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Angles
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Angles: Difficult Problems with Solutions
By Catalin David
Problem 1
∠ABC and ∠DBC are adjacent. ∠ABC has a measure of 53° and ∠DBC has a measure of 29°. The measure of ∠ABD is
24°
82°
Solution:
Since the angles are adjacent, they share vertex B and side BC. ∠ABD=∠ABC+∠DBC=53°+29°=82°.
Problem 2
∠ABC and ∠DBC are adjacent. ∠DBC has a measure of 85° and ∠ABD has a measure of 127°. The measure of ∠ABC is:
42°
212°
Solution:
∠ABC and ∠CBD are adjacent angles. Thus, ∠ABC=∠ABD-∠DBC=127°-85°=42°.
Problem 3
AC is the bisector of ∠BAD. If ∠BAC has a measure of 52°, then ∠BAD has a measure of
26°
104°
90°
156°
.
Solution:
Since AC is the bisector of ∠BAD, this angle is two times greater than ∠BAC. The measure of ∠BAD is 104°.
Problem 4
∠AOB and ∠BOC are adjacent and complementary.
If ∠AOB = x°+8° and ∠BOC = 2x° - 17°, what is the measure of ∠AOB?
33°
57°
41°
51°
Solution:
If ∠AOB and ∠BOC are adjacent and complementary, then
∠AOB + ∠BOC = 90°
x° + 8° + 2x° -17° = 90°
3x° - 9° = 90°
3x° = 99°
x° = 33°
Thus, ∠AOB = 41°
Problem 5
∠AOB and ∠BOC are adjacent and supplementary. If OE is the bisector of ∠BOC and m∠EOC = 30°, what is the measure of ∠AOB?
60°
90°
120°
80°
Solution:
OE is the bisector, then ∠BOC = 2∠EOC = 60°. ∠AOB and ∠BOC are supplementary, so ∠AOB = 180° - 60° = 120°.
Problem 6
If lines a and b are parallel, lines c and d are also parallel and m∠13 = 40°, what is the measure of ∠2?
40°
140°
150°
160°
Solution:
∠13 and ∠11 are alternate interior angles. Since lines a and b are parallel, the measures of these angles are equal. Thus, ∠11 also has 40°. ∠11 and ∠3 are corresponding. Since lines c and d are parallel, the measures of these angles are equal. Thus, ∠3 has ∠40°. ∠3 and ∠2 are adjacent and supplementary. Thus, ∠2 = 180° - 40° = 140°.
Problem 7
Point O is the intersection of lines AB and CD. If ∠AOC has a measure of x° and ∠BOC has a measure of 4x°, what is the measure of ∠AOD?
36°
144°
120°
Solution:
∠AOC and ∠BOC are adjacent and supplementary. Thus, ∠AOC + ∠BOC = 180°.
x° + 4x° = 180°
5x° = 180°
x°=36°
Thus, m∠AOC = 36° and m∠BOC = 144°. ∠AOD and ∠BOC are opposite angles, so m∠AOD = 144°.
Problem 8
Point O is the intersection of AB and CD. If m∠BOC = x° and
m∠AOC = x° + 20°, what is the measure of ∠BOD?
80°
100°
90°
Solution:
∠AOC and ∠BOC are adjacent and supplementary. Thus, ∠AOC + ∠BOC = 180°.
x + 20° + x = 180°
2x + 20° = 180°
2x = 160°
x = 80°
Thus, m∠BOC = 80° and m∠AOC = 100°. ∠BOD and ∠AOC are opposite angles, so its measure is 100°.
Problem 9
∠AOB, ∠BOC, ∠COD are adjacent and supplementary. If ∠BOC has 80°, what is the measure of ∠AOB?
60°
120°
58°
48°
Solution:
Since they are supplementary, their sum is 180°.
2x° + 10° + 80° + 3x° -20° =180°
5x° + 60° = 180°
5x° = 120°
x° = 24°
Thus, ∠AOB = 58°.
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