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Determine unknown angles in triangles

The angle measures in every triangle sum to 180∘180^\circ, so an unknown interior angle can be determined from the other two by representing the relationship numerically or with a simple equation. Reasoning includes recognizing equal angles in isosceles triangles and using supplementary angles or the exterior-angle relationship when shown in a diagram, while avoiding the misconception that apparent visual size determines measure. Trigonometric methods and more advanced geometric proofs are not included.

Detailed Explanation: Determine unknown angles in triangles

The three interior angles of every triangle add to 180∘180^\circ.

To find an unknown angle:

  1. Add the two known angles.
  2. Subtract that sum from 180∘180^\circ.
  3. Check that the three angles add to 180∘180^\circ.

Example: In triangle ABCABC, ∠A=48∘\angle A=48^\circ, ∠B=67∘\angle B=67^\circ, and ∠C=x∘\angle C=x^\circ. Find xx.

First, write the angle-sum equation:

48+67+x=18048+67+x=180

Add the known angles:

115+x=180115+x=180

Subtract 115115 from both sides:

x=65x=65

So, the unknown angle is

65∘\boxed{65^\circ}

Check:

48∘+67∘+65∘=180∘48^\circ+67^\circ+65^\circ=180^\circ

Remember that a drawing may make one angle look larger or smaller, but the markings and given measures—not its appearance—determine the angle measure.

Learn by doing: Determine unknown angles in triangles

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