Download 1 - CamarenMath

File: GeomA 2011 Answers 8.10.11
No Calculators:
Updated: August 10, 2011
(1.02)
5) If B is between A and C, AB = 7 and AC = 22,
what is the value of BC?
7
BC
Use the following graph to answer the following questions
1) If B is between A and C, AB = 9 and AC = 27,
what is the value of BC?
9
BC
27
AB + BC = AC
9 + BC = 27
-9
-9
BC = 18
2) If B is between A and C, AB = 10 and AC = 28,
what is the value of BC?
10
BC
22
AB + BC = AC
7 + BC = 22
-7
-7
BC = 15
6) If B is between A and C, AB = 6x and BC = 8x,
and AC = 42,
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
6x
8x
42
28
AB + BC = AC
10 + BC = 28
-10
-10
BC = 18
3) If B is between A and C, AB = 8 and AC = 21,
what is the value of BC?
8
BC
21
AB
6x
+
+
BC = AC
8x = 42
14x
= 42
14
14
x
= 3
AB = 6(x)
= 6(3)
= 18
6
2
BC = 8(x)
= 8(3) (Substitution Property of Equality)
= 24
7) If B is between A and C, AB = 2x and BC = 4x,
and AC = 42, what is the value of x?
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
2x
4x
AB + BC = AC
8 + BC = 21
-8
-8
BC = 13
4) If B is between A and C, AB = 9 and AC = 24,
what is the value of BC?
9
BC
24
AB + BC = AC
9 + BC = 24
-9
-9
BC = 15
42
AB
2x
+
+
BC = AC
4x = 42
6x
= 42
6
6
x
= 7
AB = 2(x)
= 2(7)
= 14
BC = 4(x)
= 4(7) (Substitution Property of Equality)
= 28
8) If B is between A and C, AB = 5x and BC = 3x,
and AC = 32,
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
5x
3x
11) If B is between A and C, AB= 5x + 4, BC = 7x - 3,
and AC = 20x - 7, find the value of x, AB, BC, and
AC?
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
(d) What is the value of AC?
7x – 3
5x + 4
32
AB
5x
+
+
BC =
3x =
8x
=
8
x
=
AB = 5(x) BC =
= 5(4)
=
= 20
=
AC
32
32
8
4
3(x)
3(4) (Substitution Property)
12
9) If B is between A and C, AB = 3x and BC = 4x,
and AC = 49,
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
3x
4x
20x – 7
AB
+
BC = AC
(5x + 4) + (7x – 3) = 20x – 7
12x + 1
= 20x – 7
-12x
-12x
+7
+7
8
= 8x
8
8
1 = x
x = 1
AB = 5(x) + 4
= 5(1) + 4
= 9
BC = 7(x) – 3
= 7(1) – 3 (Substitution Property)
= 4
AC = 20(x) – 7
= 20(1) – 7
= 13
49
AB
3x
+
+
BC = AC
4x = 49
7x
= 49
7
7
x
= 7
AB = 3(x)
BC = 4(x)
= 3(7)
= 4(7)
= 21
= 28
10) If B is between A and C, AB = 8x and BC = 3x,
and AC = 33, what is the value of AB?
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
8x
3x
AC = AB +
= 9 +
= 13
BC
4
12) If B is between A and C, AB = 3x – 1, BC = 2x + 4,
and AC = 38,
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
3x – 1
2x + 4
38
AB
+
BC =
(3x – 1) + (2x + 4) =
5x + 3
=
-3
5x
= 35
5
5
x
33
AB
+
BC = AC
8x
+
3x = 33
11x
= 33
11
11
x
= 3
AB = 8(x)
BC = 3(x)
= 8(3)
= 3(3) (Substitution Property)
= 24
= 9
or
AC
38
38
-3
= 7
AB = 3(x) – 1
= 3(7) – 1
= 20
BC = 2(x) + 4
= 2(7) + 4 (Substitution Property)
= 18
13) If B is between A and C, AB = 2x – 1, BC = 3x + 5,
and AC = 24,
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
2x – 1
(1.08)
Use the following picture to answer the questions.
3x + 5
24
AB
+
BC =
(2x – 1) + (3x + 5) =
5x + 4
=
-4
5x
= 20
5
5
x
15) If the measure of angle DBC is 39 degrees and
the measure of angle ABC is 62 degrees. Find the
measure of angle ABD.
AC
24
24
-4
= 4
AB = 2(x) – 1
= 2(4) – 1
= 7
BC
= 3(x) + 5
= 3(4) + 5
= 17
14) If B is between A and C, AB = 10x - 1, BC = 8x + 5
and AC = 17x + 7.
(a) What is the value of x?
(b) What is the value of AB?
(c) What is the value of BC?
(d) What is the value of AC?
(e) What term best describes point B? Why?
10x – 1
L ABD + L DBC = L ABC
L ABD + 39 = 62
-39
-39
L ABD
= 23
16) If the measure of angle DBC is 42 and the
measure of angle ABC is 88 degrees. Find the
measure of angle ABD.
8x + 5
17x + 7
AB
+
BC = AC
(10x – 1) + (8x + 5) = 17x + 7
18x + 4
= 17x + 7
-17x
-17x
-4
-4
x
=
AB = 10(x) – 1
= 10(3) – 1
= 29
AC = 17(x) – 1
= 17(3) – 1
= 58
3
BC
or
= 8(x) + 5
= 8(3) + 5
= 29
L ABD + L DBC = L ABC
L ABD + 42 = 88
-42
-42
L ABD
= 46
17) If the measure of angle DBC is 16 and the
measure of angle ABC is 89 degrees. Find the
measure of angle ABD.
AC = AB + BC
= 29 + 29
= 58
Point B is therefore the midpoint of AC
L ABD + L DBC = L ABC
L ABD + 16 = 89
-16
-16
L ABD
= 73
18) If the measure of angle ABD is 36 and the
measure of angle ABC is 74 degrees. Find the
measure of angle DBC.
L ABD + L DBC = L ABC
36 + L DBC = 74
-36
-36
L DBC = 38
19) If the measure of angle ABD is represented by 2x,
the measure of angle DBC is represented by 5x
and the measure of angle ABC is 91 degrees,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
21) If the measure of angle ABC = 60; the measure of
angle ABD is represented by 5x + 7 and the
measure of angle DBC is represented by 9x – 3,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
L ABD + L DBC = L ABC
5x + 7 + 9x – 3 = 60
14x + 4
= 60
-4
-4
14x
=
56 8
14
14 2
x
=
L DBC = 9(x) – 3
= 9(4) – 3
= 33
L ABD = 5(x) + 7
= 5(4) + 7
= 27
L ABD + L DBC = L ABC
2x + 5x
= 91
7x
= 91
7
7
x = 13
L ABD = 2(x)
L DBC = 5(x)
= 2(13)
= 5(13)
= 26
= 65
20) If the measure of angle ABD is represented by 4x,
the measure of angle DBC is represented by 3x
and the measure of angle ABC is 77 degrees,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
4
22) If the measure of angle ABC = 57; the measure
of angle ABD is represented by 2x – 1 and the
measure of angle DBC is represented by 5x + 2,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
L ABD + L DBC = L ABC
2x – 1 + 5x + 2 = 57
7x + 1
= 57
-1
-1
7x
= 56
7
7
x
L ABD + L DBC = L ABC
4x + 3x
= 77
7x
= 77
7
7
x = 11
L ABD = 4(x)
L DBC = 3(x)
= 4(11)
= 3(11)
= 44
= 33
L ABD = 2(x) – 1
= 2(8) – 1
= 15
=
8
L DBC = 5(x) + 2
= 5(8) + 2
= 42
23) If ray BD bisects angle ABC; the measure of angle
ABD is represented by 7x – 2 and the measure of
angle DBC is represented by 5x + 10,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
(d) What is the measure of angle ABC?
2 (L ABD)
2(4x – 4)
8x – 8
-7x
+8
x
L ABD
= L DBC
7x – 2 = 5x + 10
-5x
- 5x
+2
+2
2x
2
x
=
=
12
2
6
L ABD = L DBC = 7(x) – 2 or
= 7(6) – 2
= 40
= 5(x) + 10
= 5(6) + 17
= 40
L ABC = 2(L ABD) or = 2(L DBC)
= 2(40)
= 80
24) If ray BD bisects angle ABC; the measure of
angle ABD is represented by 7x + 8 and the
measure of angle DBC is represented by 9x – 2,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
(d) What is the measure of angle ABC?
L ABD
= L DBC
7x + 8 = 9x – 2
-7x
- 7x
+2
+2
= L ABC
= 7x + 4
= 7x + 4
- 7x
+8
=
12
L ABD = L DBC = 4(x) – 4
= 4(12) – 4
= 44
L ABC = 2(L ABD) or = 7(x) + 4
= 2(44)
= 7(12) + 4
= 88
= 88
26) If the measure of angle ABD is represented by 6x + 1;
the measure of angle DBC is represented by 4x – 7
and the measure of angle ABC is 9x + 3,
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
(d) What is the measure of angle ABC?
L ABD + L DBC = L ABC
6x + 1 + 4x – 7 = 9x + 3
10x – 6
= 9x + 3
-9x
- 9x
+6
+6
10 = 2x
2
2
x = 5
L ABD = L DBC = 7(x) + 8 or
= 7(5) + 8
= 43
25) If ray BD bisects angle ABC; the measure of angle
ABD is represented by 4x – 4 and the measure of
angle ABC is represented by 7x + 4.
(a) What is the value of x?
(b) What is the measure of angle ABD?
(c) What is the measure of angle DBC?
(d) What is the measure of angle ABC?
x
= 9(x) – 2
= 9(5) – 2
= 43
L ABC = 2(L ABD) or = 2(L DBC)
= 2(43)
= 86
L ABD = 6(x) + 1
= 6(9) + 1
= 55
=
9
L DBC = 4(x) – 7
= 4(9) – 7
= 29
L ABC = 9(x) + 3 or L ABC = AB + BC
= 9(9) + 3
= 55 + 29
= 84
= 84
(1.05)
27) Find the exact distance between the points (3,0)
and (-1, 5).
31) Find the exact distance between the points (-2, 0)
and (5, 3).
32) Find the midpoint between the points (3, 0) and (-1, -4).
Midpoint = (Average of X’s, Average of Y’s)
(X, Y)
= (X1 + X2), (Y1 + Y2)
_________________
2
= (3) + (-1),
2
28) Find the exact distance between the points (-1,4)
and (-2, -9).
=
2
2
,
=
(1, -2)
_________________
2
(0) + (-4)
2
-4
2
33) Find the midpoint between the points (-1, 4) and (-2, -9).
Midpoint = (Average of X’s, Average of Y’s)
(X, Y)
= (X1 + X2), (Y1 + Y2)
_________________
2
= (-1) + (-2),
2
=
29) Find the exact distance between the points (2, -4)
and (5, -1).
=
_________________
2
(4) + (-9)
2
-3 ,
-5
2
2
(-1 ½, -2 ½)
34) Find the midpoint between the points (2, -4) and (5, -2).
Midpoint = (Average of X’s, Average of Y’s)
(X, Y)
= (X1 + X2), (Y1 + Y2)
_________________
2
(-4) + (-2)
2
=
-6
2
=
30) Find the exact distance between the points (4, -3)
and (-6, 2).
_________________
2
= (2) + (5),
2
7 ,
2
(3 ½, -3)
35) Find the midpoint between the points (4, -3) and (-6, 2).
Midpoint = (Average of X’s, Average of Y’s)
(X, Y)
= (X1 + X2), (Y1 + Y2)
_________________
2
= (4) + (-6),
2
=
-2 ,
2
=
(-1, - ½ )
_________________
2
(-3) + (2)
2
-1
2
36) Find the midpoint between the points (-2, 0) and (5, 3).
Midpoint = (Average of X’s, Average of Y’s)
(X, Y)
=
(X1 + X2),
(Y1 + Y2)
_________________
_________________
2
= (-2) + (5),
2
=
3 ,
2
2
(0) + (3)
2
3
2
=
40) M is the midpoint of segment AB. Find the coordinates
of B given A = (-1, 4) and M = (3, -2).
Fast Way: Since M is the midpoint then the distance
from A to M is the same distance from M to B.
(1 ½ , 1 ½)
Use the following graph to answer the following questions
Bx = 3 + 4
=7
By = -2 - 6
= -8
B = (7, -8)
37) M is the midpoint of segment AB. Find the coordinates
of B given A = (2, 1) and M = (5, -2).
41) M is the midpoint of segment AB. Find the coordinates
of B given A = (-5, 1) and M = (-2, 4).
Fast Way: Since M is the midpoint then the distance
from A to M is the same distance from M to B.
Fast Way: Since M is the midpoint then the distance
from A to M is the same distance from M to B.
Bx = 5 + 3
=8
Bx = -2 + 3
=1
By = -2 - 3
= -5
B = (1, 7)
B = (8, -5)
38) M is the midpoint of segment AB. Find the coordinates
of B given A = (-1, 3) and M = (-3, 5).
Fast Way: Since M is the midpoint then the distance
from A to M is the same distance from M to B.
Bx = -3 - 2
= -5
By = 5 + 2
=7
B = (-5, 7)
39) M is the midpoint of segment AB. Find the coordinates
of B given A(0, 3) and M(-3, 7).
Fast Way: Since M is the midpoint then the distance
from A to M is the same distance from M to B.
Bx = -3 - 3
= -6
B = (-6, 11)
By = 4 + 3
=7
By = 7 + 4
= 11
42) If M is the midpoint of segment AB, AM = 4x and
AB = 7x + 9.
(a) What is the value of x?
(b) What is the value of AM?
(c) What is the value of MB?
(d) What is the value of AB?
4x
7x + 9
Since M is the midpoint then,
MB = AM and
2(AM) = AB
2(4x) = 7x + 9
AM = MB = 4(x)
8x = 7x + 9
= 4(9)
-7x -7x
= 36
x =9
AB = 2(AM) or = 7(x) + 9
=2 (36)
= 7(9) + 9
= 72
= 72
(1.07)
43) Given the following choices for angle ABC above:
(Interior, Exterior, Vertex, and On the angle), state
where each point lies on the angle.
A On the angle
B Vertex
C On the angle
(D Interior of the angle
E Exterior of the angle
44) Describes the measure of each of the following angles.
(a) Acute An angle measure < 90 degrees.
(b) Right
An angle measure = 90 degrees.
(c) Obtuse An angle measure > 90 degrees.
(d) Straight An angle measure = 180 degrees.
47) Given the following information:
L 1 = 6y + 29, L 2 = 4x + 31, and L 4 = 6x - 11
(a) Solve for the value of x?
(b) What is the measure of angles 1 and 2?
(c) What is the measure of angles 3 and 4?
(d) Solve for the value of y?
L2 = L4
4x + 31 = 6x – 11
- 4x
- 4x
+ 11
+ 11
(Vertical Opposites are
)
42 = 2x
2
2
x = 21
L 2 = L 4 = 4(x) + 31 or = 6(x) – 11
= 4(21) + 31
= 6(21) – 11 (Substitution Property)
= 115
= 115
L 1 is supplementary to = L 2
(or L 4)
L 1 + L 2 = 180
(Linear Pair Ls are supplementary)
L 1 + 115 = 180
(Substitution Property)
- 115 - 115
L1
= 65
If
L 1 = 65 and L 1 = 6y + 29
then 6y + 29 = 65
(Transitive Property of Equality)
– 29 – 29
6y
= 36
6
6
y = 6
45) Given the figure above
(a) State two straight angles
ABC, BED
(b) State two acute angles
ADB, ADC
(c) State two obtuse angles
ABD, CED
(d) State one right angle
ACD
(e) In which region of angle ADC does point E lie? Interior
(f) In which region of angle CDB does point A lie? Exterior
(1.09)
Use the following picture to answer the following questions
48) Given the following information:
L 1 = 6y + 14, L 2 = 4x + 30, and L 3 = 12y - 34
(a) Solve for the value of y?
(b) What is the measure of angles 1 and 2?
(c) What is the measure of angles 3 and 4?
(d) Solve for the value of x?
L1 = L3
(Vertical Opposites are )
6y + 14 = 12y – 34
- 6y
- 6y
+ 34
+ 34
48 = 6y
6
6
y = 8
L 1 = L 3 = 6(y) + 14 or = 12(y) – 34
= 6(8) + 14
= 12(8) – 34 (Substitution Property)
= 62
= 62
46) Given the following choices for the figure above:
(Vertical Opposites, Linear Pair, Supplementary,
Complementary and Right angle), state all choices that
apply to the following:.
(a) Angles 3 & 4
Linear Pair & Supplementary
(b) Angles 1 & 3
Vertical Opposites
(c) Angle 5
Right angle
(d) Angles 1 & 4
Linear Pair & Supplementary
(e) Angles 2 & 4
Vertical Opposites
L 1 is supplementary to = L 2
(or L 4)
L 1 + L 2 = 180
(Linear Pair Ls are supplementary)
62 + L 2 = 180
(Substitution Property)
- 62
- 62
L 2 = 118
If
L 2 = 118 and L 2 = 4x + 30
then 4x + 30 = 118
(Transitive Property of Equality)
– 30
– 30
4x = 88
4
4
x = 22
(1.11)
(2.10 – but belongs here in Section 2.01)
Use the following picture to answer the following questions
49)
Find the measures (in degrees) of the following
angles:
(a) angle AFB
45
(b) angle EFB
135
(c) angle AFC
80
(d) angle EFD
40
(e) angle AFD
140
(2.01)
Use the following picture to answer the questions in
this section.
Use the following picture to answer the following questions
53) Which of the following can be used to prove line m is
parallel to line n? (If “Yes” state relationship how.)
Yes Corresponding are 
(a) angles 4 & 8
No
(b) angles 7 & 8
Yes Same side interior angles are
(c) angles 3 & 5
supplementary
(d) angles 1 & 4
(e) angles 2 & 7
(f) angles 1 & 7
No
Yes
Yes
angles 5 & 6
angles 4 & 5
angles 1 & 5
angles 2 & 4
No
Yes
Yes
No
(g)
(h)
(i)
(j)
Alternate-ext. angles are supplementary
Same side exterior angles are
supplementary
Alternate-int. angles are supplementary
Corresponding are

(2.02)
54) In the picture below if BC is parallel to line EF,
(a) What is the value of x?
(b) What is the measure of angle ABC?
(c) What is the measure of angle BED?
(d) Label all the angle measurements.
50)
Complete the table below using the picture above.
Angle
L1
L2
L3
L4
L5
L6
L7
L8
51)
Vertical
Opposite
6
5
8
7
2
1
4
3
Corresponding
angle
3
4
1
2
7
8
5
6
Using the picture above describe the pair of angles
indicated in the table below.
Angles
L1&L4
L2&L3
L1&L8
L2&L7
L3&L6
L4&L5
L5&L8
L6&L7
52)
Linear
Pairs
2&5
1&6
4&7
3&8
1&6
2&5
3&8
4&7
Description
Same side exterior angles
Same side interior angles
Alternate exterior angles
Alternate interior angles
Alternate interior angles
Alternate exterior angles
Same side exterior angles
Same side interior angles
(a) a || c is read as?
(b) a  c is read as?
a is parallel to c
a is perpendicular to c
L ABC = L BEF (Corresponding angles are )
3x – 12 = 2x + 12
- 2x
- 2x
+ 12
+ 12
x
=
24
L ABC = L BEF = 3(x) – 12 or = 2(x) + 12
= 3(24) – 12 or = 2(24) + 12 (Substitution)
= 60
= 60
L BED + L BEF = 180 (Linear Pairs are supplementary)
L BED + 60
= 180
(Substitution Property)
- 60
- 60
L BED
= 120
55) In the picture below if BC is parallel to line EF,
(a) What is the value of x?
(b) What is the measure of angle BCF?
(c) What is the measure of angle GFH?
(d) Label all the angle measurements.
4x + 7 + 7x – 14
11x – 7 =
+ 7
11x
=
11
x
=
= 180
180
+7
187
11
17
(Corresponding & Linear Pair)
One Example for finding all angles
L ACD = 4(x) + 7
= 4(17) + 7 (Substitution Property)
= 75
L BCF = L EFH
(Corresponding angles are )
3x + 1 + 7x – 11 = 180 (Linear Pair Ls are supplementary)
10x – 10 = 180 (Combine like terms)
+ 10
+ 10
10x
10
x
L ACD + L DCF =
75 + L DCF =
- 75
L DCF =
180 (Linear Pairs are supplementary)
180
- 75
105
= 190
10
=
19
L BCF = L EFH = 3(x) + 1
= 3(19) + 1 (Substitution Property)
= 58
L BCF + L GFH = 180 (Linear Pairs are supplementary)
58 + L GFH = 180 or L GFH = 7(x) – 11
- 58 + L GFH = - 58
= 7(19) – 11
L GFH = 122
= 122
56) In the picture below if BD is parallel to line EG,
(a) What is the value of x?
(b) What is the measure of angle ACD?
(c) What is the measure of angle DCF?
(d) Label all the angle measurements.
7x – 14 = 6x + 3
- 6x
- 6x
+ 14
+ 14
x
=
17
(Vertical Opposites are
)
or
4x + 7 + 6x + 3 = 180 (Corresponding & Linear Pair)
10x + 10 = 180
– 10 – 10
10x
= 170
10
10
x
= 17
or (Continued on next page)
(2.03)
57) (a) If the slope of a line is -2/3, what is the slope of a
line that is: (i) parallel; and (ii) perpendicular to it?
(No Work needs to be shown):
(i) A parallel slope would be the same: = – 2/3
(ii) A perpendicular slope would be the negative
reciprocal of the original: = + 3/2
(b) If the slope of a line is 1/3, what is the slope of a line
that is: (i) parallel; and (ii) perpendicular to it?
(i) A parallel slope would be the same: = 1/3
(ii) A perpendicular slope would be the negative
reciprocal of the original: = – 3/1 = – 3
(c) If the slope of a line is undefined, what is the slope
of a line that is: perpendicular to it?
An undefined slope means that the denominator is 0
(Since it is not possible to divide by 0. A perpendicular
slope would be the negative reciprocal, so the
numerator would be 0). Thus the slope would = 0
(Since 0 divided by a number still equals 0)
58) (a) What is the slope of the line that passes through the
points (-3,1) and (0, -5)?
Slope = (y1) – (y2)
(x1) – (x2)
= (1) – ( - 5)
(- 3) – (0)
= 6
-3
= -2
(b) What is the slope of the line that passes through the
points (2, -1) and (2, 0)?
Slope = (y1) – (y2)
(x1) – (x2)
= (-1) – (0)
(2) – (2)
= -1
0
= Undefined (Not Possible to divide by 0)
59) (a) What is the slope of the line that passes through the
points (3, -2) and (-1, 0)?
Slope = (y1) – (y2)
(x1) – (x2)
= ( - 2) – (0)
(3) – ( - 1)
= -2
4
= -1
2
(b) What is the slope of the line that passes through the
points (3, -1) and (-1, -1)?
Slope = (y1) – (y2)
(x1) – (x2)
= ( - 1) – ( - 1)
(3) – ( - 1)
= 0
4
= 0 (0 divided by a number still equals 0)
60) A line contains the points (3, -1) and (-1, 2). Another
line graphed in the same coordinate plane contains
the points (2,0) and (-2,3). How do these two lines
relate to each other?
Slope = (y1) – (y2)
= (y1) – (y2)
(x1) – (x2)
(x1) – (x2)
= ( - 1) – (2)
= ( 0) – (3)
(3) – ( - 1)
(2) – ( - 2)
= -3
= -3
4
4
Therefore the two lines are parallel,
since they have the same slope.
(4 pts 2.04)
63) Write the equation 3x – y = 5 in slope-intercept form.
- 3x + y = - 5 (Make y positive by multiplying by -1)
+3x
+ 3x (Eventually get in the form: y = mx + b)
y = 3x – 5
64) Write the equation 8x – 2y = 10 in slope-intercept form.
- 8x + 2y = - 10 (Make y positive by multiplying by -1)
+ 8x
+ 8x (Eventually get in the form: y = mx + b)
2y = 8x – 10
2y = 8x – 10
2
2
y = 4x – 5
65) Write the equation -4x + y = -3 in slope-intercept form.
- 4x + y = - 3
+4x
+ 4x (Eventually get in the form: y = mx + b)
y = 4x – 3
66) Write the equation 2x - 3y = 6 in slope-intercept form.
-2x + 3y = - 6 (Make y positive by multiplying by -1)
+2x
+2x (Eventually get in the form: y = mx + b)
3y = 2x – 6
3y = 2x – 6
3
3
y = 2/3x – 2
67) Graph and label each of the following. Plus
indicate two points for each graph.
(a) x = 3
(b) y = 4
(c) y = 3/4 x – 3
(d) y = - 1/2 x – 2
61) A line contains the points (0, -2) and (1, 3). Another
line graphed in the same coordinate plane contains
the points (3, -1) and (-2, 0). How do these two lines
relate to each other?
Slope = (y1) – (y2)
= (y1) – (y2)
(x1) – (x2)
(x1) – (x2)
= ( - 2) – (3)
= ( - 1) – (0)
(0) – ( 1)
(3) – ( - 2)
= -5
= -1
-1
5
= 5
Therefore the two lines are perpendicular,
since the slopes are the negative reciprocal
of each other.
62) A line contains the points (0, -1) and (-1, 2). Another
line graphed in the same coordinate plane contains
the points (2, 0) and (-2, 3). How do these two lines
relate to each other?
Slope = (y1) – (y2)
= (y1) – (y2)
(x1) – (x2)
(x1) – (x2)
= ( - 1) – (2)
= (0) – (3)
(0) – ( - 1)
(2) – ( - 2)
= -3
= -3
1
4
Therefore the two lines will intersect, but
they are neither perpendicular nor parallel.
Graph
(a) x = 3
(b) y = 4
(c) y = 3/4x – 3
(d) y = - 1/2x – 2
Point 1
(3,0)
(0, 4)
(0, -3)
(0, -2)
Point 2
(3, 2)
(2, 4)
(4, 0)
(2, -3)
68) In slope-intercept form what is the equation of the
following lines below?
(a)
y = -2x + 2
3
(b)
y = 4x – 2
3
(2.05)
69) In slope-intercept form write the equation of a line
with a slope of -2 that passes through (-1, 3).
Solve both (a) mathematically and (b) graphically.
(a) Mathematical Method
y = mx + b
(3) = (- 2)(- 1) + b
3 = 2+ b
-2
-2
b= 1
y = - 2x + 1
(b) Graphic Method
71) In slope-intercept form write the equation of a line
that passes through (0, -2) and (3, 4)?
Solve both (a) mathematically and (b) graphically.
(a) Mathematical Method
Slope (m) = Y
X
y = mx + b
m = (y1) – (y2)
(4) = (2)(3) + b
(x1) – (x2)
4 = 6 + b
= ( - 2) – (4)
-6
-6
_
(0) – ( 3) - 2 = b
= -6
-3
y = mx + b
m= 2
y = 2x – 2
(b) Graphic Method
72) In slope-intercept form write the equation of a line
that passes through (-4, 5) and (4, 1)?
(a) Mathematical Method
Slope (m) = Y
X
y = mx + b
m = (y1) – (y2)
(5) = (-½ )(-4) + b
(x1) – (x2)
5 = 2+ b
= (5) – ( 1)
-2 -2
_
(- 4) – ( 4)
3 = b
= 4
-8
y = mx + b
m=-½
(b) Graphic Method
70) Write the equation of a line in slope-intercept form:
(a) with an undefined slope passing through (2, -4).
Since slope = Y
X
Undefined = Not Possible to divide by 0.
So there is no difference in the X’s, so X is always 2.
Example (2,0), (2, 4). Thus: X = 2
(b) with a slope = 0 and passing through (-3, 5).
Since slope = Y
X
Slope = 0 (0 divided by a number = 0)
So there is no difference in the Y’s, so Y is always 5.
Example (0,5), (4, 5). Thus: Y = 5
y= -½x + 3
73) Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
4x - 2y = 8 and contains the point (-6, 2).
Solve both: (a) mathematically and (b) graphically.
(a) Mathematical Method
- 4x + 2y = - 8 (Make y positive by multiplying by -1)
+ 4x
+ 4x (Eventually get in the form: y = mx + b)
2y = 4x – 8
2
2
y = 2x – 4
Parallel Slope Perpendicular Slope
y = 2x + b
y = - ½ x + b [Substitute in (–6,2)]
(2) = 2(-6) + b (2) = - ½ (-6) + b
2 = - 12 + b
2 = 3 + b
+ 12 + 12
_ -3
-3
_
14 = b
-1 = b
y = 2x + 14
y =(b) Graphic Method
½x – 1
74) Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
3x - y = - 2 and contains the point (1, 3).
Solve mathematically only
- 3x + y = 2 (Make y positive by multiplying by -1)
+ 3x
+ 3x (Eventually get in the form: y = mx + b)
y = 3x + 2
Parallel Slope
y = 3x + b
(3) = 3(1) + b
3= 3 + b
-3 -3
_
0 = b
y = 3x + 0
y = 3x
Perpendicular Slope
y = - 1/3 x + b [Substitute in (1,3)]
(3) = - 1/3(1) + b
3 =
- 1/3 + b
+ 1/3 _
+ 1/3
3 1/3 = b
y = - 1/3 x + 3 1/3
75) Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
2x + 3y = 9 and contains the point (6, 4).
Solve mathematically only
2x + 3y = 9 (Make y positive by multiplying by -1)
- 2x
- 2x (Eventually get in the form: y = mx + b)
3y = - 2x + 9
3
3
y =
- 2/3 x + 3
Parallel Slope Perpendicular Slope
y = - 2/3 x + b
y = 3/2 x + b [Substitute in (6,4)]
(4) = - 2/3(6) + b
4= -4 + b
+ 4 + 4
_
8 = b
y =
- 2 /3 x + 8
(4) = 3/2 (6) + b
4 = 9 + b
-9
-9
_
-5 = b
y = 3 /2 x – 5
(4 pts 2.08)
76) Match (i –v) with the correct property.
(a)
(b)
(c)
(d)
(e)
Distributive Property for Real Numbers
Reflexive Property of Equality
Substitution Property of Equality
Symmetric Property of Equality
Transitive Property of Equality
___
___
___
___
___
i)
ii)
iii)
iv)
v)
If x = -6, then -6 = x
-7x = -7x
If x = -2, then x + 5 = -2 + 5
If x = -2 and -2 = 1 - 3, then x = 1 - 3
2(x - 3) = 2x - 6
(a)
(b)
(c)
(d)
(e)
Distributive: 2(x - 3) = 2x – 6
Reflexive: -7x = -7x
Substitution: If x = -2, then x + 5 = -2 + 5
Symmetric: If x = -6, then -6 = x
Transitive: If x = -2 and -2 = 1 - 3, then x = 1 – 3
(v)
( ii )
( iii )
(i)
( iv )
(4 pts 3.02)
77) Match (i –vii) with the following: A triangle that contains:
(a) An isosceles triangle is defined as
_____
(b) A scalene triangle is defined as
_____
(c) An obtuse triangle is defined as
_____
(d) An equilateral triangle is defined as
_____
(e) An acute triangle is defined as
_____
(f) A right triangle is defined as
_____
(i) three congruent sides.
(ii) exactly one 90 degree angle
(iii) two congruent sides.
(iv) no congruent sides.
(v) three congruent angles.
(vi) exactly one angle between 90 and 180 degrees
(vii) three angles less than 90 degrees.
(a)
(b)
(c)
(d)
(e)
(f)
An isosceles triangle is defined as
A scalene triangle is defined as
An obtuse triangle is defined as
An equilateral triangle is defined as
An acute triangle is defined as
A right triangle is defined as
( iii )
( iv )
( vi )
(i)
( vii )
( ii )
78) Classify each triangle below by angles then sides.
(a)
(b)
(c)
(d)
(e)
(f)
Right, scalene triangle
Obtuse, scalene triangle
Obtuse, isosceles triangle
Acute, isosceles triangle
Acute, equilateral triangle
Right, isosceles triangle
81) Find the measure of angles b and c, If the measure
of angle a = 85 and the measure of angle d = 3b - 5.
L c + L d = 180 (Linear Pair Ls are supplementary)
L c + 3b - 5 = 180
(Substitution Prop.)
- 3b + 5 - 3b + 5
L c = - 3b + 185
L a + L b + L c = 180
(Sum of L’s for a  = 180)
L 85 + L b – 3b + 185 = 180 (Substitution Prop.)
270 – 2b = 180
(Combine like terms)
- 270
- 270
- 2b =
- 90
-2
-2
L b = 45
L a + L b + L c = 180 (Sum of L’s for a  = 180)
85 + 45 + L c = 180
(Substitution Property)
130 + L c = 180
- 130
- 130
L c = 50
82) Find the measure of all the angles 1-16, if the
measure of angle 2 is 30 degrees. State a reason for
how you determined what each angle is.
(4 pts 3.03)
Use the following picture to answer the following questions
79) Find the measure of angle a, If the measure of angle
b = 35 and the measure of angle d is 120.
L c + L d = 180 (Linear Pair Ls are supplementary)
L c + 120 = 180
(Substitution Property)
- 120 - 120
L c = 60
L a + L b + L c = 180 (Sum of L’s for a  = 180)
L a + 35 + 60 = 180 (Substitution Property)
L a + 95 = 180 (Combine like terms)
- 95
- 95
La
= 85
80) Find the measure of angle a, If the measure of angle
b = 70 and the measure of angle d = 140.
L c + L d = 180 (Linear Pair Ls are supplementary)
L c + 140 = 180
(Substitution Property)
- 140
- 140
L c = 40
L a + L b + L c = 180 (Sum of L’s for a  = 180)
L a + 70 + 40 = 180 (Substitution Property)
L a + 110 = 180 (Combine like terms)
- 110
- 110
La
=
70
L 2 = 30, then
L 7 = 30
L16 = 30
L15 = 150
L 3 = 60
L 6 = 60
L 5 & L12 = 60
L 9 = 60
L 4 = 120
L11 & L13 = 120
L 10 = 120
L1 & L14 = 90
Vertical Opposites
L ‘s 2 & 7 are 

Corresponding L ‘s 7 & 16 are
Linear Pair L ‘s 16 & 15 = 180
L ‘s 2 & 3 = 90
L ‘s 3 & 6 are 

Vertical Opposites L ‘s 5 & 9 are 
Linear Pair L ‘s 4 & 5 = 180
Corresponding L ‘s 4, 11, 13 are 
Vertical Opposites L ‘s 4 & 10 are 
Given a  m = 90 (and m || n )
Vertical Opposites
Corresponding L ‘s 3, 5, 12 are
(3.05)
83) In triangle ABC, angle C is (7x-3) degrees and
segment AB = 3x+5 (cm).
(a) What value of x would make  ABC equilateral?
(b) What is the perimeter of the equilateral triangle?
(a) If ABC is an equilateral triangle, then each angle
would equal 60 degrees.
If:
L c = 60 and L c = 7x – 3
Then: 7x – 3 = 60
(Transitive Property of Equality)
+ 3 + 3
7x
= 63
7
7
x = 9
Perimeter = 3 (side)
= 3 (3x + 5)
= 3 [3(9) + 5]
= 3 (32)
Perimeter = 96 cm
Use the following picture to answer the following questions
84) (a) Find the value of angles x and y, if the measure
of angle a is 100 degrees.
a + b = 180 (Linear Pair L ‘s = 180)
100 + b = 180 (Substitution Property)
- 100
- 100
b = 80
y = 80 (If sides are , then L ‘s opp. are )
L b + L y + L x = 180 (Sum of L’s for a  = 180)
80 + 80 + L x = 180 (Substitution Property)
160 + L x = 180 (Combine like terms)
- 160
- 160
L x = 20
(b) Find the value of angles x and y, if the measure
of angle a is 125 degrees.
a + b = 180 (Linear Pair L ‘s = 180)
125 + b = 180 (Substitution Property)
- 125
- 125
b = 55
y = 55 (If sides are , then L ‘s opp. are )
L b + L y + L x = 180
55 + 55 + L x = 180
110 + L x = 180
- 110
- 110
L x = 70
Use the following picture to answer the following questions.
(This is not drawn to scale)
85) Find the value of angles b, y and z,
(a) If the measure of angle a is 50 degrees.
L a + L b + L c = 180 (Sum of L’s for a  = 180)
La=Lc
(If sides are , then L ‘s opp. are )
50 + L b + 50 = 180 (Substitution Property)
L b + 100 = 180
(Combine like terms)
- 100 - 100
Lb
= 80
L x = 50
(Vertical Opposites L ‘s c & x are )
Ly=Lz
(If sides are , then L ‘s opp. are )
L x + L y + L z = 180 (Sum of L’s for a  = 180)
50 + L y + L y = 180 (Substitution Property)
2y + 50 = 180
- 50 - 50
2y
= 130
2
2
L y = 65
(Combine like terms)
and
L z = 65
(b) If the measure of angle a is 40 degrees.
L a + L b + L c = 180 (Sum of L’s for a  = 180)
La=Lc
(If sides are , then L ‘s opp. are )
40 + L b + 40 = 180 (Substitution Property)
L b + 80 = 180
(Combine like terms)
- 80 - 80
Lb
= 100
L x = 40
(Vertical Opposites L ‘s c & x are )
Ly=Lz
(If sides are , then L ‘s opp. are )
L x + L y + L z = 180 (Sum of L’s for a  = 180)
40 + L y + L y = 180 (Substitution Property)
2y + 40 = 180
- 40 - 40
2y
= 140
2
2
L y = 70
(Combine like terms)
and
L z = 70
(3.06)
86) State a rule for determining the relative lengths of a
triangle given the following information. Which are the
shortest and longest sides of the triangle below?
(Sum of L’s for a  = 180)
(Substitution Property)
(Combine like terms)
The shortest side is always opposite the smallest angle.
Therefore side a is the shortest side. The longest side is
always opposite the largest angle; therefore side c is the
longest side.
(b)
Solve for x, BD and AE
Given: BD = x + 3 and AE = 4x – 2.
If B is the midpoint of AC then CB  BA and
D is the midpoint of CE then CD  DE.
Therefore the ratio of CBD to CAE is 1:2
Thus: (2) BD = AE
2 (x + 3) = (4x – 2) (Substitution Prop)
87) (a) If two sides of a triangle are 6 and 25, what
are the possible length for the third side?
The possible lengths (x)
(25 – 6) < x < (25 + 6)
19 < x < 31
(b) If two sides of a triangle are 4 and 10, what
are the possible length for the third side?
The possible lengths (x)
(10 – 4) < x < (10 + 4)
6 < x
2x + 6 = 4x – 2
- 2x + 2
- 2x + 2
8 = 2X
2
2
x = 4
< 14
(4.01)
Use the following picture to answer the following questions.
BD = x + 3
BD = (4) + 3
BD = 7
AE = 4x – 2
AE = 4(4) – 2 (Substitution Prop)
AE = 14
(4.03)
Use the following picture to answer the following questions
88) Segment CB is congruent to segment BA; and
segment CD is congruent to DE.
(a) Solve BD given that segment AE is 14.
If: CB  BA and CD  DE; then the ratio of CBD
to CAE is 1:2
Therefore: (2) BD = AE
(2) BD = 14 (Substitution Property)
2
2
BD = 7
(b) Solve for x, BD and AE
Given: BD = 4x – 2 and AE = 7x – 1.
If B is the midpoint of AC then CB  BA and
D is the midpoint of CE then CD  DE.
Therefore the ratio of CBD to CAE is 1:2
Thus: (2) BD = AE
2 (4x – 2) = (7x – 1) (Substitution Prop)
8x – 4 = 7x – 1
- 7x + 4 - 7x + 4
x
=
3
BD = 4x – 2
BD = 4(3) – 2
BD = 12 – 2
BD = 10
AE = 7x – 1
AE = 7(3) – 1 (Substitution Prop)
AE = 21 – 1
AE = 20
89) Segment CB is congruent to segment BA; and
segment CD is congruent to DE.
(a) Solve BD given that segment AE is 18.
If: CB  BA and CD  DE; then the ratio of CBD
to CAE is 1:2
Therefore: (2) BD = AE
(2) BD = 18 (Substitution Property)
2
2
BD = 9
90) (a) If L BDC is (4x + 6) and segment BD is the altitude
of triangle ABC; Solve for the value of x.
If BD is the altitude then L BDC = 90. Therefore:
If L BDC = 90 and L BDC = 4x + 6, then:
4x + 6 = 90 (Transitive Property of Equality)
-6 -6
4x
= 84
4
4
x = 21
(b) If D is the midpoint of AC; BA = 13; BC = 2x – 3
and AD = 5; Solve for the value of AC.
If D is the midpoint of AC and AD = 5, then:
2(AD) = AC.
2(5) = AC (Substitution Property)
AC = 10
To solve for x it is not necessary to use the
information: BA = 13; BC = 2x – 3.
(c) If DB bisects L ABC and L CBD = 3x + 4 and
the measure of L ABD = 4x – 8; Solve for the
value of x, L ABC and L CBD.
If DB bisects L ABC, then L ABD  L CBD.
Therefore: 4x – 8 = 3x + 4 (Substitution Prop.)
- 3x + 8
- 3x + 8
x
=
12
L ABD  L CBD = 4x – 8 or 3x + 4
= 4(12) – 8 or 3(12) + 4
= 40
= 40
91) (a) If L BDC is (4x – 2) and segment BD is the altitude
of triangle ABC; Solve for the value of x,
If BD is the altitude then L BDC = 90. Therefore:
If L BDC = 90 and L BDC = 4x – 2, then:
4x – 2 = 90 (Transitive Property of Equality)
+2 +2
4x
= 92
4
4
x = 23
(b) If segment BD bisects AC; AD = 2x – 2 and
AC = 24: Solve for the value of x.
If BD bisects AC and AC = 24, then AD = 12.
Therefore if AD = 12 and AD = 2x – 2, then:
2x – 2 = 12 (Transitive Property of Equality)
+2 +2
2x = 14
2
2
x = 7
4.04 and 4.05
92) A sail boat is 36 feet long and has a mast that is 28.5
feet tall. A scale model of this sailboat is 12 inches long.
How tall is the mast of the model sailboat?
Model
12 in. =
x
_
Actual
36 ft
28.5 ft
x = (12 in) (28.5 ft)
36 ft
x = 9.5 in
93) A sail boat is 45 feet long and has a mast that is 37.5
feet tall. A scale model of this sailboat is 9 inches long.
How tall is the mast of the model sailboat?
Mast
37.5 ft =
x _
Length
45 ft.
9
x = (37.5 7.5 ft) (9 in)
45 5
x = 7.5 in
94) Pete, who is 6 feet in height, stands at the base of the
Eiffel Tower. Pete's shadow at 4.5 feet is far shorter than
the Eiffel Tower's 207 feet shadow. What is the height of
the Eiffel Tower?
6 ft = x
_
4.5 ft
207 ft
x = (6 ft) (207 ft)
4.5 ft
95) Pete, who is 6 feet in height, stands at the base of a
building. Pete's shadow at 2.5 feet is far shorter than the
building's 20 feet shadow. What is the height of the
building?
6 ft = x
2.5 ft
20 ft
_
x = (6 ft) (20 ft)
2.5 ft
x = (60 12 ft) (20 4)
25 5
x = 48 ft
96) A man 5 feet 10 inches tall casts a shadow 7 feet in length.
A nearby building casts a shadow that is 24 feet. What is
the height (in feet) of the building?
5ft 10 in = 5(12) +10 in 7 ft = 84 in
= 70 inches
70 in = x
_
7 ft
24 ft
x = (70 in) (24 ft)
7 ft
x = (70 10) (24 ft)
7
x = 240 20 in
1 ft 1
12 in
x = 20 ft
97) A basket ball player is 7 feet 4 inches tall casts a shadow
8 feet in length. A nearby building casts a shadow that is
36 feet. What is the height (in feet) of the building?
7ft 4 in = 7(12) + 4 in
7 ft = 84 + 4 in
x = 396 in
x = 396 66 in
1
1 ft 12 2in
x = 33 ft
98) A woman 5 feet tall casts a shadow 7 feet in length. A
nearby building casts a shadow that is 18 feet. How tall is
the building exactly?
x = (60 4 ft) (207)
45 3
x = (4 ft) (207 69)
31
x = 276 ft
= 88 inches
88 in = x
_
8 ft
36 ft
x = (88 11 in) (36 ft)
8 ft
5 ft
7 ft
=
x
18 ft
x = (5 ft)(18 ft)
7 ft
x = 90/7
x = 12 6/7 ft
_
99) A man 6 feet tall casts a shadow 5 feet in length. A nearby
building casts a shadow that is 21 feet. How tall is the
building exactly?
6 ft =
x
_
5 ft
21 ft
(4.08)
103)
If SRT is similar WVX. Find SU and ST
x = (6 ft)(21 ft)
5 ft
x = 126/5
_
x = 25 1/5 ft
(4.06)
100)
Find the exact value of x.
a
= 4
_
(a + 6)
8
(8)(a) = (4)(a + 6)
8a
= 4a + 24
- 4a
- 4a
_
4a
=
24
4
4
SU = a
= 6
ST = a + 6 = 12
104)
PUE is similar to PSR. If PU = 6; US = 3;
and PR = 18 and Find PE and ER.
_
ii & Rt
i & Rt
=
x
5
=_ 9 _
x
x2
x2
= (9) (5)
= 45
x2
= 45
101) (a) What is the Geometric Mean of 9 and 16?
Geometric mean is the square root of the
product of the numbers. Therefore:
(a) The Geometric Mean = (3)(4)
= 12
(b) What is the Geometric Mean of 25 and 49?
(b) The Geometric Mean = (5)(7)
= 35
102) (a) What is the Geometric Mean of 16 and 25?
Geometric mean is the square root of the
product of the numbers. Therefore:
(a) The Geometric Mean = (4)(5)
= 20
(b) What is the Geometric Mean of 25 and 36?
(b) The Geometric Mean = (5)(6)
= 30
9(2) = 18
Therefore PE = 6(2)
PE = 12
(4.09)
105)
PE + ER = 18
12 + ER = 18
- 12
- 12
ER = 6
Classify the triangles as acute, right, or obtuse.
(a) A triangle with the sides: 8, 15, 17
172 ? 82 + 152
289 ? 64 + 225
289
= 289 (Therefore a Right )
(b) A triangle with the sides: 8, 10, 13
132 ? 82 + 102
169 ? 64 + 100
169
> 164 (Therefore an Obtuse )
(c) A triangle with the sides: 5, 8, 9
92 ? 52 + 82
81 ? 25 + 64
81
< 89 (Therefore an Acute )
106) Find the missing leg of the following right triangles:
(a) If one leg = 7, and the hypotenuse = 25.
For Rt ’s one can use the Pythagorean theorem.
108) (a) Use the Pythagorean theorem to demonstrate
the relationship for all 30-60-90 Right Triangles.
Example below is using side length of 10.
or
a2 + 72 = 252
a2 + 49 = 625
- 49
- 49
a2
= 576
a = 24
(b) If one leg = 6, and the hypotenuse = 10.
For Rt ’s one can use the Pythagorean theorem.
The side opposite the 30o is the same as half the
hypotenuse. The side opposite the other 60o is the
same as the 30o side multiplied by the square root
of 3.
(b) Use the following picture to find the exact value of:
m, n, p, q, r and s.
or
a2 + b2
a2 + 62
a2 + 36
- 36
a2
a
= c2
= 102
= 100
- 36
= 64
= 8
(4.10)
107) (a) Use the Pythagorean theorem to demonstrate the
relationship for all 45-45-90 Right Triangles.
Example below is using side length of 10.
(4.11)
109)
The side opposite the 45o is the same as the side
opposite the other 45o. The side opposite the 90o
is the same as the 45o side multiplied by the
square root of 2. (If one is finding a leg side, divide
the hypotenuse by square root of 2).
(b) Use the following picture to find the exact value of:
(p,q); (r,s) and (t,u)
Use trigonometric functions to find the exact value of
the sine, cosine and tangent of the indicated 45o
angle shown in the triangle below.
110)
111)
Use trigonometric functions to find the exact value of
the sine, cosine and tangent of the indicated 30o
angle shown in the triangle below.
Which theorem can be used (if any) to prove
AB?
114)
SSS Theorem (common side must be  )
Which theorem can be used (if any) to prove
AB?
115)
Not enough information given to prove 
Which theorem can be used (if any) to prove
AB?
116)
SAS Theorem
Which theorem can be used (if any) to prove
AB?
Find the missing sides and angles of triangle ABC.
Angle A, side AB and side BC.
La+Lb +Lc
a + 65 + 90
a + 155
- 155
a
112)
(3.09)
113)
= 180
= 180
(Sum of L’s for a  = 180) L
= 180
- 155
= 25
sin 65o = Opp/Hyp
sin 65o = 15/Hyp
(Hyp)(sin 65o) = 15
tan 65o = Opp/Adj
tan 65o = 15/ Adj
(Adj)(tan 65o) = 15
(Hyp)(sin 65o) = 15
sin 65o
sin 65o
(Adj)(tan 65o) = 15 __
tan 65o
tan 65o
Hyp = 16.55
Adj = 6.99
AB = 16.55
BC = 6.99
Not enough information given to prove .
(Sides do not match – On one 10 is a leg
and on the other it is the hypotenuse.)
117)
Which theorem can be used (if any) to prove
AB?
Find the angle  and  on the triangles below.
Round to the nearest whole degree.
SSS Theorem (common side must be  )
tan  =
tan  =
(tan-1)tan  = (tan-1)(6/16.6)
 = 19.98
 = 20o
h = 17.55,  = 70 o
Opp/
Adj
6/
16.6
sin  = Opp/Hyp
sin  = 15/23.3
(sin-1)sin  = (sin-1)(15/23.3)
 = 40.07
 = 40o
h = 23.3, a = 17.8
118)
119)
Which theorem can be used (if any) to prove
AB?
124)
(x+ 7)+(x+3)+(2x+10)+(x+8)+(x+12)+(2x+24) = 4(180)
8x + 64 = 720
- 64 - 64
8x = 656
8
8
x = 82
SAS Theorem
Which theorem can be used (if any) to prove
AB?
125)
120)
121)
Hypotenuse-Leg Theorem
sWhich theorem can be used (if any) to
prove AB?
Not enough information given to prove 
Which theorem can be used (if any) to prove
AB?
S
SAS Theorem
122)
Given RT ES and RS  ET. Which theorem
can be used (if any) to prove TRESER ?
SSS Theorem
123)
Given A E and L ACB  L ECD and C is the
midpoint of segment AE. Which theorem can be
used (if any) to prove ACBECD ?
ASA Theorem
Solve for x (Not drawn to scale)
Solve for x (Not drawn to scale)
(2x)+(x+20)+(x+16) = 1(180)
4x + 36 = 180
- 36 - 36
4x = 144
4
4
x = 36
126)
Solve for x in the regular octagon below
(Not drawn to scale)
8(6x+33) = 6(180)
8(6x+33) = 6(180)
8
8
6x + 33 = 135
- 33 - 33
6x
= 102
6
6
x = 17
146)
Foil: (x + 8)(x – 3)
x2 + 5x – 24
127)
Foil: (x + 4)(x + 4)
x2 + 8x + 16
128)
Foil: (x + 5)(x + 5)
x2 + 10x + 25
147)
Foil: (x – 2)(x + 5)
x2 + 3x – 10
129)
Foil: (x + 6)(x + 6)
x2 + 12x + 36
148)
Foil: (x – 4)(x + 6)
x2 + 2x – 24
Foil: (x – 1)(x + 5)
x2 + 4x – 5
----------------------------------------------------
----------------------------------------------------
130)
Foil: (x + 3)(x + 5)
x2 + 8x + 15
149)
131)
Foil: (x + 4)(x + 6)
x2 + 10x + 24
150)
Foil: (x – 7)(x + 1)
x2 – 6x – 7
132)
Foil: (x + 2)(x + 4)
x2 + 6x + 8
151)
Foil: (x – 5)(x + 3)
x2 – 2x – 15
152)
Foil: (x – 8)(x + 3)
x2 – 5x – 24
----------------------------------------------------
----------------------------------------------------
133)
Foil: (x – 4)(x – 4)
x2 – 8x + 16
134)
Foil: (x – 5)(x – 5)
x2 – 10x + 25
153)
Foil: (x – 6)(x – 6)
x2 – 12x + 36
Foil: (3x – 2)(2x + 4)
6x2 + 8x – 8
154)
Foil: (x – 3)(x – 5)
x2 – 8x + 15
Foil: (5x – 2)(x – 1)
5x2 – 5x + 2
155)
Foil: (x – 4)(x – 6)
x2 – 10x + 24
Foil: (2x + 5)(3x – 2)
6x2 + 11x – 10
156)
Foil: (x – 2)(x – 4)
x2 – 6x + 8
Foil: (3x + 2)(x – 4)
3x2 – 10x – 8
157)
Foil: (2x + 2)(3x – 5)
6x2 – 4x – 10
135)
----------------------------------------------------
----------------------------------------------------
136)
137)
138)
139)
Foil: (x – 4)(x – 5)
x2 – 9x + 20
----------------------------------------------------
158)
140)
Foil: (x + 4)(x – 4)
x2 – 16
141)
Foil: (x + 10)(x – 10)
x2 – 100
142)
Foil: (3x + 5)(3x – 5)
9x2 – 25
143)
Foil: (4x + 6)(4x – 6)
16x2 – 36
----------------------------------------------------
144)
Foil: (x + 7)(x – 1)
x2 + 6x – 7
145)
Foil: (x + 5)(x – 3)
x2 + 2x – 15
Factor Completely: x2 + 6x + 8
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 6x is positive). Answer: (++)
(x + ) (x + ) Factors of 8: (1, 8), (2, 4)
Which factor above when added
will equal 6?) Answer: (2, 4).
(x + 2) (x + 4)
----------------------------------------------------
159)
Factor Completely: x2 + 10x + 21
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 10x is positive). Answer: (++)
(x + ) (x + ) Factors of 21: (1, 21), (3, 7)
Which factor above when added
will equal 10?) Answer: (3, 7).
(x + 3) (x + 7)
160)
Factor Completely: x2 + 8x + 15
166)
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 8x is positive). Answer: (++)
(x + ) (x + ) Factors of 15: (1, 15), (3, 5)
Which factor above when added
will equal 8?) Answer: (3, 5).
(x + 3) (x + 5)
161)
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 12x is positive). Answer: (++)
(x + ) (x + ) Factors of 35: (1, 35), (5, 7)
Which factor above when added
will equal 12?) Answer: (5, 7).
(x + 5) (x + 7)
162)
Factor Completely: x2 + 7x + 12
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 7x is positive). Answer: (++)
(x + ) (x + ) Factors of 12: (1, 12), (2, 6), (3,
4)
Which factor above when added
will equal 7?) Answer: (3, 4).
(x + 3) (x + 4)
163)
164)
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will give
a Positive (the 6x is negative). Answer: (– –)
(x – ) (x – ) Factors of 8: (1, 8), (2, 4)
Which factor above when added will equal 6?)
Answer: (2, 4).
(x – 2) (x – 4)
2
Factor Completely: x + 12x + 35
Factor Completely: x – 9x + 20
167)
Factor Completely: x2 – 11x + 24
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will give
a Positive (the 11x is negative). Answer: (– –)
(x – ) (x – ) Factors of 24:(1, 24), (2, 12), (3, 8), (4, 6)
Which factor above when added will equal 11?)
Answer: (3, 8).
(x – 3) (x – 8)
----------------------------------------------------
168)
Factor Completely: x2 + 6x – 7
The -7 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the positive to make a + 6x
(x + ) (x – ) Factors of 7: (1, 7)
Will the factor above when subtracted will equal 6?)
Answer: (1, 7).
(x + 7) (x – 1)
2
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 9x is negative). Answer: (– –)
(x – ) (x – ) Factors of 20: (1, 20), (2, 10),
(4, 5)
Which factor above when added
will equal 9?) Answer: (4, 5).
(x – 4) (x – 5)
169)
Factor Completely: x2 – 14x + 33
170)
Factor Completely: x2 – 9x + 14
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 9x is negative). Answer: (– –)
(x – ) (x – ) Factors of 14: (1, 14), (2, 7)
Which factor above when added
will equal 9?) Answer: (2, 7).
(x – 2) (x – 7)
Factor Completely: x2 + 3x – 10
The -10 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the positive to make a + 3x
(x + ) (x – ) Factors of 10: (1, 10), (2, 5)
Will the factor above when subtracted will equal 3?)
Answer: (2, 5).
(x + 5) (x – 2)
The 2nd Pos can be obtained by a (++) or (– –).
Signs the same. Which of these when added will
give a Positive (the 14x is negative). Answer: (– –)
(x – ) (x – ) Factors of 33: (1, 33), (3, 11)
Which factor above when added
will equal 14?) Answer: (3, 11).
(x – 3) (x – 11)
165)
Factor Completely: x2 – 6x + 8
Factor Completely: x2 + 3x – 28
The -28 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the positive to make a + 3x
(x + ) (x – ) Factors of 28: (1, 28), (2, 14), (4, 7)
Will the factor above when subtracted will equal 3?)
Answer: (4, 7).
(x + 7) (x – 4)
171)
Factor Completely: x2 + 6x – 16
The -16 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the positive to make a + 6x
(x + ) (x – ) Factors of 16: (1, 16), (2, 8), (4, 4)
Will the factor above when subtracted will equal 6?)
Answer: (2, 8).
(x + 8) (x – 2)
172)
Factor Completely: x2 + 5x – 36
The -36 can only be obtained by a (+ –).
178)
Recognize that ½ the middle number squared
equals the last number. (½ 8)2 = 16; 42 = 16
Signs are opposite. Therefore when subtracted the
bigger number must be the positive to make a + 5x
(x + ) (x – ) Factors of 36: (1, 36), (2, 18), (4, 9), (6, 6)
Will the factor above when subtracted will equal 5?)
Answer: (4, 9).
(x + 9) (x – 4)
(x + 4) (x + 4)
or
179)
or
180)
Factor Completely: x2 – 5x – 6
The - 6 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the negative to make a - 5x
(x + ) (x – ) Factors of 6: (1, 6), (2, 3)
Which factor above when subtracted will equal 5?)
Answer: (1, 6).
or
181)
182)
183)
Signs are opposite. Therefore when subtracted the
bigger number must be the negative to make a - 4x
(x + ) (x – ) Factors of 45: (1, 45), (5, 9)
Which factor above when subtracted will equal 4?)
Answer: (5, 9).
184)
Signs are opposite. Therefore when subtracted the
bigger number must be the negative to make a - 3x
(x + ) (x – ) Factors of 40: (1, 40), (2, 20) (4, 10), (5, 8)
Which factor above when subtracted will equal 3?)
Answer: (5, 8).
(x + 5) (x – 8)
177)
(x + 6) (x – 7)
----------------------------------------------------
Factor Completely: x2 – 100
Factor Completely: x2 – 16
Recognize that there is no middle term and that
this is the difference of two squares.
(x + 4) (x – 4)
Factor Completely: 9x2 – 25
Recognize that there is no middle term and that
this is the difference of two squares.
(3x + 5) (3x – 5)
185)
Factor Completely: 16x2 – 36
Recognize that there is no middle term and that
this is the difference of two squares.
(4x + 6) (4x – 6)
----------------------------------------------------
186)
Factor Completely: x2 – x – 42
The - 42 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the negative to make a - 1x
(x + ) (x – ) Factors of 42: (1, 42), (2, 21), (3, 14), (6, 7)
Which factor above when subtracted will equal 1?)
Answer: (6, 7).
Factor Completely: x2 – 36
Recognize that there is no middle term and that
this is the difference of two squares.
(x + 10) (x – 10)
Factor Completely: x2 – 4x – 45
The - 45 can only be obtained by a (+ –).
Factor Completely: x2 – 3x – 40
The - 40 can only be obtained by a (+ –).
(x + 6) 2
Recognize that there is no middle term and that
this is the difference of two squares.
(x + 6) (x – 6)
(x + 5) (x – 9)
176)
Factor Completely: x2 + 12x + 36
(x + 6) (x + 6)
(x + 1) (x – 6)
175)
(x + 5) 2
Recognize that ½ the middle number squared
equals the last number. (½ 12)2 = 36; 62 = 36
(x + 1) (x – 5)
174)
Factor Completely: x2 + 10x + 25
(x + 5) (x + 5)
Factor Completely: x2 – 4x – 5
The - 5 can only be obtained by a (+ –).
Signs are opposite. Therefore when subtracted the
bigger number must be the negative to make a - 4x
(x + ) (x – ) Factors of 5: (1, 5)
Which factor above when subtracted will equal 4?)
Answer: (1, 5).
(x + 4) 2
Recognize that ½ the middle number squared
equals the last number. (½ 10)2 = 25; 52 = 25
----------------------------------------------------
173)
Factor Completely: x2 + 8x + 16
Factor Completely: 5x2 – 7x + 2
(5x – ) (x – ) Factors of 2: (1, 2)
Now trial and error method. ADD
(5x – 1) (x – 2) Inside = -1x; Outside = -10x ; NO
(5x – 2) (x – 1) Inside = -2x; Outside = -5x ; YES!
187)
Factor Completely: 3x2 – 10x – 8
(3x
) (x
) Factors of 8: (1, 8), (2, 4)
Now trial and error method. .
(3x – 2) (x – 4) Inside = 2x; Outside = -12x ; YES!
188)
In the figure below what is the value of x ?
191)
192)
189)
In the figure below what is the value of x ?
If: AE = ED = DC = CB
If sides are , then L ‘s opp. are . And if L E and
L C = 90o, then the other angles must = 45o as
shown below.
193)
If the measure of arc XY = 50o, what is the sum of
the degree measures of angle a + b + c?
Sum of L’s = L a + L b + L c
= ½ (50) + ½ (50) + ½ (50)
= 25 + 25 + 25
Sum of L’s = 75o
If two secants intersect on a circle, the measure of
an inscribed angle is half the measure of the arc.
If the measure of L AOC = 60o and the radius = 18,
what is the length of arc AC? (
60o
AC _
360o
2r
x = (60o) (2r)
360o
x = (60o) (2 18)
360o
x = 6
What is the sum of xo + yo + zo ?
xo + yo + zo = 360o
The sum of external angles on any polygon is
always 360o (The 20o angle and the length
measurement of 9 are irrelevant values).
L ADE + L x + L BDC = 180 (A line is 180o.
45 + L x + 45 = 180 (Substitution Property)
L x + 90 = 180
(Combine like terms)
- 90 - 90
Lx
= 90
190)
194)
If the measure of L ABC = 6o and the radius = 5,
what is the length of arc AC?
If L ABC is 43o and O is the center of the circle,
what is degree measure of L AOC?
L AOC = 86o
If two secants intersect on a circle, the measure of
an inscribed angle is half the measure of the angle
measured from the center.
Degree measure L B = 12o
12o
AC _
360o
2r
x = (12o) (2r)
360o
x = (12o) (2 5)
360o 3
x = 
3
195)
If O is the center of the circle, length OC = 9 and
the measure of L AOC = 36o, what is the area of
sector AOC?
36o
360o
196)
197)
area AOC _
r 2
x = (36o) (r 2)
360o
x = (36o) ( 81)
360o 10
x = 81
10
x = 8.1 
Find the ratio of the area of ABE to ACD.
The sum of internal angles on any quadrilateral
is always 360o (Two triangles = 2(180) = 360).
Therefore:
x + y + 70 + 135 = 360
x + y + 205
= 360 (Combine like terms)
– 205
– 205
x + y
= 155
198)
area ABE _
area ACD
(base BE) (height AB) (1/2)_
(base CD) (height AC) (1/2)_
2
In the figure below which is not drawn to scale,
if the four lines intersect as shown, what is the
value of: x + y
2
1
( /3) ( /3) ( /2)
(1) (1) (1/2)
x = 4/9
If segment WZ and segment XY are diameters
of the circle with a length of 12, what is the
area of the shaded region?
The internal angle of the triangles adjacent to the
135o angle must be 45o. (linear pair)
Thus Ls X and Y must be 45o (Sum of L’s on )
The distance from X to the center = 6 (½ diameter)
or a Faster Method is:
Area is u2
So if each u = 2/ 3 that u2 = (2/ 3)2 = 4/ 9
The shaded section = a square, which area can be
found by:
Area = (diagonal 1)(diagonal 2)
2
= (6)(6)
2
= 18