In the diagram below, if angle BAC is a right angle, is triangle ADC a

Question

In the diagram below, if angle BAC is a right angle, is triangle ADC an equilateral triangle?

(1) ADC = 60 degrees
(2) EF is parallel to BC

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Official Answer

C

LighthousePrep · Answer

Solution in the video below:

https://www.youtube.com/watch?v=5tMBn-s_GGQ

SiffyB · Answer

To find: Whether triangle ADC is equilateral

To do this, we can either determine whether AD = DC = AC or that each interior angle = 60 degree

Statement 1: ADC = 60 degrees

At least one angle is equal to 60 degree, but we don't know whether AD = DC, so we can't say ADC is equilateral.
Insufficient (BCE)

Statement 2: EF is parallel to BC

This is a very interesting statement.
Let the point where EF and AD intersect be X
If EF is parallel to BC, then ADC = AXF (Corresponding angles)
Also, DFA = 90 and EAF = 90

Thus, EFA = b (since DAF has to be equal to EFA)

So, we have established that: in triangle AXF, XAF = XFA = b and AXF = ADC.
If we get to know the value of ADC, we will be able to determine whether triangle ADC is equilateral.
But, without concrete numbers, we can't say.
Insufficient (CE)

Let's combine the two statements now:

From statement 1 we get ADC = 60 degree
Thus, AXF = ADC = 60 degree

Applying angle sum property in triangle AXF, we have
2 b + AXF = 180
2b + 60 = 180
2b = 120
b = 60.

Now, in triangle ADC, 
Angle ADC = 60 (acc to statement 1)
Angle DAC = b = 60 (just inferred by combining statements 1 and 2)

Thus, the third angle (ACD) will also be 60.
We can say for sure that triangle ADC is an equilateral triangel.
 Sufficient
(C)

JeffTargetTestPrep · Answer

We need to answer the question:

Is triangle ABC equilateral?

We can rephrase the question into:

Is x = b = 60 ?

Statement One Alone:

=> ADC = 60 degrees

From triangle ADC, we have:

x + b +60 = 180

x + b = 120

If x = b = 60, then the answer is Yes. Whereas, if x = 50 and b = 70, then the answer is No. Statement one is not sufficient. Eliminate answer choices A and D.

(Note: Both x and b must be less than 90 degrees.)

Statement Two Alone:

=> EF is parallel to BC.

Since EAF = 90 = AFD, we have:

a + b = a + AFE

b = AFE

Since EF is parallel to BC, we have:

AFE = x

If we combine the two inferences, we have:

x = b

If x = b = 60, then the answer is Yes. Whereas, if x = b = 50, then the answer is No. Statement two is not sufficient. Eliminate answer choice B.

(Note: Both x and b must be less than 90 degrees.)

Statements One and Two Together:

x + b = 120 AND x = b, so we have:

x + x = 120

x = b = 60

Therefore, the answer to our rephrased question is a definite Yes. The two statements together are sufficient.

Answer: C

In the diagram below, if angle BAC is a right angle, is triangle ADC a

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In the diagram below, if angle BAC is a right angle, is triangle ADC a

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