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Quizzes > Mathematics

Measuring Angles and Arcs Quiz (Geometry 10-2)

Quick, free quiz with instant results. Tackle central angle problems.

Editorial: Review CompletedCreated By: Roberto LiberoUpdated Aug 28, 2025
Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art representing a trivia quiz on Angles Arcs Challenge for high school students.

This quiz helps you practice measuring angles and arcs in circles, including central and inscribed angles. Answer 20 quick questions and get instant feedback on what to review next. For deeper practice, try our angle relationships quiz, explore arcs and chords geometry, or take a types of angles quiz.

What is an acute angle?
An angle between 90° and 180°
An angle exactly 90°
An angle exactly 180°
An angle less than 90°
An acute angle is defined as one that measures less than 90°. This concept is one of the basic building blocks in geometry.
What is the measure of a straight angle?
360°
45°
180°
90°
A straight angle forms a straight line and measures exactly 180°. This is a fundamental geometric definition.
Which of the following represents a right angle?
90°
180°
45°
120°
A right angle is one that measures exactly 90°. This is a common and essential concept in geometry.
What is the total degree measure of a circle?
180°
270°
90°
360°
A full circle is comprised of 360 degrees. This fact is key to understanding angles and arcs in circle geometry.
What is an arc in a circle?
An angle formed by two radii
A line segment connecting two points inside the circle
The distance from the center to the circle
A continuous part of the circle's circumference
An arc is a portion of the circumference of a circle. Understanding arcs is crucial for measuring parts of a circle.
If two angles are complementary, what is the sum of their measures?
270°
180°
360°
90°
Complementary angles add up to 90 degrees. This relationship is fundamental when dealing with angle measurements.
An inscribed angle intercepts an arc measuring 80°. What is the measure of the inscribed angle?
80°
160°
20°
40°
The measure of an inscribed angle is half that of its intercepted arc. Therefore, an intercepted arc of 80° yields an angle of 40°.
What is a central angle in a circle?
An angle formed by a chord and a tangent
An angle with its vertex on the circle
An angle supplementary to an inscribed angle
An angle formed by two radii with its vertex at the center
A central angle is defined by two radii of a circle and has its vertex at the center. This distinguishes it from inscribed angles.
If an arc of a circle measures 45°, what fraction of the circle's circumference does it represent?
1/2
1/4
1/8
1/6
Since a full circle measures 360 degrees, an arc of 45° represents 45/360, which simplifies to 1/8 of the circle.
Which of the following best describes vertical angles?
Adjacent angles that form a straight line
Angles that add up to 90°
Angles supplementary to their adjacent angles
Angles opposite each other at the intersection of two lines
Vertical angles are the pairs of opposite angles formed when two lines intersect. They are always equal in measure.
Which property of an exterior angle of a triangle is always true?
It always measures 180°
It is equal to its adjacent interior angle
It is always 90°
It is equal to the sum of the two remote interior angles
The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. This is a key result in triangle geometry.
When two chords intersect inside a circle, the measure of the angle formed is:
Half the difference of the intercepted arcs
Equal to one of the intercepted arcs
Half the sum of the intercepted arcs
Double the measure of one intercepted arc
The angle formed by two intersecting chords inside a circle is equal to half the sum of the measures of the intercepted arcs. This theorem is commonly applied in circle geometry problems.
How is a chord related to its corresponding arc in a circle?
The chord is always longer than its intercepted arc
The chord's measure is twice the measure of the arc
The chord and the arc have equal lengths
The chord's endpoints determine an arc whose measure equals the central angle between the radii
The endpoints of a chord create a central angle, and the intercepted arc has a degree measure equal to that angle. This relationship is pivotal in circle geometry.
What defines a diameter in a circle?
A chord that passes through the center of the circle
Any chord shorter than the circumference
A segment connecting any two points on the circle
A radius extended to the circle's boundary
A diameter is a special type of chord that passes directly through the center of the circle, making it the longest possible chord. Recognizing the diameter is important in many circle problems.
How do complementary angles differ from supplementary angles?
Both complementary and supplementary angles add to 90°
Complementary angles add to 180° while supplementary angles add to 90°
Complementary angles add to 90° while supplementary angles add to 180°
Both complementary and supplementary angles add to 180°
Complementary angles sum to 90° and supplementary angles sum to 180°. This clear distinction is essential for solving a variety of angle problems.
A circle with a radius of 10 cm has a central angle of 80°. What is the length of the intercepted arc?
40Ï€/9 cm
20Ï€/9 cm
80Ï€/9 cm
10Ï€/9 cm
The arc length is calculated using the formula: (θ/360) Ã- 2Ï€r. Substituting 80° for θ and 10 cm for r produces (80/360) Ã- 20Ï€, which simplifies to 40Ï€/9 cm.
Two chords of equal length in a circle subtend arcs of equal measure. Which theorem supports this statement?
Inscribed Angle Theorem
Chord Bisector Theorem
Tangent-Secant Theorem
Equal chords subtend equal arcs theorem
The equal chords subtend equal arcs theorem states that chords of equal length in a circle will intercept arcs of equal measure. This theorem reinforces the symmetry present in circles.
If two secants intersect outside a circle, how is the angle formed related to the intercepted arcs?
It is twice the difference of the intercepted arcs
It is half the sum of the intercepted arcs
It is equal to the larger intercepted arc
It is half the difference of the intercepted arcs
The angle formed by two secants intersecting outside a circle is half the difference between the measures of the intercepted arcs. This theorem is a more advanced concept in circle geometry.
A circle has a circumference of 31.4 cm. An arc corresponding to a 120° central angle is drawn. What is the length of the arc?
8.38 cm
12.56 cm
15.70 cm
10.47 cm
The arc length is calculated with the formula: (θ/360) Ã- circumference. Here, (120/360) simplifies to 1/3, so the arc length is 1/3 of 31.4 cm, which is approximately 10.47 cm.
What is the relationship between an angle formed by a tangent and a chord and the intercepted arc?
It is equal to the measure of the intercepted arc
It is half the measure of the intercepted arc
It is the difference between the intercepted arc and 90°
It is twice the measure of the intercepted arc
According to the tangent-chord theorem, the angle formed between a tangent and a chord is equal to half the measure of its intercepted arc. This relation is frequently used in solving circle geometry problems.
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Study Outcomes

  1. Analyze key geometric concepts related to angles and arcs.
  2. Identify different types of angles and their properties.
  3. Measure angles and arcs accurately using appropriate tools and methods.
  4. Apply geometric formulas to compute missing angle and arc measurements.
  5. Interpret and solve real-world problems involving angles and arcs.

10-2 Skills: Measuring Angles & Arcs Cheat Sheet

  1. Central Angles and Their Arcs - When you place your protractor at the circle's center, the angle you measure matches the arc it cuts out! Think of it as the circle's way of saying "What I see, I show." Use this to breeze through segment calculations.
  2. Inscribed Angles and Intercepted Arcs - Grab any two points on the circle, connect them to another point on the edge, and voila, the angle you see is always half the arc it spans! This nifty trick shrinks big arcs into bite-sized angles for easy problem-solving.
  3. Arc Length Formula - Picture wrapping a string around the circle: its length along the edge is θ (in radians) times the radius! This formula turns angles into real-world distances so you can map out any curved path.
  4. Sector Area Calculation - A circle slice (sector) area is half the radius squared times the angle in radians: ½ r² θ. Whether you're calculating pizza slices or space segments, this formula serves up the exact area.
  5. Types of Arcs - Arcs can be "minor" (under 180°) or "major" (over 180°), like choosing between a snack or a full meal! Recognizing the difference keeps your circle descriptions accurate and your solutions on point.
  6. Angles with Vertex Inside the Circle - When two chords cross inside, the angle you get equals half the sum of the arcs they intercept. It's like averaging the power of two secret circle messages! Master this to uncover hidden angles.
  7. Angles with Vertex Outside the Circle - If secants or tangents meet outside, the angle equals half the difference between intercepted arcs. Think of it as subtracting one arc from another for an "outside scoop." This is key for external angle puzzles.
  8. Inscribed Quadrilaterals - In a four-sided shape sitting on a circle, opposite angles always add up to 180°. It's a circle's way of keeping its inscribed friends in balance! This property is a lifesaver in cyclic quadrilateral problems.
  9. Arc Addition Postulate - When two arcs share an endpoint, the whole arc's measure is just their sum! This simple rule helps you piece together complex arc puzzles one segment at a time.
  10. Double Angle Formulas - In trigonometry, sin(2θ) = 2 sin(θ) cos(θ) and cos(2θ) = cos²(θ) - sin²(θ), giving you shortcuts to crunch angles twice over. These identities supercharge your solving speed on angle problems.
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