Question

quadrilateral efgh is a parallelogram. what is m∠f?

m∠f =
(type an integer or a decimal.)

(the image shows parallelogram efgh with angle at e labeled (y + 35)° and angle at g labeled (2y - 47)°)

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Also, adjacent angles (like ∠E and ∠F) are supplementary? Wait, no, in parallelogram EFGH, sides EF and EH: Wait, actually, in a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Wait, looking at the diagram, ∠E is \( (y + 35)^\circ \) and ∠G is \( (2y - 47)^\circ \). Wait, no, in a parallelogram, opposite angles are equal. Wait, ∠E and ∠G? Wait, no, EFGH: vertices E, F, G, H. So sides EF || HG and EH || FG. So ∠E and ∠G are opposite? Wait, no, ∠E and ∠G: Wait, E is connected to F and H, F to E and G, G to F and H, H to G and E. So ∠E and ∠G are opposite? Wait, no, opposite angles in a parallelogram are equal. So ∠E = ∠G? Wait, no, maybe ∠E and ∠F are consecutive, so they are supplementary? Wait, no, let's correct: In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \( 180^\circ \)). Wait, but in the diagram, ∠E is \( (y + 35)^\circ \) and ∠G is \( (2y - 47)^\circ \). Wait, maybe ∠E and ∠G are opposite? Wait, no, E and G: E is at the bottom left, F at top left, G at top right, H at bottom right? Wait, maybe the angles at E and G are opposite? Wait, no, in parallelogram EFGH, ∠E and ∠G are opposite? Wait, no, ∠E and ∠F are adjacent, ∠F and ∠G are adjacent, etc. Wait, maybe the given angles are ∠E and ∠G, which are opposite? Wait, no, the problem is to find m∠F. Wait, maybe ∠E and ∠G are equal? Wait, no, let's re-examine. Wait, the problem says EFGH is a parallelogram. So ∠E and ∠G: Wait, no, maybe ∠E and ∠F are consecutive, but the angles given are \( (y + 35)^\circ \) (∠E) and \( (2y - 47)^\circ \) (∠G). Wait, maybe ∠E and ∠G are equal? Wait, no, opposite angles are equal. So ∠E = ∠G? Wait, no, maybe ∠E and ∠F are consecutive, but the angles given are ∠E and ∠G. Wait, perhaps the angles at E and G are equal? Wait, let's set \( y + 35 = 2y - 47 \) to solve for y.

Step2: Solve for y

Set \( y + 35 = 2y - 47 \)
Subtract y from both sides: \( 35 = y - 47 \)
Add 47 to both sides: \( y = 35 + 47 = 82 \)

Step3: Find m∠E

Now, m∠E = \( y + 35 = 82 + 35 = 117^\circ \)

Step4: Find m∠F

Since in a parallelogram, consecutive angles are supplementary (∠E and ∠F are consecutive), so \( m\angle F = 180^\circ - m\angle E \)
\( m\angle F = 180 - 117 = 63^\circ \)

Wait, but let's check again. Wait, maybe ∠E and ∠F are consecutive, so they sum to 180. Alternatively, maybe ∠F is equal to ∠H, and ∠E equal to ∠G. Wait, but we solved y by setting ∠E = ∠G (opposite angles). Then ∠E is 117, so ∠F, being consecutive to ∠E, is 180 - 117 = 63.

Answer:

63