Key takeaway: The triangle angle sum theorem states that the three interior angles of any triangle add up to exactly 180 degrees, whatever the size or shape of the triangle.
Grade 88.G.A.5Geometry and Measurement
Definition of Triangle Angle Sum Theorem
The triangle angle sum theorem says that if you measure the three inside corners of any triangle and add the results, you always get \(180^{\circ}\). Skinny, wide, tilted or tiny, every triangle obeys it.
Here is why. Through one vertex, draw a line parallel to the opposite side. The two angles that appear beside that vertex match the other two corners of the triangle (alternate interior angles), and all three angles lie side by side on a straight line. So a triangle with two known angles always has a third angle you can calculate.
Formula and notation
Name the interior angles \(A\), \(B\) and \(C\), or use the measures \(m\angle A\), \(m\angle B\), \(m\angle C\). The triangle angle sum theorem reads:
\[m\angle A + m\angle B + m\angle C = 180^{\circ}\]
To find a missing angle, subtract the two known angles from \(180^{\circ}\).
Key facts
- The sum is 180 degrees for acute, right and obtuse triangles alike.
- A triangle has at most one right angle and at most one obtuse angle.
- In an equilateral triangle each angle is 60 degrees.
- The two acute angles of a right triangle add up to 90 degrees.
Triangle Angle Sum Theorem example
A triangular pennant has angles that measure \((5x – 3)^{\circ}\), \((2x + 17)^{\circ}\) and \((3x + 6)^{\circ}\). Find \(x\) and the size of each angle.
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The three angles must total 180: \((5x – 3) + (2x + 17) + (3x + 6) = 180\).
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Collect terms: \(10x + 20 = 180\), so \(x = 16\).
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Angles: \(5(16) – 3 = 77\), \(2(16) + 17 = 49\), \(3(16) + 6 = 54\).
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Check: \(77 + 49 + 54 = 180\).
\(x = 16\), and the pennant angles are \(77^{\circ}\), \(49^{\circ}\) and \(54^{\circ}\).
Common mistake
Watch out
Tomás knows two angles of a triangle are \(64^{\circ}\) and \(49^{\circ}\) and computes the third as \(360 – (64 + 49) = 247^{\circ}\).
Why it is wrong
He used 360, which is the total for a four-sided figure. For a triangle the total is 180, so the third angle is \(180 – (64 + 49) = 67\). An answer of \(247^{\circ}\) should raise a flag at once, because no angle inside a triangle can reach \(180^{\circ}\).
Related terms and lessons
- Exterior Angle Theorem
- Supplementary Angles
- Alternate Interior Angles
- Isosceles Triangle Theorem
- Triangle Inequality Theorem
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