We are asked to draw CA, CF, and CE. To draw CA, we connect the points C and A with a segment.
Now let's do the same for CF and CE.
We are asked how many triangles were formed after drawing the three segments. Let's look at the diagram and count the triangles.
As we can see, four triangles were formed after drawing CA, CF, and CE.
Now we want to make a conjecture about which triangles are congruent and test it by measuring the sides and angles of the triangles. Let's take a look at the diagram.
It seems like △ ABC and △ EDC are congruent. The triangles CAF and CEF seem to be congruent as well!
Now let's test our conjecture! First, we will measure the sides of △ CAF and △ CEF.
As we can see, CA and CE, and AF and EF are pairs of congruent sides. We also know that FC is congruent to itself. Now, let's verify that the corresponding angles of the two triangles are congruent as well.
We can see that there are three pairs of congruent angles: ∠ ACF and ∠ ECF, ∠ CAF and ∠ CEF, and ∠ CFA and ∠ CFE. Since there are also three pairs of congruent sides, the triangles △ CAF and △ CEF are congruent.
Next we will verify that the triangles △ ABC and △ EDC are congruent. First, let's measure the sides of both triangles.
As we can see, AB and ED, BC and DC, and CA and CE are pairs of congruent sides of the two triangles. Now let's measure the angles of △ ABC and △ EDC.
We can see that ∠ ACB and ∠ ECD, ∠ CAB and ∠ CED, and ∠ ABC and ∠ EDC all form pairs of congruent angles. Together with the fact that the sides of △ ABC and △ EDC form congruent pairs, this tells us that △ ABC and △ EDC are congruent.
