Which sides are congruent? Do we know, or do we have to investigate it.
x=5
Practice makes perfect
Before we determine which values of x that make the triangles congruent, we note that the triangles share a side, BC. Therefore, we can by the Reflexive Property of Congruence say that this side is congruent in our triangles.
We recognize that the exercise says all values of x which tells us that there could be more then one value of x that makes these triangles congruent. Essentially, we have two cases, either AC≅ BD and AB≅ CD, or we have AC≅ CD and AB≅ BD. We illustrate these below.
Let's go through the cases one at a time.
Case 1
By equating the sides marked as congruent, we can solve for x. If both equations give the same answer for x, the marked sides are congruent.
Congruent sides
Equation
Solve for x
AB≅ DC
5x=3x+10
x=5
AC≅ BD
5x-2=4x+3
x=5
Since both equations give the same answer, the two pairs of sides are congruent when x=5.
Case 2
We will repeat the procedure for the second case.
Congruent sides
Equation
Solve for x
AB≅ BD
5x=4x+3
x=3
AC≅ CD
5x-2=3x+10
x=4
Since the equations give different answers, the triangles are not congruent for the second case.
