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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees

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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees  [#permalink]

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New post 20 Feb 2018, 20:45
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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

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E GEO.png
E GEO.png [ 14.27 KiB | Viewed 625 times ]

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Re: In the trapezoid above, BC is parallel to AD, angle A is 45 degrees  [#permalink]

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New post 20 Feb 2018, 21:58
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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



given cd = 20

hieght is 10 , we have a right angle triangle.. hypotunese 20 side= 10

now we know side1^2 + side2^2 = hypotunese^2 ( pythagoras therom)

10^2 + x^2 = 20^2 i.e x^2 = 400 -100 = 300

X= rt300 = 10 rt 3

now sides are 10, 10rt3 and 20 ratio 1:rt3:2 .. the angles are 90-60-30

the smaller angle has the smaller side i.e angle D = 30

now x+y+ angle a + angle b = 360 ( angle in a quadilateral)

x+y = 360 - a - b = 360 - 45 -30 =285

(D) imo
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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees  [#permalink]

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New post 20 Feb 2018, 22:15
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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


We have -

\(X + A = 180\). Given, \(A = 45 --> X = 135\)
\(Y + D = 180..... Y = 30\) as Sin Y = \(\frac{10}{20}. Hence Y = 150\)
Hence, \(X + Y = 150 + 135 = 285\)
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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees  [#permalink]

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New post 22 Feb 2018, 21:02
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
E GEO ed.png [ 16.23 KiB | Viewed 502 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
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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees   [#permalink] 22 Feb 2018, 21:02
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