Halve an Intercepted Arc for an Inscribed Angle
An inscribed angle has its vertex on the circle.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: An inscribed angle intercepts an arc of 40°. Find the angle. Candidate answers: (A) 19 ° (B) 20 ° (C) 21 ° (D) 40 °.
- Independent check inputs: An inscribed angle intercepts an arc of 44°. Find the angle. Candidate answers: (A) 21 ° (B) 22 ° (C) 23 ° (D) 44 °.
Learn the idea
Circle angle theorems distinguish central, inscribed and tangent configurations. A sector or arc occupies a stated fraction of the whole circle, but its area and length use different whole-circle measures and units. An inscribed angle has its vertex on the circle. Its rays intercept an arc, and the angle’s measure is half that arc’s angular measure.
Here is the method on the supplied teaching case. An inscribed angle is half the measure of its intercepted arc. Divide 40 by 2 to get 20.0°. Its vertex lies on the circle; a central angle has a different relationship.
Locate the angle vertex before choosing a theorem. A central angle has vertex at the center and equals its arc measure; an inscribed angle uses one half instead.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: An inscribed angle intercepts an arc of 40°. Find the angle. Candidate answers: (A) 19 ° (B) 20 ° (C) 21 ° (D) 40 °. Solution: An inscribed angle is half the measure of its intercepted arc. Divide 40 by 2 to get 20.0°. Its vertex lies on the circle; a central angle has a different relationship. The final answer is 20 °. Locate the angle vertex before choosing a theorem. A central angle has vertex at the center and equals its arc measure; an inscribed angle uses one half instead.
Check your reasoning
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Helpful tips
- An inscribed angle has its vertex on the circle.
- Locate the angle vertex before choosing a theorem.
See the answer and explanation
22 °
An inscribed angle is half the measure of its intercepted arc. Divide 44 by 2 to get 22.0°. Its vertex lies on the circle; a central angle has a different relationship.
Take it further
Changed-input task: An inscribed angle intercepts an arc of 68°. Find the angle. Candidate answers: (A) 33 ° (B) 34 ° (C) 35 ° (D) 68 °. Show the mathematical reason, not only the selected answer. Expected result: 34 °. An inscribed angle is half the measure of its intercepted arc. Divide 68 by 2 to get 34.0°. Its vertex lies on the circle; a central angle has a different relationship.
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