If triangle ABC is an isosceles triangle

If triangle ABC is an isosceles triangle, what is ∠ABC?

(1) ∠CAB = 45 degrees. The sum of the remaining two angles is 180-45=135 degrees. Triangle ABC can be 45-45-90 right isosceles triangle OR 67.5-67.5-45 isosceles triangle. Angle B can be 45, 90, or 67.5 degrees. Not sufficient.

(2) ∠BCA = 90 degrees. This statement says that we have 45-45-90 isosceles right triangle, where C=90 degrees, therefore the remaining two angles must be 45 degrees each. Sufficient.

Answer: B.

Hope it's clear.

As we know by the properties of Isosceles ,
90-45-45 ,
From A : we know about angle CAB is 45 , but we do not have an information about the other angles if any of them has a 90 degree measure. Other angles can be anything to form 180 degree
From B : We know Angle BCA has a measure of 90 degree , Hence we can derive a conclusion that it follows an essential Isosceles pattern of 90 degree . other two sides must be 45 deg & 45 deg .

I'm sure there will be better explanations than this , but I chose B from this logic.

Still cant understand why A cannot be an answer

The first statement in no sufficient because from (1) we could have more than one value of ∠ABC.

Hey shringi87 ,

Let me explain you. A says angle CAB is 45.

But we don't know whether this angle is actually one of the two equal angles.

If it is, then we will say

either angle CBA is 45 => Angle B is 45.

or angle ACB is 45 => Angle B is 90

If it is not, then we will say ,

Other two angles must be 67.5 , thereby making angle B as 67.5

Hence, we have three possibilities for angle B from this statement, hence insufficient.

Does that make sense?

ABC = Isosceles triangle so possible ways <A = <B or <A = <C or <B = <C

[1] : <A = 45

we can form <A + <B + <C = 180 [ <B = <C ] - one possibility
45 + <B + <B = 180
<B = 135/2

<A + <B + <C = 180 [ <A = <B ]

so directly <B = 45 - so two values of <B is possible - Hence A is not sufficient

[2] <C = 90
<A + <B + <C = 180 { < B = <C ]
<A + 90 + 90 = 180
< A = 0 : This is not possible as we know ABC is a triangle

So <C has to be the largest of all the angles
<A + <B + <C = 180 [ <A = <B ]
2<B + 90 = 180
<B = 45 - Hence B is the answer

Says Isosceles. Not isosceles RIGHT TRIANGLE. That's what they are testing here.



Posted from my mobile device

Target question: What is ∠ABC?

Given: Triangle ABC is an isosceles triangle
So, ∆ABC has 2 equal angles

Statement 1: ∠CAB = 45 degrees
Since ∆ABC has 2 equal angles, there are two possible triangles that satisfy statement 1:
Case a: ∠CAB = 45, ∠BCA = 45 and ∠ABC = 90. In this case, the answer to the target question is ∠ABC = 90 degrees
Case b: ∠CAB = 45, ∠BCA = 90 and ∠ABC = 45. In this case, the answer to the target question is ∠ABC = 45 degrees
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: ∠BCA = 90 degrees
This statement is much different.
We know that ∆ABC has 2 equal angles, but we also know that the 2 equal angles cannot both be 90 degrees, since all 3 angles must add to 180.
So, if the 2 equal angles were 90 degrees each, the third angle would have to be 0 degrees, which is impossible.
Since the 2 equal angles cannot both be 90 degrees, it must be the case that the OTHER 2 angles are the equal angles.
In other words, ∠BCA = 90, ∠CAB = 45 and ∠ABC = 45. So, the answer to the target question is ∠ABC = 45 degrees
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

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