Connect an Exterior Angle to Remote Interior Angles
A triangle’s interior angles total 180 degrees.
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: A triangle has two remote interior angles 35° and 50°. What is the exterior angle at its third vertex? Candidate answers: (A) 170 ° (B) 84 ° (C) 85 ° (D) 86 °.
- Independent check inputs: A triangle has two remote interior angles 36° and 51°. What is the exterior angle at its third vertex? Candidate answers: (A) 174 ° (B) 86 ° (C) 87 ° (D) 88 °.
Learn the idea
A scale drawing links represented and actual lengths through a fixed factor. Geometric relationships connect angles through straight lines and triangles, so a justified equation can replace guessing from a sketch. A triangle’s interior angles total 180 degrees. The exterior angle next to the third interior angle is supplementary to it, so the remote interior angles sum to the exterior.
Here is the method on the supplied teaching case. The third interior angle is 180−35−50=95°. Its adjacent exterior angle is 180−(95)=85°, also the sum of the remote angles.
Use the two remote interior angles rather than including the adjacent third angle in their sum. Draw and label the extended side so the exterior angle’s position remains unambiguous.
- 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: A triangle has two remote interior angles 35° and 50°. What is the exterior angle at its third vertex? Candidate answers: (A) 170 ° (B) 84 ° (C) 85 ° (D) 86 °. Solution: The third interior angle is 180−35−50=95°. Its adjacent exterior angle is 180−(95)=85°, also the sum of the remote angles. The final answer is 85 °. Use the two remote interior angles rather than including the adjacent third angle in their sum. Draw and label the extended side so the exterior angle’s position remains unambiguous.
Check your reasoning
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Helpful tips
- A triangle’s interior angles total 180 degrees.
- Use the two remote interior angles rather than including the adjacent third angle in their sum.
See the answer and explanation
87 °
The third interior angle is 180−36−51=93°. Its adjacent exterior angle is 180−(93)=87°, also the sum of the remote angles.
Take it further
Changed-input task: A triangle has two remote interior angles 42° and 57°. What is the exterior angle at its third vertex? Candidate answers: (A) 100 ° (B) 198 ° (C) 98 ° (D) 99 °. Show the mathematical reason, not only the selected answer. Expected result: 99 °. The third interior angle is 180−42−57=81°. Its adjacent exterior angle is 180−(81)=99°, also the sum of the remote angles.
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