Exterior Angle Theorem Math Example 2

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Example 2

medium
In △ABC\triangle ABC, the exterior angle at CC is 130°130°. If ∠A=5x°\angle A = 5x° and ∠B=3x+2°\angle B = 3x + 2°, find the value of xx and both interior angles.

Solution

  1. 1
    Step 1: By the Exterior Angle Theorem, the exterior angle at CC equals ∠A+∠B\angle A + \angle B: 130°=5x+(3x+2)130° = 5x + (3x + 2).
  2. 2
    Step 2: Simplify: 130=8x+2130 = 8x + 2, so 8x=1288x = 128, giving x=16x = 16.
  3. 3
    Step 3: ∠A=5(16)=80°\angle A = 5(16) = 80° and ∠B=3(16)+2=50°\angle B = 3(16) + 2 = 50°.
  4. 4
    Step 4: Check: 80°+50°=130°80° + 50° = 130°. ✓ Also, ∠C=180°−80°−50°=50°\angle C = 180° - 80° - 50° = 50°, and the exterior angle =180°−50°=130°= 180° - 50° = 130°. ✓

Answer

x=16x = 16; ∠A=80°\angle A = 80°, ∠B=50°\angle B = 50°.
The Exterior Angle Theorem gives a direct equation linking the exterior angle to the two remote interior angles. Setting up this equation and solving algebraically finds the unknown variable and then the individual angles. Always verify by checking both the exterior angle relationship and the angle sum.

About Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

Learn more about Exterior Angle Theorem →

More Exterior Angle Theorem Examples

Example 1 easy

In a triangle, two interior angles are [formula] and [formula]. An exterior angle is formed at the t

Example 3 easy

An exterior angle of a triangle measures [formula]. One of the remote interior angles is [formula].

Example 4 hard

Prove that an exterior angle of a triangle is always greater than either of the two remote interior

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