31. The answer is C. 71°.
According to the Angles of Intersecting Chords Theorem, the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle, since the two chords RT and SU intersect inside the circle.
m∠1 = (RS + TU)/263
= (55 + TU)/2TU
= 71°
32. The answer is A. WXYZ can be a parallelogram with one pair of sides measuring 15 mm and the other pair measuring 9 mm.
Since one of the important properties of parallelograms is that the opposite sides are congruent:
WX = ZY
x+8 = 15
x = 7
Therefore, WX = x + 8 = 15
XY = WZ
2x - 5 = x + 2
x = 7
Hence, XY = 2x - 5 = 2*7 - 5 = 9 and WZ = x + 2 = 9
27. The answers are B. The triangle is a right triangle and D. The triangle is an isosceles triangle.
The triangle is actually an isosceles right triangle because it has two equal sides and has angles of 45°, 45°, and 90°.
28. The answer is D. m∠PQR = 96° and m∠PSR = 84°.
Since the sum of the opposites angles of an inscribed quadrilateral is 180°, we can solve for the value of x:
m∠PQR + m∠PSR = 180
7x - 2 + 5x + 14 = 180
x = 14
Therefore, the angles can be calculated as
m∠PQR = 7x - 2 = 7*14 - 2 = 96°
m∠PSR = 5x + 14 = 5*14 + 14 = 84°