Question

find ( mangle spr )
in the diagram, quadrilateral ( pqrs ) has diagonals intersecting at ( t ). angles at ( s ) and ( r ) are ( (7x - 37)^circ ) and ( (2x + 1)^circ ) respectively.

Explanation:

Step1: Set angles equal (diagonals bisect)

Since the diagonals of a rectangle bisect each other, $\triangle STR$ is isosceles, so $\angle SRT = \angle RST$.
$$7x - 37 = 2x + 1$$

Step2: Solve for x

Subtract $2x$ and add 37 to both sides.
$$7x - 2x = 1 + 37$$
$$5x = 38$$
$$x = \frac{38}{5} = 7.6$$

Step3: Calculate $\angle SPR$

$\angle SPR$ is equal to $\angle SRT$ (alternate interior angles for parallel sides $PQ$ and $SR$ with transversal $PR$). Substitute $x$ into $2x+1$:
$$m\angle SPR = 2(7.6) + 1 = 15.2 + 1 = 16.2$$

Step4: Calculate $m\angle QTR$

$\angle QTR$ is supplementary to $\angle STR$. First find $\angle STR = 180 - 2(16.2) = 147.6^\circ$, so:
$$m\angle QTR = 180 - 147.6 = 32.4^\circ$$

Answer:

$m\angle SPR = 16.2^\circ$, $m\angle QTR = 32.4^\circ$