Angles: Problems with Solutions

Solution:
The angle measures the opening between two lines that intersect. The lines are infinite, they don't affect the measure of the angle.

Solution:
Angles are measured using degrees.

Solution:
The vertex of the angle is the point shared by the two lines
In this case, point A.

Solution:
The letter that designates an angle represents the vertex of that angle.

Solution:
If an angle is designated by 3 letters, then the middle one represents the vertex of the angle. The other two represent two points, each on one of the sides.

Solution:
No, because the middle letter is not the same, so the angles don't have the same vertex.

Solution:
Yes, because the middle letter is the same, so they have the same vertex. Moreover, points B and D are on one of the sides of the angle and points C and E are on the other side.

Solution:
Less than 90°

Solution:
∠BAC is acute because its measure is less than 90°.

Solution:
The measure of a right angle is 90°. Its sides are perpendicular.

Solution:
The sides are perpendicular, so they form a right angle. A right angle measures 90°.

Solution:
∠ABC is an acute angle because its measure is less than 90°. It can be included inside a right angle.

Solution:
The measure of an obtuse angle is greater than 90°.

Solution:
∠ABC is obtuse because its measure is greater than 90°.

Solution:
∠ABC is obtuse because its measure is greater than 90°. A right angle can be included inside it.

Solution:
A straight angle.

Solution:
The measure of a straight angle is 180° because two right angles can fit inside it.

Solution:
The bisector splits the angle into 2 equal parts. In the figure, AC is the bisector of ∠BAD.

Solution:
The bisector forms two equal angles, each having a measure of 36°.

Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, these requirements are not met, so the angles are not adjacent.

Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, the angles only share the same vertex, so they are not adjacent.

Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, the angles only share one of their sides, so they are not adjacent.

Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, all requirements are met, so the angles are adjacent.

Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case the angles share interior points, so they are not adjacent.

Solution:
Yes, since ∠ABC and ∠CBD are adjacent angles.
Thus, ∠ABD = ∠ABC + ∠CBD = 42° + 36° = 78°.

Solution:
Yes, since ∠ABC and ∠CBD are adjacent angles. Thus, ∠DBC=∠ABD-∠ABC=100°-56°=44°.

Solution:
Two angles are complementary if the sum of their measures is 90°.

Solution:
The sum of their measures is 43° + 47° = 90°, so the angles are complementary.

Solution:
If OA and OC are perpendicular, then ∠AOC has a measure of 90°. ∠AOB and ∠BOC are adjacent. ∠AOC=∠AOB+∠BOC=90°. The angles are complementary.

Solution:
∠AOB and ∠BOC are adjacent and complementary.
In this case, ∠AOB + ∠BOC = 90°.
∠AOB + 24° = 90°.
∠AOB = 90° - 24° = 66°

Solution:
The sum of the measures of complementary angles is 90°. If one of the angles has a measure of 40°, the measure of the other one is 90°-40°=50°.

Solution:
The sum of the measures of supplementary angles is 180°.

Solution:
∠AOB and ∠BOC are adjacent. Thus, ∠AOB+∠BOC=∠AOC. But ∠AOC's sides are on the same line, so its measure is 180°. Hence, ∠AOB and ∠BOC are supplementary.

Solution:
∠AOB and ∠BOC are adjacent and supplementary. Thus m∠AOB + m∠BOC = 180°.
45° + m∠BOC = 180°.
m∠BOC = 180° - 45° = 135°.

Solution:
The angles are adjacent two by two. Thus, ∠AOB+∠BOC+∠COD=∠AOD. But ∠AOD's sides are on the same line, so its measure is 180°. ∠AOB and ∠COD are complementary, so ∠AOB+ ∠COD=90°. ∠BOC + 90° = 180°, so ∠BOC = 90°.

Solution:
If ∠AOB and ∠BOC are complementary, then ∠AOB+∠BOC=90°. If ∠BOC and ∠COD are complementary, then ∠BOC+∠COD=90°.
∠AOB+∠BOC=∠BOC+∠COD. Thus, ∠AOB=∠COD, so two angles having the same complementary angle are equal.

Solution:
If ∠AOB and ∠BOC are complementary, then ∠AOB+∠BOC=90°.
If ∠BOC and ∠COD are complementary, then ∠BOC+∠COD=90°.
∠AOB+∠BOC=∠BOC+∠COD.
Thus, ∠AOB=∠COD=60°,
∠AOD=∠AOC+∠COD=90°+60°=150°.

Solution:
Yes, since vertical angles are formed by two intersecting lines.

Solution:
No, since vertical angles are formed by two intersecting lines. Points A, O and C are not on the same line. Points B, O and D are not on the same line either.

Solution:
∠AOB and ∠DOA are adjacent.
∠AOB+∠AOD=∠BOD=180°.
∠COD and ∠DOA are adjacent.
∠COD+∠DOA=∠COA=180°.
Thus, ∠AOB and ∠COD have the same supplementary and are equal.

Solution:
Since they are vertical angles, they have the same measure. ∠COD has a measure of 35°.

Solution:
∠DOA and ∠AOB are supplementary. In this case, m∠AOB + m∠DOA = 180°, so ∠DOA = 145°.

Solution:
Alternate interior angles are non-adjacent, inside the space formed by lines a and b, on one side and the other of line c (the secant line). Two angles that match these properties are 3 and 5, also 4 and 6.

Solution:
Alternate exterior angles are non-adjacent, outside the space formed by lines a and b, on one side and the other of line c (the secant line). Two angles that match these properties are 2 and 8, also 1 and 7.

Solution:
Corresponding angles are non-adjacent, one of them outside the space formed by lines a and b and the other inside, both on the same side of line c (the secant line). Two angles that match these properties are 3 and 7.
Other pairs are 2 and 6, 1 and 5, 4 and 8.

Solution:
The two angles are alternate exterior angles. They have the same measure, so lines a and b are parallel.

Solution:
The two angles are alternate interior angles, but they have different measures. In this case, lines a and b are not parallel.

Solution:
The two angles are alternate exterior angles. Since lines a and b are parallel, the angles will have the same measure, so m∠7 = 75°.

Solution:
The two angles are corresponding angles. Since lines a and b are parallel, the angles will have the same measure, so ∠6 also has 120°.

Solution:
Since lines a and b are parallel, ∠2 and ∠8 have the same measure, so ∠8 also has 35°. However, ∠7 and ∠8 are adjacent supplementary angles, so the measure of
∠7 is 180° - 35° = 145°.

Solution:
Vertical angles are equal. Adjacent supplementary angles add up to [tex]180^\circ[/tex].
[tex]180^\circ-30^\circ=150^\circ[/tex]
The other angles are [tex]150^\circ[/tex], [tex]30^\circ[/tex] and [tex]150^\circ[/tex].

Solution:
The two angles adjacent to [tex]\alpha[/tex] are vertical angles, so they are equal: each is [tex]120^\circ\div 2=60^\circ[/tex].
[tex]\alpha=180^\circ-60^\circ=120^\circ[/tex]

Solution:
[tex]\alpha=0. 25\beta=\frac{\beta}{4}[/tex]
Adjacent supplementary angles add up to [tex]180^\circ[/tex].
[tex]\frac{\beta}{4}+\beta=180^\circ \Rightarrow \frac{5\beta}{4}=180^\circ \Rightarrow \beta=144^\circ[/tex]

Solution:
[tex]\angle AOC=x[/tex], [tex]\angle BOC=4x[/tex]
[tex]x+4x=150^\circ \Rightarrow 5x=150^\circ \Rightarrow x=30^\circ[/tex]
[tex]\angle AOC=30^\circ[/tex], [tex]\angle BOC=4\times 30^\circ=120^\circ[/tex]

Solution:
The bisector divides the angle into two equal parts: [tex]\angle BOL=\frac{1}{2}\angle AOB[/tex].
[tex]\angle AOB-\frac{1}{2}\angle AOB=23^\circ \Rightarrow \frac{1}{2}\angle AOB=23^\circ \Rightarrow \angle AOB=46^\circ[/tex]

Solution:
[tex]32^\circ 43'+15^\circ 28'=47^\circ 71'[/tex]
Since [tex]1^\circ=60'[/tex], we have [tex]71'=1^\circ 11'[/tex]:
[tex]47^\circ 71'=48^\circ 11'[/tex]

Solution:
Let the smaller angle be [tex]x[/tex]:
[tex]x+(x+28^\circ)=180^\circ \Rightarrow 2x=152^\circ \Rightarrow x=76^\circ[/tex]
[tex]x+28^\circ=104^\circ[/tex]

Solution:
The two angles cannot be adjacent supplementary (their sum would be [tex]180^\circ[/tex]), so they are vertical angles: each is [tex]40^\circ[/tex].
Each of the other two angles is [tex]180^\circ-40^\circ=140^\circ[/tex].

Solution:
[tex]\beta+5\beta=180^\circ \Rightarrow \beta=30^\circ, \alpha=150^\circ[/tex]
The angle between the bisectors is [tex]\frac{\alpha}{2}+\frac{\beta}{2}=75^\circ+15^\circ=90^\circ[/tex].
In fact, it is always [tex]90^\circ[/tex]: [tex]\frac{\alpha+\beta}{2}=\frac{180^\circ}{2}=90^\circ[/tex] – the bisectors of adjacent supplementary angles are perpendicular.

Solution:
Any two neighboring angles of the quadrilateral are co-interior angles at two parallel lines, so their sum is [tex]180^\circ[/tex]: [tex]180^\circ-52^\circ=128^\circ[/tex].
The angles are [tex]52^\circ[/tex], [tex]128^\circ[/tex], [tex]52^\circ[/tex], [tex]128^\circ[/tex] (the quadrilateral is a parallelogram).