Question

What additional piece of information is needed to show that ABC ≅ XYZ by AAS?

There are two triangles, triangle ABC and triangle XYZ. Side BC is congruent to side YZ and angle ABC is congruent to angle XYZ.

Answer 1

There are several congruence criteria between triangles. The case of your figure, the congruence must be given by the equality of the sides BC and ZY.

If we have an equal side on both sides, we need 2 angles at the end of that segment to be equal to each other. Your exercise shows that angle B and angle Y are equal.

So for them to be congruent, the angle C and the Z need to be equal.

Then the answer would be the letter C.

The first thing we have to do is identify all the angles, as shown in the figure. Triangles also show lines in yellow (single and double) in both triangles and this notation means that those sides are the same length. In other words, both triangles are isoceles.

What information can we get:

- By opposite vertices we know that R1 = R2=R

- The angles of the isosceles triangles are equal at their ends, that is: P=Q and S=R2=R

We also know that the internal sum of the angles of the triangles gives 180º. This gives us 2 more equations

`\begin{gathered} S+R+44º=180º \\ Q+R+P=180º \end{gathered}`

We replace the equalities taken from the isosceles triangles

`\begin{gathered} S+S+44º=180º \\ 2S=180º-44º \\ S=(136º)/(2) \\ S=68º=R \\ P+S+P=180º \\ 2P+68º=180º \\ P=(112º)/(2) \\ P=56º=Q \end{gathered}`