Triangle Angle Sum Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
In △ABC\triangle ABC, ∠A=2x+10°\angle A = 2x + 10°, ∠B=3x−5°\angle B = 3x - 5°, ∠C=x+15°\angle C = x + 15°. Find all three angles.

Solution

  1. 1
    Step 1: Apply the angle sum: (2x+10)+(3x−5)+(x+15)=180(2x+10) + (3x-5) + (x+15) = 180.
  2. 2
    Step 2: Combine like terms: 6x+20=1806x + 20 = 180, so 6x=1606x = 160, giving x=1606=26.6‾°x = \frac{160}{6} = 26.\overline{6}°.
  3. 3
    Step 3: Compute each angle: ∠A=2(26.6‾)+10=63.3‾°\angle A = 2(26.\overline{6})+10 = 63.\overline{3}°; ∠B=3(26.6‾)−5=75°\angle B = 3(26.\overline{6})-5 = 75°; ∠C=26.6‾+15=41.6‾°\angle C = 26.\overline{6}+15 = 41.\overline{6}°.
  4. 4
    Step 4: Check: 63.3‾+75+41.6‾=180°63.\overline{3} + 75 + 41.\overline{6} = 180°. ✓

Answer

∠A≈63.3°\angle A \approx 63.3°, ∠B=75°\angle B = 75°, ∠C≈41.7°\angle C \approx 41.7°.
Setting up an equation using the angle sum property and solving for the variable is the standard approach to algebraic angle problems. After finding xx, substitute back to get each angle and verify the sum equals 180°.

About Triangle Angle Sum

The three interior angles of any triangle always sum to exactly 180°180°, so knowing two angles determines the third.

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More Triangle Angle Sum Examples

Example 1 easy

A triangle has angles [formula] and [formula]. Find the third angle.

Example 2 medium

In an isosceles triangle, the vertex angle is [formula]. Find the two base angles.

Example 3 easy

Can a triangle have angles of [formula], [formula], and [formula]? Justify your answer.

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