In the above figure, lines AB and CD are perpendicular to one another.

Question

Competition Mode Question https://gmatclub.com/forum/download/file.php?id=54653 In the above figure, lines AB and CD are perpendicular to one another. If AD is parallel to BC and ∠CAB = 2∠ABD, what is the measure of ∠ACB? (1) ∠CAD = 90 (2) ∠CBD = 60 Are You Up For the Challenge: 700 Level Questions 1.png

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • Lines AB and CD are perpendicular to each other.
• AD is parallel to BC.
• ∠ CAB =   2 ∠ ABD 
• Based on the above information lets’ draw the below diagram:
  https://www.dropbox.com/s/k9zh8e412m5wxml/quad_trap.png?dl=1 
• From the above diagram, we can see that  y + z = 90 ……Eq.(i)  
We need to find the value of ∠ACB.
 • This means we need to find the value of  (90 – 2x + y)

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  ∠CAD = 90
 • According to this statement:  2x + z = 90 
• Substituting the value of z from Eq.(i) into the above equation, we get,  2x + 90 -y = 90  
 o  ⟹ 90 -2x + y = 90   
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  ∠CBD = 60
 • According to this statement:  z +x = 60 
• Subtracting the above equation from Eq.(i), we get,  y – x = 30 
• However, we cannot find the value of  y – 2x  from this statement. 
Hence, statement 2 is NOT sufficient.
Thus, the correct answer is  Option A.

MayankSingh · Answer

IMO A

lines AB and CD are perpendicular to one another. If AD is parallel to BC and ∠CAB = 2∠ABD, what is the measure of ∠ACB

(1) ∠CAD = 90

AD || CB => ∠DAB = ∠CBA
Now, ∠CAD = ∠CAB + ∠DAB = ∠CAB + ∠CBA = 90
In Triangle, CAB, ∠ACB = 180- ( ∠CAB + ∠CBA)= 90
Sufficient.

(2) ∠CBD = 60

∠CBA = 90-x
So, ∠ACB = 180- (2x+90-x) = 90-x
But we don't know x, so not sufficient

freedom128 · Answer

AD is parallel to CB, then ∠DAB = ∠ABC = b. 
∠ABD = a, then ∠CAB = 2∠ABD = 2a.

Q. what is the measure of ∠ACB?

(1) ∠CAD = 90 = 2a + b
∠ACB = 180 - (2a+b)
= 180 - 90 
= 90
SUFFICIENT

(2) ∠CBD = 60 = a + b
∠ACB = 180 - (2a+b) 
= 180 - (a+b) - a 
= 180 - 60 - a
= 120 - a
We need to know the value of ∠ABD or a. 
NOT SUFFICIENT

FINAL ANSWER IS (A)

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kickmunch94 · Answer

AnthonyRitz

Statement 1)

Take triangle ACD.
Using angle sum property we get ACD +ADC = 90
ADC = DCB ( AD IS PARALLEL TO BC )
HENCE LACB = 90

Cant' we use the same approach for statement 2 ?

AnthonyRitz · Answer

No, you sadly cannot.

The problem is that you don't know whether angle ACD and angle BDC are equal; those are not opposite interior angles for the parallel lines AD and BC. Rather, you'd need AC and BD parallel, and we don't have that.

In the above figure, lines AB and CD are perpendicular to one another.

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