Question

In the figure above, \(\overline{F G} \| \overline{J K}, F G=J K\), and \(G H=H J\). What is the measure of angle \(J\) if \(F K=H J(\sqrt{2})\) ? A) \(90^{\circ}\) B) \(120^{\circ}\) C) \(135^{\circ}\) D) \(160^{\circ}\)

Short answer

A) \(90^{\circ}\)

Step-by-step solution

Identify angle relationships

Since FG is parallel to JK, we can see that angles FGH and JHJ are alternate interior angles and thus equal. Angle H is then the exterior angle of triangle JHK.

Find angle HFK

Use the fact that the sum of the interior angles of a triangle equals 180 degrees. The triangles GHF and FHK are right-angled triangles as their sides are proportional to the sides of a 45-45-90 degree triangle. So, we have angle F = angle H = 45 degrees. Now, angle HFK = angle F + angle H = 45 + 45 = 90 degrees.

Apply the Exterior Angle Theorem

Since angle H is the exterior angle of triangle JHK and angle HFK is 90 degrees (right angle), by the Exterior Angle Theorem, we can find angle J. The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. \[ \angle J = \angle JHK + \angle F \]

Calculate angle J

We know that angle F and angle JHK are both 45 degrees. Now, substitute these values: \[ \angle J = 45 + 45 = 90 \] The measure of angle J is 90 degrees. Therefore, the correct answer is: A) \(90^{\circ}\).