To find the values of the variables, we will first draw the diagram by carefully matching the relationship between the vertices.
Drawing the Figure
The first given piece of information tells us about the congruence of two triangles. The order of the letters in the congruence statement indicates corresponding vertices.
△ A B C≅△ D E F
We are also given the measure of several sides. Let's put all the information on the figure.
Finding x
To find x, let's focus on the segments where the measure is expressed in terms of x.
Notice that A C and D F are corresponding sides of two congruent triangles, so their measures are the same.
To find y, let's focus on the segment D E, where the measure is expressed in terms of y. Notice that the measure of the corresponding side A B of the other triangle is given.
Since these are corresponding sides of congruent triangles, their measures are the same.
We found that x=12, so AC=11+12=23. This means that the side lengths of triangle △ ABC are 7, 9, and 23. If you try to construct a triangle like this using ruler and compass, you will find that this is impossible. The measures 7 and 9 are too short to reach from A to C through B.
