Hi, there--

THE PROBLEM:
There are two triangles that are side-by-side but not touching. 

Given: line PR is congruent to line DE, line PT is congruent to line DF, angle R is congruent to 
angle E, and angle T is congruent to angle F.

Prove: triangle PRT is congruent to triangle DEF. 

A SOLUTION:
Triangle PRT is congruent to triangle DEF because by AAS (angle-angle-side.) Then two 
corresponding angles and a corresponding, but non-included side are congruent, the triangles 
are congruent. (non-included means that the side is not between to two angles.)

You have two corresponding congruent angles:
Angle R is congruent to angle E
Angle T is congruent to angle F

You actually have two corresponding non-included sides. (You only need one.)
Line segment PR is congruent to line segment DE
Line segment PT is congruent to line segment DF

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I'm wondering if you have proved the AAS congruence theorem in your class yet. If not, then 
you cannot use it in this problem. (We cannot use what we haven't proved.)
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Never fear, you can also use SAS (side-angle-side) for your proof. Here is how:

We are given two triangles PRT and DEF. Line segment PR is congruent to line segment DE. 
Line PT is congruent to line DF. Angle R is congruent to angle E, and angle T is congruent to 
angle F.

We want to show that angle P is congruent to angle D. Then we will have two pairs of 
congruent angles and an included side congruent. Then the triangles will be congruent by SAS.

Let's show that angle P is congruent to angle D.

We know that the sum of the interior angles in any triangle is 180 degrees. 

In triangle PRT, the measures of angle P + angle R + angle T = 180
In triangle DEF, the measures of angle D + angle E + angle F = 180

The measures of angle R and angle E are equal because they are congruent.
The measures of angle T and angle F are equal because they are congruent.

Substitute angle R for angle E and angle T for angle F in the second equation.

Then the measures of angle D + angle R + angle T = 180

Rearrange this equation in terms of angle D.
angle D = 180 - (angle R + angle T)

Rearrange the first equation in arms of angle P.
angle P = 180 - (angle R + angle T)

Notice that angle D and angle P are both equal to 180 - (angle R + angle T). By the transitive 
property of equality, their measures are equal.

Angle D is congruent to angle P because they have equal measure.

Line segment PT is the included side between angle P and angle T.
Line segment DF is the included side between angle D and angle F.

Therefore, triangle PRT is congruent to triangle DEF by SAS (side-angle-side).

Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com