Geometry Circle Angles and Arcs Test Answer Key and Scoring

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

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Looking for a circle angles and arcs test answer key? Then this answer key gives the solutions and the scoring guide of the Geometry test on Circles: Angles, Arcs, Chords and Tangents. Each solution shows the theorem, the equation and the final measure. The scoring notes show how partial credit is earned for the relationship, the algebra and the answer.

Standards: G-C.A.1, G-C.A.2 · Level: Grade 10 Geometry

Sommaire

Question 1 Central and Inscribed (2 points)

The inscribed angle is half the central angle on the same arc: ½ · 124° = 62°.

m∠ACB = 62°

Scoring: 1 point for the relationship, 1 point for the answer.

Question 2 Semicircle with Algebra (2 points)

∠C = 90°, so the other two angles add to 90°: 6x + 12 = 90, giving x = 13.

∠A = 35° and ∠B = 55°.

x = 13; A = 35°, B = 55°

Scoring: 1 point for the 90° sum, 1 point for x and the angles.

Question 3 Tangent Segment (2 points)

∠OTP = 90°, so PT = √(14² − 6²) = √160 = 4√10 ≈ 12.65.

PT = 4√10

Scoring: 1 point for the right triangle, 1 point for the simplified radical.

Question 4 Inscribed Quadrilateral (3 points)

(5x − 8) + (3x + 20) = 180 gives 8x = 168, so x = 21: ∠A = 97°, ∠C = 83°.

(4y + 3) + (6y − 3) = 180 gives 10y = 180, so y = 18: ∠B = 75°, ∠D = 105°.

A = 97°, B = 75°, C = 83°, D = 105°

Scoring: 1 point for each of the two supplementary equations, 1 point for the four angles.

Question 5 Arcs and Inscribed Angles (3 points)

Arc DA = 88°.

(a) Arc DA = 360° − (64° + 96° + 112°) = 88°.

(b) ∠BAD intercepts arc BCD = 96° + 112° = 208°, so ∠BAD = 104°.

(a) 88° (b) 104° (c) 32°

Scoring: 1 point per part.

Question 6 Congruent Chords with Algebra (3 points)

6x + 4 = 8x − 22, so 26 = 2x and x = 13.

Arc AB = 6(13) + 4 = 82°. The inscribed angle is ½ · 82° = 41°.

x = 13; arc = 82°; inscribed angle = 41°

Scoring: 1 point for the equation, 1 point for x and the arc, 1 point for the inscribed angle.

Question 7 Chord in a Fountain (3 points)

The perpendicular from the center bisects the chord, so half the chord is 12 ft. Then d = √(18² − 12²) = √180 = 6√5 ≈ 13.42.

6√5 ≈ 13.4 ft

Scoring: 1 point for bisecting the chord, 1 point for the Pythagorean equation, 1 point for the answer.

Question 8 Two Tangents (2 points)

∠OAP = ∠OBP = 90°, so ∠AOB = 180° − 44° = 136°. The minor arc is 136°, so the major arc is 360° − 136° = 224°.

∠AOB = 136°; major arc = 224°

Scoring: 1 point for each answer.

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