Question

the hl theorem
in the diagram, \\( \overline { b c } \cong \overline { e f } \\) and \\( \angle a \\) and \\( \angle d \\) are right angles.
for the triangles to be congruent by hl, what must be the value of \\( x \\)?

Explanation:

Step1: Apply the HL Theorem

The HL (Hypotenuse - Leg) Theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. Given \(BC\cong EF\) (hypotenuses) and \(\angle A=\angle D = 90^{\circ}\), for \(\triangle ABC\) and \(\triangle DEF\) to be congruent by HL, the legs \(AB\) and \(DE\) must be congruent.
We know \(AB = 34\) (from the left - hand triangle) and \(DE=4x + 2\) (from the right - hand triangle).

Step2: Set up the equation

Set \(AB=DE\), so \(34=4x + 2\).
Subtract 2 from both sides of the equation: \(34−2=4x+2−2\).
We get \(32 = 4x\).

Step3: Solve for \(x\)

Divide both sides of the equation \(32 = 4x\) by 4. \(\frac{32}{4}=\frac{4x}{4}\).

Answer:

\(x = 8\)