Circle Angles and Arcs Worksheet
High School Geometry
Circle Angles & Arcs
Name:
Date: Period:
Core Theorem: The measure of a minor arc equals its central angle: \(m\widehat{AB} = m\angle AOB\) Semicircle: \(180^\circ\) Full Circle: \(360^\circ\)
Part 1: Minor Arcs & Central Angles (Direct Relationships) Find the indicated arc or angle measure.
1
Given: \(m\angle AOB = 68^\circ\)
O A B 68°
Find the measure of minor arc \(\widehat{AB}\).
\(m\widehat{AB} =\)
2
Given: \(m\widehat{CD} = 115^\circ\)
P C D
Given that \(m\widehat{CD} = 115^\circ\), find central angle \(m\angle CPD\).
\(m\angle CPD =\)
3
\(\overline{EG}\) is a diameter
K E G F 48°
Given \(m\angle FKG = 48^\circ\), find the measure of minor arc \(\widehat{FG}\).
\(m\widehat{FG} =\)
4
Given: \(m\widehat{KL} = 56^\circ\)
Q K M L
Diameter \(\overline{KM}\) has arc \(m\widehat{KL} = 56^\circ\). Find \(m\angle LQM\).
\(m\angle LQM =\)
5
\(\overline{OA} \perp \overline{OB}\)
O A B
Given \(\overline{OA} \perp \overline{OB}\), find the measure of minor arc \(\widehat{AB}\).
\(m\widehat{AB} =\)
6
Given: \(m\widehat{ST} = 142^\circ\)
C R T S
Diameter \(\overline{RT}\) has arc \(m\widehat{ST} = 142^\circ\). Find central angle \(m\angle RCS\).
\(m\angle RCS =\)
Circle Angles & Arcs Practice Page 1 of 5
Part 2: Adjacent Arcs & Semicircles
Using Linear Pairs and Angle Addition
Questions 7–12
Arc Addition Postulate: If point \(B\) is on \(\widehat{AC}\), then \(m\widehat{AB} + m\widehat{BC} = m\widehat{ABC}\). A diameter divides the circle into two \(180^\circ\) semicircles.
7
\(\overline{AD}\) is a diameter
O A B C D
Given diameter \(\overline{AD}\) with \(m\widehat{AB} = 52^\circ\) and \(m\widehat{BC} = 64^\circ\), find \(m\widehat{CD}\).
\(m\widehat{CD} =\)
8
Given: \(m\widehat{WX} = 74^\circ\)
P W X Y Z
Intersecting diameters \(\overline{WY}\) and \(\overline{XZ}\) have \(m\widehat{WX} = 74^\circ\). Find \(m\widehat{XY}\).
\(m\widehat{XY} =\)
9
\(\overline{JL}\) is a diameter
C 75° J K L
Given diameter \(\overline{JL}\) and \(m\angle KCL = 75^\circ\), find minor arc \(m\widehat{JK}\).
\(m\widehat{JK} =\)
10
\(\overline{PS}\) is a diameter
O 38° 47° P Q R S
Given \(m\angle POQ = 38^\circ\) and \(m\angle ROS = 47^\circ\), find \(m\widehat{QR}\).
\(m\widehat{QR} =\)
11
Three Radii
M A B C
Given \(m\widehat{AB} = 110^\circ\) and \(m\widehat{BC} = 120^\circ\), find minor arc \(m\widehat{AC}\).
\(m\widehat{AC} =\)
12
\(\overline{AC}\) is a diameter
O 68° A B C
Given diameter \(\overline{AC}\) and \(m\angle AOB = 68^\circ\), find \(m\angle BOC\).
\(m\angle BOC =\)
Circle Angles & Arcs Practice Page 2 of 5
Part 3: Major Arcs & Full Circles
Three-Letter Major Arc Notation
Questions 13–18
Major Arc Rule: Major arcs measure greater than \(180^\circ\) and are named with three letters (e.g., \(\widehat{ACB}\)). \(m(\text{major arc}) = 360^\circ - m(\text{minor arc})\)
13
Major Arc Target
O 105° C A B
Given \(m\angle AOB = 105^\circ\), find major arc \(m\widehat{ACB}\).
\(m\widehat{ACB} =\)
14
Given: \(m\widehat{DFE} = 290^\circ\)
P F D E
Given major arc \(m\widehat{DFE} = 290^\circ\), find central angle \(m\angle DPE\).
\(m\angle DPE =\)
15
\(\overline{AC}\) is a diameter
O 65° A B C D
Given \(m\angle BOC = 65^\circ\), find the measure of major arc \(\widehat{ABD}\).
\(m\widehat{ABD} =\)
16
Minor to Major Arc
Q F G H
Given \(m\angle GQH = 90^\circ\), find the measure of major arc \(\widehat{GFH}\).
\(m\widehat{GFH} =\)
17
Given: \(m\widehat{JLK} = 312^\circ\)
C J K L
Given major arc \(m\widehat{JLK} = 312^\circ\), find minor arc \(m\widehat{JK}\).
\(m\widehat{JK} =\)
18
\(\overline{RT}\) is a diameter
O 50° R S T U
Given diameter \(\overline{RT}\) and \(m\angle ROS = 50^\circ\), find major arc \(m\widehat{STU}\).
\(m\widehat{STU} =\)
Circle Angles & Arcs Practice Page 3 of 5
Part 4: Algebraic Problem Solving
Solving for Unknown Variable \(x\)
Questions 19–24
Strategy: Set up an equation using geometric properties: congruent angles/arcs (\(=\)), supplementary linear pairs (\(= 180^\circ\)), or vertical angles. Solve for \(x\).
19
Given: \(m\widehat{AB} = 76^\circ\)
O 4x+12° A B
If \(m\angle AOB = (4x + 12)^\circ\) and \(m\widehat{AB} = 76^\circ\), find \(x\).
\(x =\)
20
Solve for \(x\)
P 100° C D
If \(m\angle CPD = 100^\circ\) and \(m\widehat{CD} = (7x - 5)^\circ\), find \(x\).
\(x =\)
21
\(\overline{AB}\) is a diameter
O 5x+10° 3x+10° A C B
If \(\angle AOC = (5x + 10)^\circ\) and \(\angle BOC = (3x + 10)^\circ\), find \(x\).
\(x =\)
22
Vertical Arcs
P W X Y Z
Diameters \(\overline{WY}\) and \(\overline{XZ}\) intersect at \(P\). If \(m\widehat{WX} = (6x - 14)^\circ\) and \(m\widehat{ZY} = (4x + 18)^\circ\), find \(x\).
\(x =\)
23
Solve for \(x\)
O 6x+22° E F
If \(m\angle EOF = (6x + 22)^\circ\) and \(m\widehat{EF} = (9x - 8)^\circ\), find the value of \(x\).
\(x =\)
24
\(\overline{JK}\) is a diameter
C J L K
Given diameter \(\overline{JK}\) with \(m\widehat{JL} = (11x - 3)^\circ\) and \(m\widehat{LK} = (4x + 3)^\circ\), find \(x\).
\(x =\)
Circle Angles & Arcs Practice Page 4 of 5
Part 5: Multi-Step Mastery
Solve for \(x\) and Find the Target Measure
Questions 25–30
Two-Step Instructions: (1) Write and solve an equation for \(x\). (2) Substitute \(x\) back into the expression to find the specified arc or angle.
25
\(\overline{AB}\) is a diameter
O 3x+15° 2x+5° A C B
If \(\angle AOC = 3x+15\) and \(\angle BOC = 2x+5\), find \(x\) and \(m\widehat{AC}\).
\(x =\)
\(m\widehat{AC} =\)
26
Solve & Substitute
O 5x-4° M N P
If \(m\angle MON = (5x - 4)^\circ\) and \(m\widehat{MN} = (3x + 28)^\circ\), find \(x\) and major arc \(m\widehat{MPN}\).
\(x =\)
\(m\widehat{MPN} =\)
27
Full Circle Sum \(360^\circ\)
C A B D
If \(m\widehat{DA} = 2x+10\), \(m\widehat{AB} = 3x+20\), and \(m\widehat{BD} = 5x-30\), find \(x\) and \(m\widehat{AB}\).
\(x =\)
\(m\widehat{AB} =\)
28
\(\overline{RT}\) is a diameter
P R S T
Given diameter \(\overline{RT}\) with \(m\widehat{RS} = (4x + 2)^\circ\) and \(m\widehat{ST} = (7x - 9)^\circ\), find \(x\) and \(m\widehat{ST}\).
\(x =\)
\(m\widehat{ST} =\)
29
Minor & Major Arcs
O C A B
If minor arc \(m\widehat{AB} = (2x + 30)^\circ\) and major arc \(m\widehat{ACB} = (5x + 50)^\circ\), find \(x\) and \(m\widehat{ACB}\).
\(x =\)
\(m\widehat{ACB} =\)
30
Intersecting Diameters
O E F G H
Diameters \(\overline{EG}\) and \(\overline{FH}\) intersect. If \(m\widehat{EF} = (3x + 20)^\circ\) and \(m\widehat{FG} = (6x - 20)^\circ\), find \(x\) and \(m\widehat{EHG}\).
\(x =\)
\(m\widehat{EHG} =\)
Circle Angles & Arcs Practice Page 5 of 5
