Prove the Angle Sum Property of a Triangle

It is common knowledge that the sum of all the interior angles of a triangle equals 180°, but how do we know that? To prove that the sum of all angles of a triangle is 180 degrees, you need to understand some common geometric theorems. Using a few of these geometric concepts, there is a simple proof that can be written.

Steps

Proving the Angle Sum Property

  1. Draw a line parallel to side BC of the triangle that passes through the vertex A. Label the line PQ. Construct this line parallel to the bottom of the triangle.[1]
  2. Write the equation angle PAB + angle BAC + angle CAQ = 180 degrees. Remember, all of the angles that comprise a straight line must be equal to 180°. Because angle PAB, angle BAC, and angle CAQ combine together to make line PQ, their angles must sum to 180°. Call this Equation 1.[2]
  3. State that angle PAB = angle ABC and angle CAQ = angle ACB. Because you constructed line PQ parallel to side BC of the triangle, the alternate interior angles (PAB and ABC) made by the transversal line (line AB) are congruent. Similarly, the alternate interior angles (CAQ and ACB) made by the transversal line AC are also congruent.[2]
    • Equation 2: angle PAB = angle ABC
    • Equation 3: angle CAQ = angle ACB
    • It is a geometric theorem that alternate interior angles of parallel lines are congruent.[2]
  4. Substitute angle PAB and angle CAQ in Equation 1 for angle ABC and angle ACB (as found in Equation 2 and Equation 3) respectively. Knowing that the alternate interior angles are equal lets you substitute the angles of the triangle for the angles of the line.[1]
    • Thus we get, Angle ABC + angle BAC + angle ACB = 180°.
    • In other words, in the triangle ABC, angle B + angle A + angle C = 180°. Thus, the sum of all the angles of a triangle is 180°.

Understanding the Angle Sum Property

  1. Define the angle sum property. The angle sum property of a triangle states that the angles of a triangle always add up to 180°.[3] Every triangle has three angles and whether it is an acute, obtuse, or right triangle, the angles sum to 180°.
    • For example, in triangle ABC, angle A + angle B + angle C = 180°.
    • This theorem is useful for finding the measure of an unknown angle when you know the other two.
  2. Study examples. To really grasp this concept, it can be helpful to study some examples. Look at a right triangle, where one of the angles is 90° and the other angles each measure 45°. Summing 90° + 45° + 45° = 180°. Study other triangles of various shapes and sizes and sum their angles. You will see that they always add up to 180°.[3]
    • For the right triangle example: angle A = 90°, angle B = 45°, and angle C = 45°. The theorem states that angle A + angle B + angle C = 180°. Adding the angles gives you 90° + 45° + 45° = 180°. Therefore, left hand side (L.H.S.) equals right hand side (R.H.S.).
  3. Use the theorem to solve for an unknown angle. Using simple algebra, you can use the angle sum theorem to solve for an unknown angle if you know the other two angles of the triangle. Rearrange the basic equation to solve for the unknown angle.
    • For example, in triangle ABC, angle A = 67° and angle B = 43°, but angle C is unknown.
    • angle A + angle B + angle C = 180°
    • 67° + 43° + angle C = 180°
    • angle C = 180° - 67° - 43°
    • angle C = 70°



Related Articles

  • Classify Triangles

Sources and Citations

  1. ↑ 1.0 1.1 http://hotmath.com/hotmath_help/topics/angle-sum-theorem.html
  2. ↑ 2.0 2.1 2.2 http://www.basic-mathematics.com/angle-sum-theorem.html
  3. ↑ 3.0 3.1 http://www.algebraden.com/angle-sum-property-of-triangle.htm
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