Final Answer:
The measure of ∠SPQ in the rhombus PQRS with diagonal PR drawn is 90 degrees.
Step-by-step explanation:
In a rhombus, opposite angles are equal, and the diagonals bisect each other at right angles. Therefore, ∠SPQ is half of the measure of ∠PSR. Additionally, ∠PSR is a right angle because the diagonals of a rhombus intersect at right angles. Hence, ∠SPQ is half of a right angle, making it 90 degrees.
Now, let's delve into the detailed explanation. In a rhombus, all sides are equal, and opposite angles are equal. Let ∠PSR be the angle formed at the intersection of diagonals PS and SR. Since the diagonals of a rhombus bisect each other at right angles, ∠PSR is a right angle, meaning it measures 90 degrees.
Since ∠SPQ is an angle in the same triangle (PSR) as ∠PSR, it is half of the measure of ∠PSR. Therefore, ∠SPQ = 1/2 * 90 degrees = 45 degrees. However, it's crucial to note that in a rhombus, opposite angles are equal. Thus, ∠SPQ is equal to the opposite angle, which is ∠SRP. Therefore, ∠SPQ = ∠SRP = 45 degrees.
In conclusion, the measure of ∠SPQ in the rhombus PQRS with diagonal PR drawn is 90 degrees. This is derived from the property of diagonals bisecting each other at right angles in a rhombus, leading to a right angle at the intersection point, and subsequently, ∠SPQ being half of that right angle.