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Triangles and TrigonometryTriangle Congruence

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Now that we can check if three sides can form a triangle, let’s think about how we would actually construct a triangle with these sides.

Draw a triangle that has sides of lengths 4cm, 5cm and 6cm.

In the box of the length, draw the longest side of the triangle, which is 6cm. Now we already have two of the three vertices of the triangle – the challenge is to find the last one.

Next, draw a circle of radius 4cm around one of the vertices, and a circle of radius 5cm around the other one.

The third vertex of the triangle is the intersectioncenterradius of the two circles. Now we can simply connect them to form a triangle.

The circles actually intersect twicethree timesinfinitely many times: once at the top and once at the bottom. We can pick either of these intersections, and the resulting two triangles are congruentequilateralperpendicular.

Congruence Conditions

But is it possible to construct a different triangle with the same three sides?

We already saw two triangles above, but they were both congruent. In fact, any two triangles that have the same three side lengths are congruent. This is called the SSS Congruence Condition for triangles (“Side-Side-Side”).

We now have two conditions for triangles: “AA” means that two triangles are similarcongruenttransformations, and “SSS” means that two triangles are congruentsimilarequal. There are a few more congruence conditions:

Two triangles are congruent if any one of the following conditions is met:

SSS

All sides are congruent.

SAS

Two sides and the included angle are congruent.

ASA

Two angles and the included side are congruent.

AAS

Two angles and one of the non-included sided.

You can think of these conditions as “shortcuts”: to check if two triangles are congruent, you just need to check one of the conditions above.

Once you know that two triangles are congruent, you know that all of their corresponding sides and angles are congruent. This is often called CPOCT, or “Corresponding Parts of Congruent Triangles are Congruent”.

It is interesting to note that all conditions consists of different values (either sides or angles)!

Constructing Triangles

At the beginning of this section, we saw how to construct a triangle if we know its three sides. Similarly, there are ways to construct triangles for each of the congruence conditions above.

SAS

COMING SOON – Animation

Draw the triangle that has sides of 5cm and 3cm, and an included angle of 40°.

Like before, we start by drawing one of the sides of the triangle.

Next, use a protractor to measure a 40° angle around one of the two vertices. Let’s mark this angle with a point.

We can connect the vertex to our measurement, to form the second side of the triangle.

We know that this side has to be 3cm long, so let’s measure that distance with a ruler and then mark the third vertex of the triangle.

Finally, we can connect the last two vertices, to complete the triangle.

Of course, we could have drawn the 3cm side first, or picked the other vertex to draw the 40° angle around. However in all those cases, the resulting triangles would have been congruent to this one.

ASA

COMING SOON – Animation

Draw the triangle that has angles of 70° and 50°, and an included side of length 5cm.

Let’s start by drawing the first side, using a ruler to measure 5cm.

Now let’s use a protractor to measure an angle of 70° around one of the endpoints of the side, and and angle of 50° around the other endpoint. (Which way round does not matter – the resulting triangles will be congruent.)

Connecting the angle marks to the endpoints completes the triangle.

AAS

COMING SOON – Animation

Draw the triangle that has angles of 40° and 50°, and an included side of length 5cm.

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.