It is known that the sum of the internal angles of a triangle is equal to 180 °, but how did this claim come about? To prove this, you need to know the common theorems of geometry. Using some of these concepts, you can simply proceed to the demonstration.
Steps
Part 1 of 2: Prove the Property of the Sum of Angles
Step 1. Draw a line parallel to the BC side of the triangle crossing vertex A
Name this segment PQ and build this line parallel to the base of the triangle.
Step 2. Write the equation:
angle PAB + angle BAC + angle CAQ = 180 °. Remember that all angles that make up a straight line must be 180 °. Since the angles PAB, BAC and CAQ all together form the segment PQ, their sum must be equal to 180 °. Define this equality as "Equation 1".
Step 3. State that the angle PAB is equal to the angle ABC and that the angle CAQ is the same as that of ACB
Since the line PQ is parallel to the side BC by construction, the alternate internal angles (PAB and ABC) defined by the transverse line (AB) are congruent; for the same reason, the alternate internal angles (CAQ and ACB) defined by the diagonal line AC are equal.
- Equation 2: angle PAB = angle ABC;
- Equation 3: angle CAQ = angle ACB.
- The equality of the alternate internal angles of two parallel lines crossed by a diagonal is a geometry theorem.
Step 4. Rewrite equation 1 by replacing angle PAB with angle ABC and angle CAQ with angle ACB (found in equation 2 and 3)
Knowing that the alternate internal angles are the same, you can replace those that make up the line with those of the triangle.
- Consequently, you can state that: angle ABC + angle BAC + angle ACB = 180 °.
- In other words, in a triangle ABC, the angle B + the angle A + the angle C = 180 °; it follows that the sum of the internal angles is equal to 180 °.
Part 2 of 2: Understanding the Property of the Sum of Angles
Step 1. Define the property of the sum of the angles of a triangle
This states that adding the internal angles of a triangle always gives the value of 180 °. Each triangle always has three vertices; regardless of whether it is acute, obtuse or rectangle, the sum of its angles is always 180 °.
- For example, in a triangle ABC, the angle A + the angle B + the angle C = 180 °.
- This theorem is useful for finding the width of an unknown angle by knowing that of the other two.
Step 2. Study some examples
To internalize the concept, it is worth considering some practical examples. Look at a right triangle where one angle measures 90 ° and the other two 45 °. Adding the amplitudes you find that 90 ° + 45 ° + 45 ° = 180 °. Consider other triangles of different sizes and types and find the sum of the internal angles; you can see that the result is always 180 °.
For the example of the right triangle: angle A = 90 °, angle B = 45 ° and angle C = 45 °. The theorem states that angle A + angle B + angle C = 180 °. Adding the amplitudes you find that: 90 ° + 45 ° + 45 ° = 180 °; consequently, equality is verified
Step 3. Use the theorem to find an angle of unknown magnitude
By performing some simple algebraic calculations, you can exploit the theorem of the sum of the internal angles of a triangle to find the value of the unknown one by knowing the other two. Change the arrangement of the terms of the equation and solve it for the unknown.
- For example, in a triangle ABC, the angle A = 67 ° and the angle B = 43 °, while the angle C is unknown.
- Angle A + angle B + angle C = 180 °;
- 67 ° + 43 ° + angle C = 180 °;
- Angle C = 180 ° - 67 ° - 43 °;
- Angle C = 70 °.
