Why is the sum of a triangle's angles will always equal 180 degrees?

Answer 1

By historical convention a complete circle is 360 degrees, so if you take a point on a straight line and rotate until you face the other way the total angle is 180 degrees.

Now if you take any two parallel lines, any distance apart, and mark a point anywhere on one line, and two points anywhere on the other line, then by varying these two lines and three points you can form any shaped triangle by joining up the three points.

Now because the lines are parallel there is no angle between them, so a straight line drawn from one to the other will provide two identical angles rotated 180 degrees about.

Call these two identical angles A. Now if you take a new line from one of the intersections to a new point on the other parallel line the same applies and we will call the two new identical angles B.

If you draw this out you will have a triangle with a base with angles A & B and for the sake of convention we will call the remaining angle at the peak of the triangle C.

The total angle at the peak of the triangle enabling one of the parallel lines to rotate and face the other way is, as we have said, 180 degrees but is also equal to C plus the rotated angles A and B. therefore A+B+C = 180 Degrees which is the same as your three angles in the triangle. ( It's much easier with diagrams). Or, you can work simpler like this: In a piece of paper draw a conveniently large scalene triangle ABC. Write 1 under the vertex A, 2 under the vertex B, and 3 under the vertex C (these numbers represent respectively the measure of angles A, B, and C).

Cut the paper, and on another piece of paper point the three vertices A, B, and C. Connect these three points, and write the corresponding numbers 1, 2, and 3 respectively at vertices A, B, and C.

Take the triangle and put the vertex C at the point A, A at C, and point B. Connect point B with others, and write the corresponding numbers under the corresponding vertices.

Take the triangle and put the vertex C at the point B, the vertex B at A, and point A. Connect point A with others, and write the corresponding numbers under the corresponding vertices again.

You can see that a straight angle is formed, whose measure is equal to the sum of the angles 1 + 3 + 2. Since the measure of a straight angle is equal to 180 degrees, then we say that the sum of the angles of a triangle is eaqual to 180 degrees

(m A = 1) + (m B = 2) + (m C = 3) = 180 degrees.