Practice
Inscribed Angles Practice
Fifty original questions on intercepted arcs, same-arc angles, semicircles, cyclic quadrilaterals, and multi-step angle reasoning.
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Question 1
Explanation
An inscribed angle is half its intercepted arc: 40/2=20 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 2
Explanation
An inscribed angle is half its intercepted arc: 66/2=33 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 3
Explanation
An inscribed angle is half its intercepted arc: 84/2=42 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 4
Explanation
An inscribed angle is half its intercepted arc: 100/2=50 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 5
Explanation
An inscribed angle is half its intercepted arc: 118/2=59 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 6
Explanation
An inscribed angle is half its intercepted arc: 140/2=70 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 7
Explanation
An inscribed angle is half its intercepted arc: 156/2=78 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 8
Explanation
An inscribed angle is half its intercepted arc: 176/2=88 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 9
Explanation
An inscribed angle is half its intercepted arc: 210/2=105 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 10
Explanation
An inscribed angle is half its intercepted arc: 260/2=130 degrees.
- Reasoning
Halve the arc only after confirming the vertex is on the circle.
Question 11
Explanation
Inscribed angles that intercept the same arc are congruent, so the second angle is 24°.
- Reasoning
Trace both chord pairs to the same arc endpoints.
Question 12
Explanation
Inscribed angles that intercept the same arc are congruent, so the second angle is 37°.
- Reasoning
Trace both chord pairs to the same arc endpoints.
Question 13
Explanation
Inscribed angles that intercept the same arc are congruent, so the second angle is 48°.
- Reasoning
Trace both chord pairs to the same arc endpoints.
Question 14
Explanation
Inscribed angles that intercept the same arc are congruent, so the second angle is 62°.
- Reasoning
Trace both chord pairs to the same arc endpoints.
Question 15
Explanation
Inscribed angles that intercept the same arc are congruent, so the second angle is 79°.
- Reasoning
Trace both chord pairs to the same arc endpoints.
Question 16
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-18=72 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 17
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-27=63 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 18
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-34=56 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 19
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-41=49 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 20
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-52=38 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 21
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-61=29 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 22
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-68=22 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 23
Explanation
Angle B intercepts the diameter and is 90°. Thus angle C=90-75=15 degrees.
- Reasoning
Mark the right angle opposite the diameter, then use the triangle sum.
Question 24
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-58=122.
- Reasoning
Use the cyclic condition, not appearance.
Question 25
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-64=116.
- Reasoning
Use the cyclic condition, not appearance.
Question 26
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-71=109.
- Reasoning
Use the cyclic condition, not appearance.
Question 27
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-83=97.
- Reasoning
Use the cyclic condition, not appearance.
Question 28
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-96=84.
- Reasoning
Use the cyclic condition, not appearance.
Question 29
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-107=73.
- Reasoning
Use the cyclic condition, not appearance.
Question 30
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-119=61.
- Reasoning
Use the cyclic condition, not appearance.
Question 31
Explanation
Opposite angles of a cyclic quadrilateral are supplementary: 180-136=44.
- Reasoning
Use the cyclic condition, not appearance.
Question 32
Explanation
The intercepted arc is twice the inscribed angle: 2(23)=46 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 33
Explanation
The intercepted arc is twice the inscribed angle: 2(31)=62 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 34
Explanation
The intercepted arc is twice the inscribed angle: 2(44)=88 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 35
Explanation
The intercepted arc is twice the inscribed angle: 2(57)=114 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 36
Explanation
The intercepted arc is twice the inscribed angle: 2(69)=138 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 37
Explanation
The intercepted arc is twice the inscribed angle: 2(82)=164 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 38
Explanation
The intercepted arc is twice the inscribed angle: 2(101)=202 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 39
Explanation
The intercepted arc is twice the inscribed angle: 2(124)=248 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 40
Explanation
The intercepted arc is twice the inscribed angle: 2(137)=274 degrees.
- Reasoning
Reverse the half-arc relationship by doubling.
Question 41
Explanation
Use 2(4x-3)=6x+10. Then 8x-6=6x+10, so x=8.
- Reasoning
Set arc equal to twice angle and validate the resulting measure.
Question 42
Explanation
Set 2(3x+5)=8x-10. Then 6x+10=8x-10, so x=10.
- Reasoning
Translate the theorem before solving.
Question 43
Explanation
Their sum is 180: 12x+12=180, so x=14. The angles are 78° and 102°.
- Reasoning
Set opposite cyclic angles to a 180-degree total.
Question 44
Explanation
The nonright angles total 90: 6x+6=90, so x=14.
- Reasoning
A diameter creates a right angle; the remaining pair is complementary.
Question 45
Explanation
Set the angles equal: 7x-9=4x+18 gives x=9, then each is 54°.
- Reasoning
Same arc means equal angle expressions.
Question 46
Explanation
The theorem works for the specified intercepted arc: half of 250° is 125°.
- Reasoning
Do not automatically replace a named major arc with its minor complement.
Question 47
Explanation
The fourth angle is opposite 91°, so it equals 180-91=89°. The other opposite pair, 68° and 112°, also totals 180°.
- Reasoning
Pair opposite, not adjacent, vertices.
Question 48
Explanation
The arc is twice the inscribed angle, 70°, and the central angle equals that arc.
- Reasoning
Use the intercepted arc as the bridge.
Question 49
Explanation
A right triangle's hypotenuse is a diameter of its circumcircle, so the radius is half of 26.
- Reasoning
Recognize the converse semicircle structure.
Question 50
Explanation
The angle is half the arc, so solving for the arc requires multiplication by 2.
- Reasoning
Identify which quantity is the half before calculating.
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Questions to review
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- Question 9Inscribed angle theoremEasy
- Question 10Inscribed angle theoremEasy
- Question 11Same-arc anglesEasy
- Question 12Same-arc anglesEasy
- Question 13Same-arc anglesEasy
- Question 14Same-arc anglesEasy
- Question 15Same-arc anglesEasy
- Question 16Semicircle right trianglesMedium
- Question 17Semicircle right trianglesMedium
- Question 18Semicircle right trianglesMedium
- Question 19Semicircle right trianglesMedium
- Question 20Semicircle right trianglesMedium
- Question 21Semicircle right trianglesMedium
- Question 22Semicircle right trianglesMedium
- Question 23Semicircle right trianglesMedium
- Question 24Cyclic quadrilateralMedium
- Question 25Cyclic quadrilateralMedium
- Question 26Cyclic quadrilateralMedium
- Question 27Cyclic quadrilateralMedium
- Question 28Cyclic quadrilateralMedium
- Question 29Cyclic quadrilateralMedium
- Question 30Cyclic quadrilateralMedium
- Question 31Cyclic quadrilateralMedium
- Question 32Intercepted arcMedium
- Question 33Intercepted arcMedium
- Question 34Intercepted arcMedium
- Question 35Intercepted arcMedium
- Question 36Intercepted arcMedium
- Question 37Intercepted arcMedium
- Question 38Intercepted arcMedium
- Question 39Intercepted arcMedium
- Question 40Intercepted arcMedium
- Question 41Inscribed angle algebraHard
- Question 42Inscribed angle algebraHard
- Question 43Cyclic algebraHard
- Question 44Semicircle algebraHard
- Question 45Same-arc algebraHard
- Question 46Major intercepted arcHard
- Question 47Cyclic structureHard
- Question 48Central-inscribed comparisonHard
- Question 49Inscribed right triangleHard
- Question 50Error analysisHard