Question 1 :
In the figure above, ray OQ bisects ∠ROP.
If ∠SOQ = 11x + 6 and ∠ROP = 8x - 12, what is the measure of ∠SOR?
A) 92
B) 96
C) 102
D) 108
Answer :
Since ray OQ bisects ∠ROP,
∠QOP = (1/2)∠ROP
∠QOP = (1/2)(8x - 12)
∠QOP = 4x - 6
∠SOQ and ∠QOP together form a linear pair.
∠SOQ + ∠QOP = 180°
11x + 6 + 4x - 6 = 180
15x = 180
x = 12
Given :
∠ROP = 8x - 12
Substitute x = 12.
∠ROP = 8(12) - 12
∠ROP = 96 - 12
∠ROP = 84°
∠SOR and ∠ROP together form a linear pair.
∠SOR + ∠ROP = 180°
∠SOR + 84° = 180°
∠SOR = 96°
The correct answer choice is (A).
Question 2 :
In the figure above, ∠PQT = 120° and ∠SQR = 135°. What is the measure ∠SQT?
A) 63
B) 68
C) 75
D) 79
Answer :
Let ∠SQT = x.
∠PQS = ∠PQT - ∠SQT
∠PQS = 120° - x
In the figure above,
∠PQS + ∠SQR = 180°
120° - x + 135° = 180°
-x + 255 = 180°
-x = -75°
x = 75°
∠SQT = 75°
The correct answer choice is (C).
Question 3 :
In the figure above, lines a, b and c are parallel. What is the value of (x + y)?
A) 160
B) 200
C) 230
D) 290
Answer :
In the figure above,
a° + b° + 70° = 360°
a + b = 290 ----(1)
x° and a° are alternate interior angles and they are equal.
x = a
And also, y° and b° are alternate interior angles and they are equal.
y = b
In (1), replace a by x and b by y.
x + y = 290
The correct answer choice is (D).
Question 4 :
In the figure above, lines a and b are parallel and QS bisects ∠PQR. What is the value of x?
A) 54
B) 60
C) 68
D) 72
Answer :
∠PQR and ∠QPT are alternate interior angles, hence they are equal.
∠PQR = ∠QPT
∠PQR = 108°
Since QS bisects ∠PQR,
∠SQR = (1/2)∠PQR
∠SQR = (1/2)(108°)
∠SQR = 54°
∠PSQ and ∠SQR are alternate interior angles, hence they are equal.
∠PSQ = ∠SQR
x° = 54°
The correct answer choice is (A).
Question 5 :
In the figure above, PQ || RS and QR || ST. What is the value of x?
A) 47
B) 51
C) 55
D) 57
Answer :
Since PQ || RS, ∠QPR and ∠SRT are corresponding angles, hence they are equal.
∠SRT = ∠QPR
∠SRT = x°
Since QR || ST, ∠QRP and ∠STR are corresponding angles, hence they are equal.
∠QRP = ∠STR
∠QRP = 65°
In the figure above,
∠QRP + ∠QRS + ∠SRT = 180°
∠QRP + ∠QRS + ∠SRT = 180°
65° + 68° + x° = 180°
133 + x = 180
x = 47
The correct answer choice is (A).
Question 6 :
In the figure above, p || q. What is the value of (a + b)?
A) 160
B) 175
C) 185
D) 200
Answer :
In the figure above, c° and 35° are vertical angles, hence they are equal.
c = 35
a + c = 90
Substitute c = 35.
a + 35 = 65
a = 30
Since p || q, b° and c° are consecutive interior angles, hence they are supplementary.
b + c = 180
b + 35 = 180
b = 145
a + b = 30 + 145
a + b = 175
The correct answer choice is (B).
Question 7 :
In the figure above, what is the value of (a + b)?
Answer :
In the given figure, draw ∠c as shown below.
In the figure above,
a + c = 360 ----(1)
b - c = 180 ----(2)
(1) + (2) :
a + b = 540
Question 8 :
In the figure above, AB is parallel to DE. What is the measure of ∠BCD?
Answer :
In the given figure, draw CF, which is parallel to AB and DE as shown below.
Since AB is parallel to CF, ∠ABC and ∠BCF are consecutive interior angles, hence they are supplementary.
∠ABC + ∠BCF = 180°
110° + ∠BCF = 180°
∠BCF = 70°
Since CF is parallel to DE, ∠FCD and ∠CDE are consecutive interior angles, hence they are supplementary.
∠FCD + ∠CDE = 180°
∠FCD + 145° = 180°
∠FCD = 35°
By the Angle Addition Postulate,
∠BCD = ∠BCF + ∠FCD
∠BCD = 70° + 35°
∠BCD = 105°
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