Triangle ABC has two congruent sides AB and AC, indicated by the single hash marks. This makes triangle ABC an isosceles triangle. In an isosceles triangle, the angles opposite the congruent sides are also congruent. Therefore, ∠ABC and ∠ACB are congruent.
Triangle DEF has two congruent sides DF and EF, indicated by the double hash marks. This makes triangle DEF an isosceles triangle. In an isosceles triangle, the angles opposite the congruent sides are also congruent. Therefore, ∠FDE and ∠FED are congruent.
We are given that ∠BAC and ∠FDE are congruent, as indicated by the identical arc markings. Since ∠FDE and ∠FED are congruent, we can conclude that ∠BAC, ∠FDE, and ∠FED are all congruent. Let's denote the measure of these congruent angles as 'y'.
We also know that ∠ABC and ∠ACB are congruent. Let's denote the measure of these congruent angles as 'z'.
Since triangle ABC and triangle DEF are both isosceles triangles with one pair of congruent angles being equal, they are similar triangles. This implies corresponding sides are proportional, and all corresponding angles are congruent. Hence, ∠ABC is congruent to ∠DEF.
Since ∠ABC and ∠DEF are congruent, and we know the expressions for their measures, we can set up an equation:
\(13x - 12 = 7x - 6\)
Subtract \(7x\) from both sides:
\(6x - 12 = -6\)
Add 12 to both sides:
\(6x = 6\)
Divide both sides by 6:
\(x = 1\)
Answered
