Proving that the angles inside a triangle – any triangle – sum up to 180° is very simple, but leaves most people unsatisfied (or unconvinced) because it depends on the properties of something called Alternate Interior Angles.

I’ll give the proof first and then explain Alternate Interior Angles

A demonstration of the angles of a triangle summing up to 180° can be found here.

Proof

Given any triangle, how can you prove that the angles inside a triangle sum up to 180°?

1) Draw a line parallel to one of the sides of the triangle that passes through the corner opposite to that side:

2) This creates two new angles. We’ll call them D and E:

3) We know by Alternate Interior Angles that angle D must be the same as angle A and angle E must be the same as angle C.

4) Replace the new angles found in step 2) with A and C:

5) We see that angle A + angle B + angle C add up to give us a straight line or 180°:

Alternate Interior Angles

When two lines intersect (cross each other), you get four angles: A, B, C, and D:

It happens that angles A and C are identical and angles B and D are identical.

If we move one of the lines so it remains parallel to its original position (in math-speak, this is called translation), what happens to the angles? Nothing. The angles remain the same. It doesn’t matter where the two lines intersect (cross). Only the angle of intersection matters.

We don’t have to move the line, we can just draw a line parallel to it – we can draw as many parallel lines as we like (but that is likely to confuse things). Each intersection (crossing) has the same angles:

We saw earlier, that A and C are the same angle (as well, as B and D):

We call the angles between the parallel lines interior angles because they are inside (in between) the parallel lines.

We notice that the angles on opposite sides of the transversal (math-speak for the line that intersects (crosses) the parallel lines) are the same. We say the angles on alternate sides of the line are identical.

Put his all together and you get Alternate Interior Angle

Knowing all this, when we take another look at the proof that all the angles in a triangle sum to 180° we see the following:

1) the base of the triangle forms one of the parallel lines and we draw the second.

2) each of the sides of the triangle acts like a transversal intersecting the parallel lines.

Because angles on alternate sides of the transversal are identical, we are able to sum all three angles and show that they sum up to 180°

The post Proving that the Angles in a Triangle Sum up to 180° appeared first on Complete, Concrete, Concise.

Here is a simple way to demonstrate that fact.

Materials

• paper
• pencil or pen
• ruler (or some sort of straight-edge)
• coloured crayon
• scissors

1) Draw a triangle on a sheet of paper. Any sort of triangle you like. Use the ruler or straight-edge to ensure the sides are straight:

2) Colour the edges of the triangles. Do not colour the inside of the triangle. If you really want to colour the inside of the triangle, then use a different colour for the inside from the edges:

3) Cut out the triangle. Make sure the edges are as straight as possible:

4) Cut the corners off the triangle. Make the corners large enough so they are easy to handle:

Notice that each corner has two coloured edges and one uncoloured edge. (I should have used a more contrasting colour than green – maybe red – or else pressed harder when I coloured.

5) Draw a straight line on a sheet of paper using the ruler (I used one of the scraps from the paper I cut the triangle out of):

6)Assemble the corners on the straight line. Ensure that (1)coloured edges touch the straight line and (2) coloured edges touch other coloured edges:

What’s Happening

Each corner that you cut off contains an angle from the triangle. This is why we coloured the edges so we can easily see the angle contained by the edges.

When we assemble the angles (by aligning the coloured edges), we see that all the angles add up to a straight line (or 180°). In other words: Angle 1 + Angle 2 + Angle 3 = 180°

The post Demonstrating that the Angles in a Triangle Sum up to 180 Degrees appeared first on Complete, Concrete, Concise.