Answer:
[tex]m\angle 2=53^{\circ}[/tex]
[tex]m\angle 7=74^{\circ}[/tex]
Step-by-step explanation:
It is given that quadrilateral GHJK is a rectangle and [tex]n\angle 3=37^{\circ}[/tex].
All interior angles of a rectangle are right angles. Diagonals are equal and bisect each other.
Now,
[tex]m\angle HKJ+m\angle HKG=90^{\circ}[/tex] (Interior angles of a rectangle are right angles)
[tex]m\angle 3+m\angle HKG=90^{\circ}[/tex]
[tex]37^{\circ}+m\angle HKG=90^{\circ}[/tex]
[tex]m\angle HKG=90^{\circ}-37^{\circ}[/tex]
[tex]m\angle HKG=53^{\circ}[/tex]
In an isosceles triangle, angle with equal sides are equal.
[tex]m\angle JGK=m\angle HKG[/tex]
[tex]m\angle 2=53^{\circ}[/tex]
Therefore, measure of angle 2 is 53 degrees.
Let the diagonals intersect each other at point O.
In triangle OGK,
[tex]m\angle OGK+m\angle OKG+m\angle GOK=180^{\circ}[/tex] (Angle su property)
[tex]53^{\circ}+53^{\circ}+m\angle GOK=180^{\circ}[/tex]
[tex]106^{\circ}+m\angle GOK=180^{\circ}[/tex]
[tex]m\angle GOK=180^{\circ}-106^{\circ}=74^{\circ}[/tex]
Vertical opposite angles are equal. So,
[tex]m\angle 7=74^{\circ}[/tex]
Therefore, measure of angle 7 is 74 degrees.
