Problem 1 :
In the figure above, what is the value of x?
Solution:
From the figure,
∠BDA + ∠ADC = 180°
x + 80 = 180°
x = 180 - 80°
x = 100°
Problem 2 :
In the figure above, AB = AC and c = 50°. What is the value of a?
A) 65 B) 70 C) 75 D) 80
Solution:
∠ABC = ∠ACB
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
b = c
∠ABC = c
b = 50°
∠ABC + ∠ACB + ∠BAC = 180°
b + c + a = 180°
50° + 50° + a = 180°
100 + a = 180
a = 80°
So, option (D) is correct.
Problem 3 :
In the figure above, ∠BAC = 20° and AB = AC. If triangles ACD and ADE are isosceles, what is the value of x?
A) 10 B) 15 C) 20 D) 25
Solution:
∠BAC = 20°
Since AB and AC are equal, it will make equal angles only.
∠ABC = ∠ACB
2∠ABC + 20 = 180
2∠ABC = 160
∠ABC = 160/2 ==> 80
∠ACD = 180 - 80 ==> 100
In triangle ACD, the other two angles must be acute angle.
∠CDA = ∠CAD (isosceles triangle)
∠CDA = ∠CAD = 40
∠ADE = 140
Since ADE is isosceles triangle and already we have obtuse angle, the other two must be equal and acute angles.
x = 20
∠DAE = 20
Problem 4 :
In the figure above, what is the value of x?
Solution:
In triangle DCE,
40 + 40 + x = 180°
80 + x = 180
x = 100°
In triangle ACB,
100 + 45 + x = 180°
145 + x = 180°
x = 180 - 145
x = 35
Problem 5 :
In the figure above, AB = BC and ∠ABC = 110°. What is the value of y?
A) 125 B) 130 C) 135 D) 145
Solution:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
∠BAC = ∠ACB
∠ABC + ∠ACB + ∠BAC = 180°
∠ABC + ∠ACB + ∠ACB = 180°
110 + ∠ACB + ∠ACB = 180°
∠ACB = 35°
∠ACB + ∠ACD = 180°
35° + y= 180°
y = 145°
So, option (D) is correct.
Problem 6 :
In the figure above, AB, BC, CD, DE and EA are line segments. What is the value of x + y?
Solution:
In triangle BAY,
40° + y + 30° = 180°
70 + y = 180°
y = 110°
30 + 100 + x = 180
x = 180 - 130
x = 50
x + y = 50 + 110
= 160
Problem 7 :
In the figure above, the measure of ∠AOB is 2/5 the measure of ∠AOC. What is the value of x?
Solution:
∠AOB = 2/5∠AOC
∠BOC = 30°
∠AOC = x + 30°
∠AOB = x
x = 2/5(x + 30°)
5x = 2(x + 30)
5x = 2x + 60
5x - 2x = 60
3x = 60
x = 60/3
x = 20
So, the value of x is 20.
Problem 8 :
In the figure above, MN is a line. What is the value of b?
Solution:
3a° + b° = 180°
5a = 180°
a = 36
3(36) + b = 180°
108 + b = 180°
b = 72°
Problem 9 :
In right triangle ABC above, what is the value of y?
Solution:
5x° + 4x° = 180°
9x° = 180°
x = 20°
2x° + y° = 90°
2(20) + y° = 90°
y = 90° - 40°
y = 50°
Problem 10 :
In the figure above, ∠BAC = 30°, ∠BCA = 45°, and AB = 8. What is the length of BC?
A) 4 B) 4√2 C) 4√3 D) 8√2
Solution:
By drawing a perpendicular bisector, the side which is opposite to 90 degree is hypotenuse. BD is the smaller side
2 smaller side = 8
BD = 4
BC^{2} = BD^{2} + DC^{2}
BC^{2} = 4^{2} + 4^{2}
BC^{2} = 16 + 16
BC^{2} = 32
BC = √32
BC = 4√2
So, option B is correct.
Problem 11 :
In the figure above, lines l and m are parallel and lines s and t are parallel. If the measure of ∠1 is 35°, what is the measure of ∠2?
A) 35° B) 55° C) 70° D) 145°
Solution:
Corresponding angles are equal.
∠1 = 35°
∠2 + ∠1 = 180°
∠2 + 35 = 180°
∠2 = 145°
So, option (D) is correct.
Problem 12 :
In the figure, line m is parallel to line n, and line t intersects both lines. What is the value of x?
A) 33 B) 57 C) 123 D) 147
Solution:
Corresponding angles are equal.
x = 180 - 33
x = 147
So, option (D) is correct.
Problem 13 :
Two isosceles triangles are shown above. If 180 - z = 2y and y = 75, what is the value of x?
Solution:
180 - z = 2y
180 - z = 2(75)
180 - z = 150
z = 30
In an isosceles triangle, the base angles are equal. So,
x = z
x = 30
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