Geometry Circles: Angles, Chords and Tangents Worksheet Solutions (PDF)
This circle angles answers for Geometry covers circles: angles, chords and tangents (Common Core G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4), with a free PDF to print.
Geometry · Common Core State Standards: G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4
Circle angles answers: exercises 1 to 10
Exercise 1 – Central angle
A central angle equals its intercepted arc.
72°
Exercise 2 – Inscribed angle
\(110^{\circ} \div 2 = 55^{\circ}\)
55°
Exercise 3 – Find the arc
\(2 \cdot 47^{\circ} = 94^{\circ}\)
94°
Exercise 4 – Angle in a semicircle
\(\angle ACB = 90^{\circ}\) because it is inscribed in a semicircle.
\(m\angle B = 180^{\circ} – 90^{\circ} – 35^{\circ} = 55^{\circ}\)
55°
Exercise 5 – Same arc, same angle
Inscribed angles on the same arc are congruent.
\(3x + 5 = 5x – 15\)
\(x = 10\)
\(3(10) + 5 = 35^{\circ}\)
x = 10; 35°
Exercise 6 – Central and inscribed
\(4x – 30 = 2(x + 20)\)
\(2x = 70\)
\(x = 35\)
Central angle: 110°; inscribed angle: 55°.
x = 35; 110° and 55°
Exercise 7 – Inscribed quadrilateral
Opposite angles are supplementary.
\(m\angle C = 180^{\circ} – 82^{\circ} = 98^{\circ}\)
\(m\angle D = 180^{\circ} – 95^{\circ} = 85^{\circ}\)
98° and 85°
Exercise 8 – Inscribed quadrilateral algebra
\((3x + 7) + (5x – 27) = 180\)
\(8x – 20 = 180\)
\(x = 25\)
A = 82° and C = 98°.
x = 25; 82° and 98°
Exercise 9 – Tangent and distance
The tangent is perpendicular to the radius, so triangle OAP is a right triangle.
\(OP = \sqrt{5^{2} + 12^{2}} = 13\)
OP = 13
Exercise 10 – Equal tangent segments
Tangent segments from one point are congruent.
\(3x + 4 = 5x – 10\)
\(x = 7\)
\(PA = 25\)
x = 7; PA = 25
Exercises 11 and up
Exercise 11 – Tangent and radius angle
\(\angle OAP = 90^{\circ}\).
\(180^{\circ} – 90^{\circ} – 64^{\circ} = 26^{\circ}\)
26°
Exercise 12 – Chord from a central angle
The central triangle has two sides of 10 and a \(60^{\circ}\) angle between them, so it is equilateral.
10
Exercise 13 – Distance to a chord
The perpendicular from the center bisects the chord: half is 12.
\(d = \sqrt{13^{2} – 12^{2}} = 5\)
d = 5
Exercise 14 – Arcs add to 360
\(80 + 95 + 110 + x = 360\)
\(x = 360 – 285 = 75\)
75°
Exercise 15 – Tangent-chord angle
\(130^{\circ} \div 2 = 65^{\circ}\)
65°
Exercise 16 – Chords crossing inside
\(\frac{70^{\circ} + 50^{\circ}}{2} = 60^{\circ}\)
60°
Exercise 17 – Circles of a triangle
Circumradius: \(10 \div 2 = 5\).
Inradius: \(\frac{6 + 8 – 10}{2} = 2\).
5 and 2
Exercise 18 – All circles are similar
Scale factor: \(\frac{8}{3}\).
Circumference ratio equals the scale factor: \(\frac{8}{3}\).
Area ratio: \(\left( \frac{8}{3} \right)^{2} = \frac{64}{9}\).
8/3; 8/3; 64/9
Exercise 19 – Ferris wheel
\(360^{\circ} \div 8 = 45^{\circ}\).
The chord is \(2 \cdot 20 \cdot \text{sin}\,22.5^{\circ} = 15.3\) feet.
45°; 15.3 ft
Exercise 20 – Round pond
\(OP = \sqrt{15^{2} + 20^{2}} = 25\)
25 feet
Key facts behind this circle angles answers
- Central angle = intercepted arc; inscribed angle = half the arc.
- An angle inscribed in a semicircle measures 90 degrees.
- Opposite angles of an inscribed quadrilateral are supplementary.
- A tangent is perpendicular to the radius at the point of tangency.
- Tangent segments from one outside point are congruent.
How to use this circle angles answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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