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655-705 (Hard)|   Geometry|      
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LakerFan24
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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees 20 Feb 2018, 20:45
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55% (hard)

Question Stats:

64% (02:24) correct 36% (02:33) wrong based on 117 sessions

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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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D

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E GEO.png [ 14.27 KiB | Viewed 3056 times ]

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Hatakekakashi
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In the trapezoid above, BC is parallel to AD, angle A is 45 degrees 20 Feb 2018, 21:58
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LakerFan24
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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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 20 Feb 2018, 22:15
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LakerFan24
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
Show more

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 22 Feb 2018, 21:02
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LakerFan24
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
Show more
Attachment:
E GEO ed.png
E GEO ed.png [ 16.23 KiB | Viewed 2836 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 12 Sep 2021, 04:44
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The two triangles are 45-45-90 and 30-60-90 triangles
10, 10, 10(root(2)) and 10, 10root(3) and 20

90+90+45+60 = 285
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