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Apply the triangle angle-sum relationship

The three interior angles of every triangle sum to 180°, regardless of the triangle’s size, shape, or orientation. This relationship supports finding an unknown angle by representing the known and unknown measures in an equation and distinguishes interior-angle measures from side lengths or appearances in a diagram. The scope is limited to angle-sum applications, not exterior-angle theorems, trigonometry, or formal deductive proofs.

Detailed Explanation: Apply the triangle angle-sum relationship

Every triangle has three interior angles that add to 180∘180^\circ:

angle1+angle2+angle3=180∘\text{angle}_1+\text{angle}_2+\text{angle}_3=180^\circ

The triangle’s size, shape, or orientation does not change this relationship. Focus on the angle measures inside the triangle, not the side lengths.

Example: Find the unknown angle xx.

The two known interior angles measure 52∘52^\circ and 67∘67^\circ.

  1. Write an equation using the triangle angle-sum relationship:

52∘+67∘+x=180∘52^\circ+67^\circ+x=180^\circ

  1. Add the known angles:

119∘+x=180∘119^\circ+x=180^\circ

  1. Subtract 119∘119^\circ from both sides:

x=61∘x=61^\circ

  1. Check the answer:

52∘+67∘+61∘=180∘52^\circ+67^\circ+61^\circ=180^\circ

Therefore, the unknown interior angle is 61∘\boxed{61^\circ}.

Learn by doing: Apply the triangle angle-sum relationship

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    Geometry of Triangles - Isosceles, Solve Angle from Double Angle

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