This geometry circles angles test answer key gives the full solution and the scoring guide for each of the 9 test questions on circles: chords, arcs and inscribed angles.
Use this answer key to score the circles test out of 20 points. Every solution is fully worked, and the scoring note after each one explains how the points are earned, including partial credit for correct reasoning with a small arithmetic slip.
Geometry Circles Angles Test Answer Key: questions 1–5
Question 1: Similar circles (2 points)
- \(\frac{15}{6}=\frac{5}{2}\). Scale factor \(\frac{5}{2}\)
- Circumferences scale by the same factor: \(12\pi\cdot\frac{5}{2}=30\pi\). \(30\pi\)
Scoring: 1 point for each part.
Question 2: Central and inscribed angles (2 points)
- A central angle equals its arc. \(124^{\circ}\)
- An inscribed angle is half its arc: \(\frac{124}{2}=62\). \(62^{\circ}\)
Scoring: 1 point for each angle.
Question 3: Inscribed angle algebra (2 points)
\(5x+2=\frac{1}{2}(12x-12)\) gives \(5x+2=6x-6\), so \(x=8\). The angle is \(5(8)+2=42^{\circ}\) (the arc is \(84^{\circ}\)).
\(x=8\) and the angle measures \(42^{\circ}\).
Scoring: 1 point for a correct equation and \(x\); 1 point for the angle measure.
Question 4: Triangle in a semicircle (2 points)
- An angle inscribed in a semicircle is a right angle. \(90^{\circ}\)
- \(BC=\sqrt{26^{2}-10^{2}}=\sqrt{576}=24\). \(BC=24\)
Scoring: 1 point for naming the right angle with a reason; 1 point for the correct length.
Question 5: Tangent and radius (3 points)
- A radius is perpendicular to a tangent at the point of tangency. \(90^{\circ}\)
- \(PT=\sqrt{15^{2}-9^{2}}=\sqrt{144}=12\). \(PT=12\)
- \(180^{\circ}-90^{\circ}-53^{\circ}=37^{\circ}\). \(37^{\circ}\)
Scoring: 1 point for each part.
Geometry Circles Angles Test Answer Key: questions 6–9
Question 6: Tangents from a point (2 points)
- The angle at \(P\) and the minor arc sum to \(180^{\circ}\): \(180^{\circ}-64^{\circ}=116^{\circ}\). \(116^{\circ}\)
- \(360^{\circ}-116^{\circ}=244^{\circ}\). \(244^{\circ}\)
Scoring: 1 point for each arc.
Question 7: Chord and center (2 points)
The perpendicular from the center bisects the chord, so half the chord is 20. \(d=\sqrt{25^{2}-20^{2}}=\sqrt{225}=15\).
The distance is 15 units.
Scoring: 1 point for using the bisected chord (half length 20); 1 point for the correct distance.
Question 8: Inscribed quadrilateral (3 points)
\(\angle A\) and \(\angle C\) are opposite: \((3x+15)+(5x-3)=180\), \(8x+12=180\), \(x=21\). \(\angle B\) and \(\angle D\) are opposite: \((4y+2)+(2y+22)=180\), \(6y+24=180\), \(y=26\).
\(\angle A=78^{\circ}\), \(\angle B=106^{\circ}\), \(\angle C=102^{\circ}\), \(\angle D=74^{\circ}\). Check: \(78+106+102+74=360\).
\(x=21\), \(y=26\); angles \(78^{\circ}\), \(106^{\circ}\), \(102^{\circ}\), \(74^{\circ}\).
Scoring: 1 point for \(x\); 1 point for \(y\); 1 point for all four angle measures.
Question 9: Skating rink (2 points)
- Noah’s angle is inscribed in a semicircle. \(90^{\circ}\)
- \(NB=\sqrt{60^{2}-36^{2}}=\sqrt{2304}=48\). 48 feet
Scoring: 1 point for the angle with its reason; 1 point for the correct distance.
Scoring summary
20 points: add the points of every question of this geometry circles angles test answer key.
How to use this geometry circles angles test answer key
Use this geometry circles angles test answer key to score the test fairly and to see which skills need more practice.
Printable PDF · Test Answer Key
Geometry Circles: Chords, Arcs and Inscribed Angles test answer key
Download the PDF
See every chapter on the Geometry math page.
