Question

what is the measure of ∠spq in this rhombus?
m∠spr = (4x + 11)°
m∠qpr = (5x − 4)°
enter your answer in the box.
m∠spq = □°

Explanation:

Step1: Recall Rhombus Angle Bisector

In a rhombus, the diagonal bisects the angles. So \( \angle SPR = \angle QPR \).
Set \( 4x + 11 = 5x - 4 \).

Step2: Solve for x

Subtract \( 4x \) from both sides: \( 11 = x - 4 \).
Add 4 to both sides: \( x = 15 \).

Step3: Find \( \angle SPR \) and \( \angle QPR \)

Substitute \( x = 15 \) into \( 4x + 11 \): \( 4(15) + 11 = 60 + 11 = 71^\circ \).
Same for \( 5x - 4 \): \( 5(15) - 4 = 75 - 4 = 71^\circ \).

Step4: Find \( \angle SPQ \)

\( \angle SPQ = \angle SPR + \angle QPR = 71 + 71 = 142^\circ \).

Answer:

\( 142 \)