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Apply properties of triangles to determine unknown measures

This understanding includes interpreting a triangle’s angle and side relationships to determine unknown measures: interior angles sum to 180°, an exterior angle equals the sum of its remote interior angles, and equal sides have equal opposite angles, including equilateral cases. Learners translate labeled diagrams and verbal conditions into equations and distinguish mathematically determined relationships from a figure’s visual appearance; trigonometric ratios and generalized laws of sines or cosines are beyond this scope.

Detailed Explanation: Apply properties of triangles to determine unknown measures

Use the facts given in the problem to write equations. Do not rely on how the triangle looks in a picture.

  • The interior angles of a triangle add to 180∘180^\circ.
  • Equal sides have equal opposite angles.
  • An exterior angle equals the sum of the two remote interior angles.

Example

In △ABC\triangle ABC, AB=ACAB=AC. Side BCBC is extended past CC to point DD. If ∠B=50∘\angle B=50^\circ, find ∠A\angle A, ∠C\angle C, and the exterior angle ∠ACD\angle ACD.

Step 1: Use the equal sides

Since AB=ACAB=AC, the opposite angles are equal:

∠B=∠C\angle B=\angle C

Because ∠B=50∘\angle B=50^\circ,

∠C=50∘\angle C=50^\circ

Step 2: Use the triangle angle sum

The three interior angles add to 180∘180^\circ:

∠A+∠B+∠C=180∘\angle A+\angle B+\angle C=180^\circ

Substitute the known measures:

∠A+50∘+50∘=180∘\angle A+50^\circ+50^\circ=180^\circ ∠A=80∘\angle A=80^\circ

Step 3: Find the exterior angle

The exterior angle ∠ACD\angle ACD has remote interior angles ∠A\angle A and ∠B\angle B. Therefore,

∠ACD=∠A+∠B\angle ACD=\angle A+\angle B ∠ACD=80∘+50∘=130∘\angle ACD=80^\circ+50^\circ=130^\circ

So the unknown measures are

∠A=80∘,∠C=50∘,∠ACD=130∘\boxed{\angle A=80^\circ,\quad \angle C=50^\circ,\quad \angle ACD=130^\circ}

Learn by doing: Apply properties of triangles to determine unknown measures

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