Triangle Angle Sum Theorem
In short: Triangle angle sum theorem: whatever the shape of a triangle, its three inside angles always total 180 degrees, so two known angles force the value of the third.
8th GradeGeometry8.G.A.5
What is Triangle Angle Sum Theorem?
Tear the three corners off a paper triangle and set them side by side with their points touching, and they fit along a straight edge, just as the triangle angle sum theorem predicts for skinny, wide, acute or obtuse triangles alike.
Calling the inside angles \(A\), \(B\) and \(C\), the statement is \(A + B + C = 180^{\circ}\). To prove it, sketch a helper line along the top vertex, level with the base; alternate interior angles then gather all three angles on that helper line.
How to spot it
- A triangle is drawn with two angle measures given and a question mark in the third corner.
- The corners carry algebraic labels such as x, 2x + 6 and 3x − 12.
- You are asked whether three given measures can be the angles of a single triangle.
- A right triangle lists one slanted corner, and the remaining corner is the target.
The method
- Set up the equation with the three angles on the left and \(180^{\circ}\) on the right.
- Plug in the measures you know, or the algebraic expressions given for each corner.
- Collect like terms, then isolate the unknown to solve.
- Re-add all three angles at the end; anything other than \(180^{\circ}\) signals an error.
Triangle Angle Sum Theorem: solved example
Problem
A pennant for a marching band has corner angles of \(x^{\circ}\), \((2x + 6)^{\circ}\) and \((3x – 12)^{\circ}\). Determine all three angles.
Solution
By the triangle angle sum theorem, \(x + (2x + 6) + (3x – 12) = 180\).
That simplifies to \(6x – 6 = 180\), so \(6x = 186\) and \(x = 31\). The other corners are \(2(31) + 6 = 68\) and \(3(31) – 12 = 81\). Since \(31 + 68 + 81 = 180\), the angles are \(31^{\circ}\), \(68^{\circ}\) and \(81^{\circ}\).
Try it
Question 1
A roof brace forms a right triangle whose sharper corner is \(35^{\circ}\). Find the other sharp corner.
Show the answer
\(180 – 90 – 35 = 55\), so the second acute angle is \(55^{\circ}\).
Question 2
Could a triangle have angles \(95^{\circ}\), \(50^{\circ}\) and \(40^{\circ}\)?
Show the answer
Those add to \(185^{\circ}\), not \(180^{\circ}\), so no such triangle exists.
Triangle Angle Sum Theorem vs. Exterior Angle Theorem
The sum rule stays inside the triangle: three interior angles, total \(180^{\circ}\). The exterior angle rule steps outside, after a side is extended, and says that outside angle equals the two far interior angles added together. Both grow from the straight-line fact, yet they answer different questions.
Keep going
Lessons
Worksheets and quizzes
- Angles in Triangles and Parallel Lines – Worksheet
- Angles in Triangles and Parallel Lines – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
- Exterior Angle Theorem
- Isosceles Triangle Theorem
- Supplementary Angles
- Alternate Interior Angles
- Pythagorean Theorem
Whole dictionary: A–Z · 8th Grade · Geometry