# 三角形与三角学三角形全等

SSS

SAS

ASA

AAS

## Constructing Triangles

### AAS

Draw the triangle that has angles of 40° and 50°, and an included side of length 5cm. 画出角度为40° 和 50°

Again, we’ll start by constructing the first side of the triangle, which is 5cm long.

And again, we’ll use a protractor to measure an angle of 40° around one of the endpoints, and draw the second side of the triangle. However, we don’t yet know where this side will end.

Instead, let’s pick any point around this line, pretend it’s the third vertex of the triangle and measure an angle of 50°.

As you can see, this doesn’t quite work: the third side does not yet link up with the vertex A. To fix this, we simply have to shift it: we draw a parallel line that goes through A. (You already learned how to construct parallel lines in a previous course.)

Now the two angles at the top are alternate angles, so they must be congruent and both 50°. We can erase the incorrect, first line to get our completed AAS triangle.

### SSA

The SSA construction is slightly different. You might have noticed that “SSA” was not in the list of congruence conditions above, so comparing SSA is two triangles is not enough to confirm they are congruent. This will show you why:

COMING SOON – Animation

Draw the triangle that has sides of 4cm and 5cm, and a non-included angle of 50°.

Like always, let’s start by drawing the first side of the triangle which is 5cm long.

Next, let’s measure an angle of 50° around one of the endpoints and draw the second side of the triangle. However, we don’t yet know where this side will end.

The third side has o be 4cm long. Using a protractor we can draw a circle of radius 4cm around the other endpoint of the original side.

The final vertex of the triangle is formed by the intersection of the circle and the second line. However, in this case, there are two intersections!

These two triangles are clearly not congruent. This means that there are two different triangles that have sides of 4cm and 5cm, as well as a non-included angle of 50°. SSA is not enough to confirm two triangles are congruent.

Archie