Math 2 Quiz: Circle Angles And Arcs

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Circle Angles And ArcsQuestion 1 of 11

In circle $O$ , secants $PAB$ and $PCD$ are drawn from external point $P$ , where $A$ and $C$ are the closer intersection points to $P$ . If $\angle APD = 40°$ and arc $BD = 140°$ , what is the measure of arc $AC$ ?

60°

80°

100°

220°

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Math 2 Quiz

Math 2 Quiz: Circle Angles And Arcs

Practice Circle Angles And Arcs in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Circle Angles And Arcs, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$60°$ (correct answer)
$80°$
$100°$
$220°$

Explanation: When two secants are drawn from an external point, the angle between them equals half the difference of the intercepted arcs. Here,

\angle APD = \frac{1}{2}(\text{arc } BD - \text{arc } AC)

. Substituting the known values:

40° = \frac{1}{2}(140° - \text{arc } AC)

. Solving:

80° = 140° - \text{arc } AC

, so

\text{arc } AC = 140° - 80° = 60°

. Choice B (

80°

) comes from incorrectly using

2 \times 40° = 80°

. Choice C (

100°

) might result from

140° - 40°

. Choice D (

220°

) could come from using

140° + 80°

Question 2

A regular hexagon is inscribed in a circle. If one side of the hexagon subtends an arc of measure $x°$ , and an inscribed angle is drawn from a vertex of the hexagon to subtend three consecutive sides, what is the measure of this inscribed angle in terms of $x$ ?

$\frac{x}{2}$
$\frac{3x}{2}$ (correct answer)
$3x$
$\frac{3x}{4}$

Explanation: In a regular hexagon inscribed in a circle, each side subtends an equal arc. Since the total circle is

360°

and there are 6 sides, each side subtends an arc of

\frac{360°}{6} = 60°

. So

x = 60°

. Three consecutive sides subtend a total arc of

3x

. An inscribed angle that intercepts this arc has measure

\frac{3x}{2}

(half the intercepted arc). Choice A (

\frac{x}{2}

) would be the inscribed angle intercepting just one side. Choice C (

3x

) incorrectly assumes the angle equals the arc. Choice D (

\frac{3x}{4}

) has no geometric basis for this relationship.

Question 3

A tangent from external point $T$ touches circle $O$ at point $A$ . A secant from $T$ passes through points $B$ and $C$ on the circle (with $B$ closer to $T$ ). If arc $AC = 140°$ and the angle between the tangent and secant is $35°$ , what is the measure of arc $AB$ ?

$210°$
$105°$
$175°$
$70°$ (correct answer)

Explanation: When you see a tangent and secant drawn from an external point, you're dealing with the angle-arc relationship for angles formed outside a circle. The key theorem states that an angle formed by two secants, two tangents, or a tangent and secant from an external point equals half the difference of the intercepted arcs. Here, the angle between tangent

TA

and secant

TBC

35°

. The intercepted arcs are arc

AC = 140°

(given) and arc

AB

(what we're finding). Using the theorem:

35° = \frac{1}{2}|

arc

AC

- arc

AB

|$$

35° = \frac{1}{2}|140° -

arc

AB

|$$

70° = |140° -

arc

AB

|$$ This gives us two cases:

140° -

arc

AB = 70°

140° -

arc

AB = -70°

Solving: arc

AB = 70°

or arc

AB = 210°

Since

B

is closer to

T

than

C

, and both points lie on the circle with the tangent at

A

, the geometric configuration requires arc

AB = 70°

. Looking at the wrong answers: (A)

210°

is the other mathematical solution but doesn't fit the geometric constraint. (B)

105°

likely comes from incorrectly using

35° × 3

. (C)

175°

might result from adding

35°

140°

instead of using the proper theorem. Remember: Always apply the geometric constraints after solving algebraically. The position of points matters in circle problems, not just the arithmetic.

Question 4

Circle $M$ has center $(3, 4)$ and radius $5$ . Points $A(-2, 4)$ and $B(8, 4)$ are on the circle. Point $C$ is also on the circle such that triangle $ABC$ is inscribed in the circle. If $C$ is in the upper half of the circle, what is the measure of $\angle ACB$ ?

$45°$
$60°$
$90°$ (correct answer)
$120°$

Explanation: First, verify that

A

and

B

are on the circle: distance from

M(3,4)

A(-2,4)

\sqrt{(-2-3)^2 + (4-4)^2} = 5

✓. Distance from

M

B(8,4)

\sqrt{(8-3)^2 + (4-4)^2} = 5

✓. Since

A(-2,4)

and

B(8,4)

have the same

y

-coordinate and the center

M(3,4)

also has

y = 4

, the chord

AB

passes through the center, making it a diameter. Any inscribed angle that intercepts a diameter (semicircle) measures

90°

. Therefore,

\angle ACB = 90°

. Choice A would be for a

90°

arc. Choice B would be for a

120°

arc. Choice D would be for a

240°

arc.

Question 5

A regular hexagon is inscribed in circle $O$ . If one side of the hexagon subtends a central angle, and $P$ is a point on the circle not at any vertex of the hexagon, what is the measure of the inscribed angle $\angle APB$ where $A$ and $B$ are adjacent vertices of the hexagon?

$120°$
$60°$
$90°$
$30°$ (correct answer)

Explanation: When you encounter problems involving regular polygons inscribed in circles, remember that inscribed angles and central angles have a fundamental relationship: an inscribed angle is always half the central angle that subtends the same arc. First, let's find the central angle. A regular hexagon has 6 equal sides, so each side subtends a central angle of

\frac{360°}{6} = 60°

. This means the arc

AB

(between adjacent vertices) measures

60°

. Now, since

P

is any point on the circle (not at a vertex), angle

\angle APB

is an inscribed angle that intercepts the same arc

AB

. By the inscribed angle theorem, this inscribed angle equals half the central angle:

\frac{60°}{2} = 30°

. Looking at the wrong answers: Choice A (

120°

) incorrectly doubles the central angle instead of halving it. Choice B (

60°

) gives you the central angle itself—this is a common mistake when students forget to apply the inscribed angle theorem. Choice C (

90°

) might seem tempting if you incorrectly think about the hexagon's internal angles, but those aren't relevant here. The correct answer is D (

30°

). Key strategy: For any regular

n

-sided polygon inscribed in a circle, each side subtends a central angle of

\frac{360°}{n}

. Any inscribed angle intercepting that same arc will be exactly half that value. This pattern appears frequently on geometry exams, so memorizing the inscribed angle theorem will save you time.

Question 6

In circle $O$ , central angle $\angle AOB = 72°$ and inscribed angle $\angle ACB$ intercepts the same arc $AB$ . Point $D$ is on the circle such that inscribed angle $\angle ADB$ intercepts arc $AB$ from the opposite side of the circle. What is the measure of $\angle ACB + \angle ADB$ ?

$72°$ (correct answer)
$108°$
$144°$
$216°$

Explanation: Since central angle

\angle AOB = 72°

, the arc

AB

measures

72°

. An inscribed angle measures half the intercepted arc, so

\angle ACB = 36°

. Since

\angle ADB

also intercepts arc

AB

(though from the opposite side), it also measures

36°

. Therefore,

\angle ACB + \angle ADB = 36° + 36° = 72°

. Choice B incorrectly adds the central angle to one inscribed angle. Choice C doubles the central angle. Choice D incorrectly uses the reflex arc measure.

Question 7

In circle $O$ , inscribed quadrilateral $ABCD$ has $\angle A = 85°$ and $\angle B = 110°$ . A diagonal $AC$ divides the quadrilateral into two triangles. What is the difference between the measures of $\angle C$ and $\angle D$ ?

$0°$
$25°$ (correct answer)
$70°$
$165°$

Explanation: In an inscribed quadrilateral, opposite angles are supplementary. So

\angle A + \angle C = 180°

and

\angle B + \angle D = 180°

. Given

\angle A = 85°

, we have

\angle C = 180° - 85° = 95°

. Given

\angle B = 110°

, we have

\angle D = 180° - 110° = 70°

. Therefore,

\angle C - \angle D = 95° - 70° = 25°

. Choice A suggests they're equal. Choice C is the value of

\angle D

. Choice D is the sum of

\angle A

and

\angle B

minus some value.

Question 8

In circle $M$ , chord $AB$ subtends a central angle of $100°$ . Point $C$ is on the major arc $AB$ , and point $D$ is on the minor arc $AB$ . What is $\angle ACB - \angle ADB$ ?

$-40°$ (correct answer)
$0°$
$40°$
$80°$

Explanation: Since the central angle is

100°

, the minor arc

AB = 100°

and the major arc

AB = 360° - 100° = 260°

. Point

C

on the major arc creates inscribed angle

\angle ACB

that intercepts the minor arc, so

\angle ACB = 100°/2 = 50°

. Point

D

on the minor arc creates inscribed angle

\angle ADB

that intercepts the major arc, so

\angle ADB = 260°/2 = 130°

. Therefore,

\angle ACB - \angle ADB = 50° - 130° = -40°

. Choice C reverses the subtraction order. Choice B assumes they're equal. Choice D adds them instead.

Question 9

Two chords $AB$ and $CD$ intersect inside circle $O$ at point $P$ . If arc $AC = 80°$ and arc $BD = 40°$ , what is the measure of $\angle APD$ ?

$40°$
$60°$ (correct answer)
$80°$
$120°$

Explanation: When two chords intersect inside a circle, the measure of each angle formed equals half the sum of the intercepted arcs.

\angle APD

intercepts arcs

AC

and

BD

, so

\angle APD = \frac{80° + 40°}{2} = 60°

. Choice A uses only one arc divided by 2. Choice C uses just one arc. Choice D uses the sum without dividing by 2.

Question 10

Circle $P$ has center at the origin. Points $A(3, 4)$ , $B(-4, 3)$ , and $C$ are on the circle. If inscribed angle $\angle BAC = 30°$ , what is the measure of the minor arc $BC$ ?

$30°$
$60°$ (correct answer)
$90°$
$120°$

Explanation: An inscribed angle measures half the intercepted arc. Since

\angle BAC = 30°

, the arc

BC

that it intercepts measures

2 \times 30° = 60°

. Choice A incorrectly equates the inscribed angle with the arc. Choice C might result from confusing this with a right angle relationship. Choice D doubles the correct answer, possibly confusing the relationship direction.

Question 11

Circle $R$ has inscribed quadrilateral $WXYZ$ . If arc $WX = 90°$ , arc $XY = 70°$ , and arc $YZ = 110°$ , what is the measure of $\angle WXY + \angle WZY$ ?

$90°$
$135°$
$180°$ (correct answer)
$270°$

Explanation: In an inscribed quadrilateral, opposite angles are supplementary.

\angle WXY

and

\angle WZY

are opposite angles, so

\angle WXY + \angle WZY = 180°

. This can also be verified by calculating: arc

ZW = 360° - 90° - 70° - 110° = 90°

\angle WXY

intercepts arc

WZY = 110° + 90° = 200°

, so

\angle WXY = 100°

\angle WZY

intercepts arc

WXY = 90° + 70° = 160°

, so

\angle WZY = 80°

. Thus

100° + 80° = 180°

. Choice A uses one arc measure. Choice B is the sum of two adjacent arc measures. Choice D uses three-quarters of the circle.