LakerFan24 wrote:

In the trapezoid above, BC is parallel to AD, angle A is 45 degrees and the length of line segment CD is 20. If the height of the trapezoid is 10, then which of the following is the value of x+y?

A) 180

B) 255

C) 270

D) 285

E) 300

Attachment:

E GEO ed.png [ 16.23 KiB | Viewed 400 times ]
Find \(x\): A base angle and upper angle on the same side of a trapezoid sum to 180°

(Parallel lines cut by a transversal create supplementary adjacent interior angles.)

45 + x = 180

x = 135

Find y:Drop an

altitude from C to base AD at X.

∆ CDX is a right triangle

Trapezoid height = 10

For right ∆ CDX, Leg CX = 10

Hypotenuse CD = 20

Let Leg DX = \(L\)

\(L^2 + 10^2 = 20^2\)

\(L^2 = 300\)

\(\sqrt{L^2} =\sqrt{100*3}\)

\(L = DX= 10\sqrt{3}\)If a right triangle has sides in ratio

\(a : a\sqrt{3} : 2a\),

angles opposite those sides, respectively, have measures 30°-60°-90°

\(10\) corresponds with \(a\)

Leg CX must be opposite a 30° angle

\(10\sqrt{3}\) corresponds with

\(a\sqrt{3}\).

Leg DX must be opposite a 60° angle

At vertex C, two angles form ∠\(y\)

∠XCD = 60

∠BCX = 90

\(y\) = (90 + 60) = 150

\(x + y = (135 + 150) = 285\)Answer D