB is congruent to segment BC _____________________________
14. ! XY ≅ AB ____________________________________________________
15. Point B bisects segment AC_______________________________________
16. 2x+5 is equal to 4x-8____________________________________________
!
17. Point A is the midpoint of segment PT_______________________________________
For 18-19, suppose ! RS is congruent to ! MN . For each set of lengths, solve for x, and find
the length of each segment. For 20-21, ! AB ≅ BC .
18. RS = 3x + 17, MN = 7x – 15
19. RS = x + 10, MN = 2x + 4
x = ______
RS = ____
MN = ____
!
2x - 8
20.
_________
x=
AB = _________
BC = _________
!
!
21.
x + 17
x = ______
RS = _____
MN = ____
AC = _________
3x - 31
x+6
x=
_________
!
!
!
BONUS
AB = _________
BC = _________
AC = _________
5(2x + 2)
3(3x – 1)
!
B
C
_________
AB = _________
BC = _________
x=
AC = _________
Geometry
Name: _________________________
Worksheet 1.2 (Congruence and Segment Addition)
Date: ________________ Period: ___
Suppose RS is congruent to MN . For each of the set of lengths, solve for x, and find
the length of each segment.
1. RS = 3x + 17, MN = 7x – 15
2. RS = x + 10, MN = 2x + 4
3. RS = 3x – 2, MN = x + 6
4. RS = 5x – 10, MN = 2x + 20
Suppose AB is congruent to BC. Solve for x, and find the length of AB, BC and AC.
5.
6.
2x − 8
A
x + 17
B
7x − 6
A
C
12 − 2x
B
C
X = ______
AB = ______
BC = ______
AC = ______
X = ______
AB = ______
BC = ______
AC = ______
X = ______
AB = ______
BC = ______
AC = ______
3 x − 31
7.
x+6
A
B
C
5 ( 2x + 2)
8.
3 ( 3x − 1)
A
B
C
X = ______
AB = ______
BC = ______
AC = ______
Suppose that PR = 47. Solve for x and find the length of segments PQ and QR.
9.
4x −1
3x − 1
P
Q
R
X = ______
PQ = ______
10.
2x + 7
2x
P
Q
R
X = ______
PQ = ______
11.
2 x − 13
x
P
Q
R
12.
P
5
x + 11
2
Q
R
QR = ______
X = ______
PQ = ______
5
x + 11
2
QR = ______
QR = ______
X = ______
PQ = ______
QR = ______
Ms. Menard
Geometry
Name: __________________
Segment Addition
Write the Segment Addition Postulate for the points described.
1. S is between D and P
2. J is between S and H
If DS = 4 and SP = 5, then DP = ____.
3. C is between Q and R
If SJ = 5 and SH = 12, then JH = ____
4. T is between M and N
If QC = 2x, RC = 3x, and QR = 15,
what is x? ___________
If NT = x + 5, MN = 3x, and MT = 7,
what is MN? __________
In the diagram of collinear points, GK = 24, HJ = 10, and GH = HI = IJ.
Find each length.
5. HI __________
6. IJ ___________
7. GH _______________
8. JK __________
9. IG ___________
10. IK ________________
Find QR in the following problems.
11. If RS = 44.6 and SQ = 68.4, find QR.
12. If RS = 33.4 and SQ = 80, find QR.
In the following problems, suppose J is between H and K.
♦ Use the Segment Addition Postulate to solve for x.
♦ Then find the length of each segment.
13. HJ = 5x
JK = 7x
KH = 96
14. HJ = 2x + 5
JK = 3x – 7
KH = 18
15. HJ = 6x - 5
JK = 4x - 6
KH = 129
16. HJ = 3x + 8
JK = 4x + 6
KH = 28
17. HJ = 4t – 15
JK = 5t – 6
KH = 15
18. HJ = 5p
JK = p
KH = 3
Kuta Software - Infinite Geometry
Name___________________________________
The Distance Formula
Date________________ Period____
Find the distance between each pair of points. Round your answer to the nearest tenth, if necessary.
1)
2)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
3)
2
4
2
4
2
4
x
4)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
5)
x
6)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
7) (−2, 3), (−7, −7)
8) (2, −9), (−1, 4)
9) (5, 9), (−7, −7)
10) (8, 5), (−1, 3)
11) (−10, −7), (−8, 1)
12) (−6, −10), (−2, −10)
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-1-
x
Worksheet by Kuta Software LLC
Find the distance between each pair of points.
13)
14)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
15)
2
4
2
4
2
4
x
16)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
17)
x
18)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
19) (0, −2), (−5, −1)
20) (6, 4), (−5, −1)
21) (3, 8), (9, 10)
22) (10, 1), (9, −4)
23) (−8, 10), (−6, 7)
24) (−5, 6), (8, −4)
x
Critical thinking questions:
25) Name a point that is
2 away from (−1, 5).
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26) Name a point that is between 50 and 60
units away from (7, −2) and state the
distance between the two points.
-2-
Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 1
Name___________________________________
The Distance Formula
Date________________ Period____
Find the distance between each pair of points.
1) (7, 3), (−1, −4)
2) (3, −5), (−3, 0)
3) (6, −7), (3, −5)
4) (5, 1), (5, −6)
5) (5, −8), (−8, 6)
6) (4, 6), (−4, −3)
7) (−7, 0), (−2, −4)
8) (−4, −3), (1, 4)
9) (−2, 2), (−6, −8)
10) (6, 2), (0, −6)
11) (−3, −1), (−4, 0)
12) (−5, 4), (3, 1)
13) (−2, 3), (−1, 7)
14) (8, −5), (−1, −3)
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-1-
Worksheet by Kuta Software LLC
15) (20, −10), (8, 6)
16) (−3, 17), (15, −7)
17) (11, 11), (−13, 8)
18) (10, 19), (−13, 9)
19) (16, −6), (1, 2)
20) (7, −10), (−10, −4)
21) (−6.8, 0.7), (−2.1, −6.2)
22) (−0.6, −0.455), (1.77, −5.3)
23) (−7.5, 1.1), (−4.1, −1.9)
24) (−7.487, 1.8), (−3.1, −1.2)
25)
(
7, 5 3 ), (−6 7, − 3 )
27) (− 2, − 2 ),
(
2, 6 2 )
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26)
(
6, −6 5 ), (2 6,
28)
(
2, −7 3 ), (4 2, 8 3 )
-2-
5)
Worksheet by Kuta Software LLC
The Pythagorean Theorem, the Distance Formula, and Slope
HSA Practice 1
1. A diagram of a baseball field is shown below. The infield is a square that measures 90
feet on each side. (HSA 2001 Public Release Question)
90 ft
30 ft
Note: The figure is not drawn to scale.
A player threw a ball from point P to third base. How far did the player throw the
ball? Round the answer to the nearest foot.
A
79 feet
B
127 feet
C
150 feet
D
210 feet
2. Triangle JKL is shown on the grid below. (HSA 2000 Public Release Question)
J (3, 4)
K (5, 2)
L (-4, -5)
What is the length of KL ? Round the answer to the nearest tenth of a unit.
F
3.2 units
G
4.0 units
H
11.4 units
J
11.7 units
HSA Geometry Activities
Page 46
Activity 2
Page 11
The Pythagorean Theorem, the Distance Formula, and Slope
HSA Practice 1 (Continued)
3. A lake is shown below. An island is located at (4, 5). A boat travels in a straight line
from (2, 0) to the island. (HSA 2001 Public Release Question)
How far does the boat travel? Round the answer to the nearest tenth of a unit.
A
3.3 units
B
3.7 units
C
5.4 units
D
7.8 units
4. An architect is designing a ramp for delivery trucks. A drawing of the ramp is shown on
the grid below. (HSA 2001 Public Release Question)
What is the slope of the ramp?
F
H
−
7
2
G
2
7
J
HSA Geometry Activities
Page 47
−
2
7
7
2
Activity 2
Page 12
The Pythagorean Theorem, the Distance Formula, and Slope
HSA Practice 2
Determine the distance between the two points, using both the Pythagorean Theorem and the
distance formula.
B
A
Pythagorean Theorem
Distance Formula
Which method do you think is better to determine the distance between two points? Use
mathematics to justify your answer.
Determine the slope between the two points.
HSA Geometry Activities
Page 48
Activity 2
Page 13
Kuta Software - Infinite Geometry
Name___________________________________
The Midpoint Formula
Date________________ Period____
Find the midpoint of each line segment.
1)
2)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
3)
2
4
2
4
2
4
2
4
x
4)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
5)
x
6)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
7)
x
8)
y
−4
y
4
4
2
2
−2
2
4
−4
x
−2
−2
−2
−4
−4
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-1-
x
Worksheet by Kuta Software LLC
Find the midpoint of the line segment with the given endpoints.
9) (−4, 4), (5, −1)
10) (−1, −6), (−6, 5)
11) (2, 4), (1, −3)
12) (−4, 4), (−2, 2)
13) (5, 2), (−4, −3)
14) (−1, 1), (5, −5)
15) (2, −1), (−6, 0)
16) (−3.1, −2.8), (−4.92, −3.3)
17) (−5.1, −2), (1.4, 1.7)
18) (4.9, −1.3), (−5.2, −0.6)
19) (5.1, 5.71), (6, 3.6)
20) (3.1, −2.1), (−0.52, −0.6)
Find the other endpoint of the line segment with the given endpoint and midpoint.
21) Endpoint: (−1, 9), midpoint: (−9, −10)
22) Endpoint: (2, 5), midpoint: (5, 1)
23) Endpoint: (5, 2), midpoint: (−10, −2)
24) Endpoint: (9, −10), midpoint: (4, 8)
25) Endpoint: (−9, 7), midpoint: (10, −3)
26) Endpoint: (−6, 4), midpoint: (4, 8)
Critical thinking questions:
27) Find the point that is one-fourth of the way
from (2, 4) to (10, 8).
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28) One endpoint of a line segment is (8, −1).
The point (5, −2) is one-third of the way
from that endpoint to the other endpoint.
Find the other endpoint.
-2-
Worksheet by Kuta Software LLC

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