Question

Quadrilateral ghjk is a rectangle. find measure <2 and <7 if m<3 = 37. image attached

Answer 1

`m\angle 2=53^(\circ)`

`m\angle 7=74^(\circ)`

Explanation:

It is given that quadrilateral GHJK is a rectangle and
`n\angle 3=37^(\circ)`.

All interior angles of a rectangle are right angles. Diagonals are equal and bisect each other.

Now,

`m\angle HKJ+m\angle HKG=90^(\circ)` (Interior angles of a rectangle are right angles)

`m\angle 3+m\angle HKG=90^(\circ)`

`37^(\circ)+m\angle HKG=90^(\circ)`

`m\angle HKG=90^(\circ)-37^(\circ)`

`m\angle HKG=53^(\circ)`

In an isosceles triangle, angle with equal sides are equal.

`m\angle JGK=m\angle HKG`

`m\angle 2=53^(\circ)`

Therefore, measure of angle 2 is 53 degrees.

Let the diagonals intersect each other at point O.

In triangle OGK,

`m\angle OGK+m\angle OKG+m\angle GOK=180^(\circ)` (Angle su property)

`53^(\circ)+53^(\circ)+m\angle GOK=180^(\circ)`

`106^(\circ)+m\angle GOK=180^(\circ)`

`m\angle GOK=180^(\circ)-106^(\circ)=74^(\circ)`

Vertical opposite angles are equal. So,

`m\angle 7=74^(\circ)`

Therefore, measure of angle 7 is 74 degrees.