Question

in the figure shown, what is the value of x?
triangle abc with ab and ac marked with one and two tick marks respectively, bc labeled 13x - 12.
triangle def with de and df marked with one and two tick marks respectively, ef labeled (7x - 6).
x = \square

Explanation:

Step1: Identify congruent triangles

The two triangles have two pairs of congruent sides (marked with ticks) and the included angles are equal (angle A and angle D seem to be corresponding equal angles, indicating SAS congruence). So their corresponding sides are equal. Thus, \(13x - 12=7x - 6\) (Wait, no, wait. Wait, looking at the sides: in triangle ABC, side BC is \(13x - 12\), and in triangle DEF (or D E F), side EF is \(7x - 6\)? Wait, no, maybe I mixed up. Wait, actually, the marked sides: in triangle ABC, AB and AC have ticks (AB has one tick, AC has two? Wait, no, maybe the triangles are congruent, so corresponding sides are equal. Wait, maybe the side \(13x - 12\) and \(7x - 6\) are corresponding sides? Wait, no, let's re - examine. Wait, the first triangle: ABC, with AB (one tick), AC (two ticks), and BC is \(13x - 12\). The second triangle: DEF (D, E, F), with DE (one tick), DF (two ticks), and EF is \(7x - 6\)? Wait, no, maybe the triangles are congruent by SAS, so the sides opposite the equal angles (or the corresponding sides) are equal. Wait, maybe I made a mistake. Wait, actually, the correct equation should be \(13x - 12 = 7x - 6\)? No, that would give negative x. Wait, no, maybe I flipped the sides. Wait, maybe it's \(13x - 12=7x + 6\)? No, the problem says EF is \(7x - 6\)? Wait, the user's image: the second triangle has side EF as \((7x - 6)\)? Wait, no, the first triangle's side BC is \(13x - 12\), the second triangle's side EF is \((7x - 6)\)? Wait, maybe the triangles are congruent, so BC = EF? Wait, no, that would be \(13x - 12=7x - 6\), solving: \(13x - 7x=-6 + 12\), \(6x = 6\), \(x = 1\)? But that seems odd. Wait, maybe I got the sides wrong. Wait, maybe the first triangle's side is \(13x - 12\) and the second's is \(7x + 6\)? No, the user wrote \((7x - 6)\). Wait, maybe the correct equation is \(13x-12 = 7x - 6\)? Let's solve it:

Step1: Set up the equation

Since the triangles are congruent (by SAS, as the included angles are equal and two sides are marked congruent), their corresponding sides are equal. So we set \(13x - 12=7x - 6\) (Wait, no, this gives \(13x-7x=-6 + 12\), \(6x = 6\), \(x = 1\). But let's check: if \(x = 1\), \(13x-12=1\), \(7x - 6=1\). Oh, that works. Wait, maybe that's correct.

Wait, let's do it step by step.

Step1: Set the corresponding sides equal

Because the two triangles are congruent (by the Side - Angle - Side (SAS) congruence criterion, as two sides are marked congruent and the included angle is equal), their corresponding sides are equal. So we have the equation:
\(13x-12 = 7x - 6\)

Step2: Solve for x

Subtract \(7x\) from both sides:
\(13x-7x-12=7x - 7x-6\)
\(6x-12=-6\)

Add 12 to both sides:
\(6x-12 + 12=-6 + 12\)
\(6x=6\)

Divide both sides by 6:
\(x=\frac{6}{6}=1\)

Wait, but let's verify. If \(x = 1\), then \(13x-12=13(1)-12 = 1\), and \(7x - 6=7(1)-6 = 1\). So the sides are equal, which makes sense for congruent triangles.

Answer:

\(x = 1\)