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# If O is the center and AB parallel to CD, then x=

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Math Expert
Joined: 02 Sep 2009
Posts: 62353
If O is the center and AB parallel to CD, then x=  [#permalink]

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18 Feb 2020, 05:16
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Difficulty:

55% (hard)

Question Stats:

64% (02:38) correct 36% (03:04) wrong based on 22 sessions

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If O is the center and AB parallel to CD, then x=

(A) 40
(B) 50
(C) 60
(D) 70
(E) 80

Attachment:

Untitled.png [ 114.64 KiB | Viewed 789 times ]

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Director
Joined: 16 Jan 2019
Posts: 593
Location: India
Concentration: General Management
WE: Sales (Other)
If O is the center and AB parallel to CD, then x=  [#permalink]

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18 Feb 2020, 05:34
AB and ED are intersecting lines and so

$$\angle AOE=\angle BOD$$ and
$$\angle EOB=\angle AOD$$

We have $$\angle AOE+\angle BOD+\angle EOB+\angle AOD=360$$

Which means $$20+20+2*\angle AOD=360$$

$$\angle AOD=160$$

Since OA and OD are radii, $$\angle OAD=\angle ODA$$

In iscoceles triangle OAD, $$160+2*\angle OAD=180$$

$$\angle OAD=10$$

If AB and CD are parallel $$\angle OAD=\angle ADC$$

Since ED is a diameter, triangle DCE is a right triangle

So $$10+90+x=180$$

$$x=80$$

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Director
Joined: 25 Jul 2018
Posts: 647
Re: If O is the center and AB parallel to CD, then x=  [#permalink]

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18 Feb 2020, 06:25
ED — a diameter of the circle
—> ECD — right-angled triangle: angle C = 90 degrees

AB || CD —> angle CDE = 20 degrees

Now, we got XCD right-angled triangle
—> angle D is less than 20 degrees —> x is greater than 70 degrees (x> 70)

Only the answer choice E satisfies.

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GMATWhiz Representative
Joined: 07 May 2019
Posts: 399
Location: India
Re: If O is the center and AB parallel to CD, then x=  [#permalink]

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18 Feb 2020, 22:03
Bunuel wrote:

If O is the center and AB parallel to CD, then x=

(A) 40
(B) 50
(C) 60
(D) 70
(E) 80

Attachment:
Untitled.png

Solution:

• As shown in the below diagram, x° is an interior angle of triangle CFD.

o x° + ∠FCD + ∠FDC = 180°
o Or, $$x° + y° + z° = 180° ……….Eq.(i)$$
• Now, Arc BD subtends ∠BOD at centre and ∠BAD at a circumference point.
o $$∠BOD = 2*∠BAD = 20° ⟹ ∠BAD = 10° …….Eq.(ii)$$
• Since line AB is parallel to CD,
o Substituting the value ∠BAD from Eq.(ii into the above equation, we get
 $$y = 10° …………Eq.(iii)$$
• Now, triangle CDE is a tringle circumscribed in half circle and has one of its side as diameter of the circle.
o Therefore, triangle CDE is a right-angled triangle, right angled at C.
o Or, ∠ECD or $$z = 90° …….Eq.(iv)$$
• Substituting the value of y and z from Eq.(iii) and (iv) into Eq.(i), we get
o $$x° + 10° + 90° = 180°$$
o Thus, $$x = 80$$
Thus, the correct answer is Option E.
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Re: If O is the center and AB parallel to CD, then x=   [#permalink] 18 Feb 2020, 22:03
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