Question

11 quadrilateral ebcf and \\(\overline{ad}\\) are drawn below, such that abcd is a parallelogram, \\(\overline{eb} \cong \overline{fb}\\), and \\(ef \perp fh\\).

if \\(m\angle e = 62^\circ\\) and \\(m\angle c = 51^\circ\\), what is \\(m\angle fhb\\)?

1. \\(79^\circ\\)
2. \\(76^\circ\\)
3. \\(73^\circ\\)
4. \\(62^\circ\\)

Explanation:

Step1: Find $\angle EFB$ in $\triangle EBF$

Since $\overline{EB} \cong \overline{FB}$, $\triangle EBF$ is isosceles with $\angle E = \angle EFB = 62^\circ$.

Step2: Calculate $\angle EBF$

Sum of angles in a triangle is $180^\circ$.
$\angle EBF = 180^\circ - 62^\circ - 62^\circ = 56^\circ$

Step3: Find $\angle EBC$ (parallelogram property)

In parallelogram $ABCD$, $\angle ABC = 180^\circ - \angle C = 180^\circ - 51^\circ = 129^\circ$

Step4: Calculate $\angle FBC$

$\angle FBC = \angle ABC - \angle EBF = 129^\circ - 56^\circ = 73^\circ$

Step5: Find $\angle EFH$

Since $EF \perp FH$, $\angle EFH = 90^\circ$. $\angle BFH = \angle EFH - \angle EFB = 90^\circ - 62^\circ = 28^\circ$

Step6: Calculate $\angle FHB$ (triangle angle sum)

In $\triangle FHB$, $\angle FHB = 180^\circ - \angle FBC - \angle BFH = 180^\circ - 73^\circ - 28^\circ = 79^\circ$

Answer:

1. $79^\circ$