Final answer:
To find the measure of ∠PQS in rhombus PQRS given that ∠QPR is four times ∠QSR, we use the properties of a rhombus to establish relationships between the angles and solve for the measure, which is approximately 102.86 degrees.
Step-by-step explanation:
In a rhombus, opposite angles are equal, and adjacent angles are supplementary. Given that the measure of ∠QPR is four times the measure of ∠QSR in rhombus PQRS, we can use the properties of a rhombus to find the measure of ∠PQS. Let's denote the measure of ∠QSR as 'x'. Then, the measure of ∠QPR would be '4x'. Since the sum of the angles around any point is 360 degrees, for point Q in the rhombus, we can write the following equation: x + 4x + 2x (twice the measure of ∠PQS which is equal to ∠QSR, because ∠PQS and ∠QSR are opposite angles in the rhombus) equals 360 degrees.
This gives us 7x = 360 degrees, and solving for x gives us x = 360/7 degrees. Therefore, the measure of ∠PQS, being twice the measure of ∠QSR, is 2x = 2(360/7) degrees, which simplifies to 720/7 or approximately 102.86 degrees.