Question

Complete the proof that the sum of a triangle's interior angle measures is 180 degrees.

Answer 1

The sum of a triangle's interior angles is 180 degrees, which can be proven by extending a line parallel to one side of the triangle and using alternate interior angles to show that these angles, when extended, form a straight line that measures 180 degrees.

Step-by-step explanation:

To complete the proof that the sum of a triangle's interior angles is 180 degrees, we must first understand that a triangle has three sides and is a three-sided figure lying on a plane. Every triangle, no matter the type, has interior angles that add up to a specific measure. The concept behind this proof is that if you take a line parallel to one side of the triangle and draw lines from the other two vertices to form alternate interior angles, each corresponding pair of these new angles and the ones in the triangle will be equal due to the properties of parallel lines and transversals. This effectively 'straightens out' the triangle's interior angles, aligning them along a straight line, which by definition is 180 degrees.

Here's a step-by-step explanation:

1. Draw a triangle ABC.
2. Extend a line from point C parallel to the base AB.
3. Label the intersections of this line with the extensions of sides AC and BC as points D and E, respectively.
4. Angles CAD and EBC are equal to angles CAB and CBA by alternate interior angles, as CD is parallel to AB.
5. The angle ACB is also there and is the same.
6. The angles CAD, ACB, and EBC form a straight line, which measures 180 degrees.
7. Therefore, CAB + ACB + CBA = CAD + ACB + EBC = 180 degrees, proving the sum of the interior angles of a triangle is 180 degrees.

This proof relies on the understanding that if one angle is more than another, and a third angle is more than the second, then the first is also more than the third. This knowledge is used when establishing the equality of alternate interior angles.

Answer 2

The sum of a triangle's interior angle measures is 180 degrees because each angle corresponds to the amount of rotation needed to move from one side to the next. By dividing a full rotation of 360 degrees equally among the three angles, we can determine that each angle of a triangle is 120 degrees. However, in terms of the angles measured with respect to the plane on which the triangle lies, each angle is the supplement of the corresponding angle measured around a point. Therefore, each angle in a triangle is 60 degrees, and the sum of all three angles is 180 degrees.

Step-by-step explanation:

To understand why the sum of a triangle's interior angle measures is 180 degrees, we can start by looking at a triangle as a three-sided figure lying on a plane. Each angle in the triangle corresponds to the amount of rotation needed to move from one side to the next.

By definition, a full rotation around a point is 360 degrees. So, if we divide this rotation equally among the three angles of a triangle, each angle would be 360/3 = 120 degrees. Therefore, the sum of all the angles in a triangle is 120 + 120 + 120 = 360 degrees.

However, when we say that the sum of a triangle's interior angles is 180 degrees, we are referring to the angles measured with respect to the plane on which the triangle lies. So, each angle is measured as the supplement of the corresponding angle when measured around a point. Therefore, each angle in a triangle is 180 - 120 = 60 degrees. And the sum of all three angles is 60 + 60 + 60 = 180 degrees. This is why the sum of a triangle's interior angle measures is always 180 degrees.