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Grade 10 · lesson 2 of 4

Use the Right Angle at Tangency

A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 22°. Find its third angle. Candidate answers: (A) 136 ° (B) 67 ° (C) 68 ° (D) 69 °.
  • Independent check inputs: A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 23°. Find its third angle. Candidate answers: (A) 134 ° (B) 66 ° (C) 67 ° (D) 68 °.

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. A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent. Their intersection therefore creates an exact right angle.

Here is the method on the supplied teaching case. Radius and tangent are perpendicular at the contact point, giving 90°. The triangle angle total gives the remaining angle 180−90−22=68°.

The radius must end at the point of tangency. A different radius through another point does not establish this same angle relationship; use the actual contact point as the shared endpoint.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 22°. Find its third angle. Candidate answers: (A) 136 ° (B) 67 ° (C) 68 ° (D) 69 °. Solution: Radius and tangent are perpendicular at the contact point, giving 90°. The triangle angle total gives the remaining angle 180−90−22=68°. The final answer is 68 °. The radius must end at the point of tangency. A different radius through another point does not establish this same angle relationship; use the actual contact point as the shared endpoint.

Your turn

Check your reasoning

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Helpful tips
  • A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.
  • The radius must end at the point of tangency.
A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 23°. Find its third angle.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

67 °

Radius and tangent are perpendicular at the contact point, giving 90°. The triangle angle total gives the remaining angle 180−90−23=67°.

Take it further

Changed-input task: A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 29°. Find its third angle. Candidate answers: (A) 122 ° (B) 60 ° (C) 61 ° (D) 62 °. Show the mathematical reason, not only the selected answer. Expected result: 61 °. Radius and tangent are perpendicular at the contact point, giving 90°. The triangle angle total gives the remaining angle 180−90−29=61°.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 24°. Find its third angle.
Need a hint?
  • A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.
  • The radius must end at the point of tangency.
2. A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 25°. Find its third angle.
Need a hint?
  • A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.
  • The radius must end at the point of tangency.
3. A triangle includes a radius and the tangent at its endpoint, forming one right angle. Its other known angle is 26°. Find its third angle.
Need a hint?
  • A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.
  • The radius must end at the point of tangency.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.