Triangle Sum Theorem
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn the Triangle Sum Theorem and how to use it to find missing angle measures in triangles.
The Concept
Section titled “The Concept”The Triangle Sum Theorem states that the sum of the measures of the interior angles in any triangle is always 180°.
This is true for all triangles: equilateral, isosceles, scalene, right, acute, or obtuse.
If you know two angles in a triangle, you can find the third by subtracting their sum from 180°.
You can also use this theorem to check if given angles can form a triangle (their sum must be exactly 180°).
Worked Example
Section titled “Worked Example”-
In a triangle, two angles measure 50° and 60°. What is the third angle?
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In △DEF, ∠D = 90° and ∠E = 35°. What is ∠F?
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Can angles of 70°, 80°, and 40° form a triangle?
70° + 80° + 40° = 190°. No, because the sum is greater than 180°.
Real-World Application
Section titled “Real-World Application”The Triangle Sum Theorem is used in many practical situations:
- Roof design: Roof trusses form triangles, and knowing the angles helps calculate slopes and cuts.
- Construction: When building frames or shelves, you ensure angles add up correctly for stability.
- Navigation and surveying: Triangulation uses the fact that angles in triangles sum to 180° to determine distances and positions.
- Art and design: When drawing perspective or laying out patterns, understanding triangle angles helps maintain proportions.
This theorem is one of the most frequently used tools in basic geometry.