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Grade 10 · lesson 3 of 4

Combine Equal Base Angles with the Triangle Total

Equal sides in an isosceles triangle face equal base angles.

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: An isosceles triangle has vertex angle 40°. Find either equal base angle. Candidate answers: (A) 140 ° (B) 69 ° (C) 70 ° (D) 71 °.
  • Independent check inputs: An isosceles triangle has vertex angle 42°. Find either equal base angle. Candidate answers: (A) 138 ° (B) 68 ° (C) 69 ° (D) 70 °.

Learn the idea

Parallel-line and triangle theorems connect geometric conditions to numerical equations. Some side and angle values are impossible, so reasoning must check the defining configuration rather than rely only on arithmetic. Equal sides in an isosceles triangle face equal base angles. Subtract its vertex angle from the triangle total, then divide the remaining measure equally between those base angles.

Here is the method on the supplied teaching case. The two base angles are equal. They total 180−40=140°, so each is half, 70.0°. The vertex angle is the angle between the equal sides.

The vertex angle lies between the equal sides. Dividing the entire 180 degrees by two ignores that vertex; identify the known angle’s position before applying the equal-angle property.

  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: An isosceles triangle has vertex angle 40°. Find either equal base angle. Candidate answers: (A) 140 ° (B) 69 ° (C) 70 ° (D) 71 °. Solution: The two base angles are equal. They total 180−40=140°, so each is half, 70.0°. The vertex angle is the angle between the equal sides. The final answer is 70 °. The vertex angle lies between the equal sides. Dividing the entire 180 degrees by two ignores that vertex; identify the known angle’s position before applying the equal-angle property.

Your turn

Check your reasoning

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Helpful tips
  • Equal sides in an isosceles triangle face equal base angles.
  • The vertex angle lies between the equal sides.
An isosceles triangle has vertex angle 42°. Find either equal base angle.

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

See the answer and explanation

69 °

The two base angles are equal. They total 180−42=138°, so each is half, 69.0°. The vertex angle is the angle between the equal sides.

Take it further

Changed-input task: An isosceles triangle has vertex angle 54°. Find either equal base angle. Candidate answers: (A) 126 ° (B) 62 ° (C) 63 ° (D) 64 °. Show the mathematical reason, not only the selected answer. Expected result: 63 °. The two base angles are equal. They total 180−54=126°, so each is half, 63.0°. The vertex angle is the angle between the equal sides.

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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. An isosceles triangle has vertex angle 44°. Find either equal base angle.
Need a hint?
  • Equal sides in an isosceles triangle face equal base angles.
  • The vertex angle lies between the equal sides.
2. An isosceles triangle has vertex angle 46°. Find either equal base angle.
Need a hint?
  • Equal sides in an isosceles triangle face equal base angles.
  • The vertex angle lies between the equal sides.
3. An isosceles triangle has vertex angle 48°. Find either equal base angle.
Need a hint?
  • Equal sides in an isosceles triangle face equal base angles.
  • The vertex angle lies between the equal sides.

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

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