Q: does quadrilateral ABCD have a 60 degree angle?

(1) ABCD has two right angles.

Well, there's 180 degrees left, but we have no idea how it's split up: insufficient.

(2) One angle is double another angle.

We could have 120/60, or we could have 100/50: insufficient.

Together:

It's very temping to say that once we remove our two 90 degree angles we're left with 180 and that, building in rule (2), the two remaining angles have to be 120/60, giving us a "YES" answer.

However, there's another possibility. Instead of our double values being 120/60, our double values could be 90/45. A quadrilateral with angles of 135/90/90/45 satisfies both statements and gives us a "NO" answer.

So, even after combining, we can get both a "YES" and a "NO": choose (E), not enough information.

thanks, Stuart!

First, I chose C, but didn't understand what was wrong with that. But after your explanation, everything is clear!
Thank you
Feruza

IMO E

Stuart Can you please show me how to construct an quadrilateral with two 90 degree and other two are not 90 each? I am asking for a real construction.

Quad ABCD
Angle ABC = 90; Angle BCD = 45; Angle CDA = 90; Angle DAB = 135.

Sketch this one out, and you see why the answer is E.

Stuart Kovinsky wrote:Q: does quadrilateral ABCD have a 60 degree angle?

(1) ABCD has two right angles.

Well, there's 180 degrees left, but we have no idea how it's split up: insufficient.

(2) One angle is double another angle.

We could have 120/60, or we could have 100/50: insufficient.

Together:

It's very temping to say that once we remove our two 90 degree angles we're left with 180 and that, building in rule (2), the two remaining angles have to be 120/60, giving us a "YES" answer.

However, there's another possibility. Instead of our double values being 120/60, our double values could be 90/45. A quadrilateral with angles of 135/90/90/45 satisfies both statements and gives us a "NO" answer.

So, even after combining, we can get both a "YES" and a "NO": choose (E), not enough information.

Staurt,

Didn't mean to question your solution.

I would like to go with B as the ans.

Stmt 2 states - The degree measure of ABC is twice the measure of BCD.

It is clearly stating the sum of the angles on the straight line on one side of the quadrilateral.

This gives x + 2x = 180 ==> x = 60.

This is sufficient to say that one angle of the quadrilateral is 60.

mrsmarthi wrote:

Stuart Kovinsky wrote:Q: does quadrilateral ABCD have a 60 degree angle?

(1) ABCD has two right angles.

Well, there's 180 degrees left, but we have no idea how it's split up: insufficient.

(2) One angle is double another angle.

We could have 120/60, or we could have 100/50: insufficient.

Together:

It's very temping to say that once we remove our two 90 degree angles we're left with 180 and that, building in rule (2), the two remaining angles have to be 120/60, giving us a "YES" answer.

However, there's another possibility. Instead of our double values being 120/60, our double values could be 90/45. A quadrilateral with angles of 135/90/90/45 satisfies both statements and gives us a "NO" answer.

So, even after combining, we can get both a "YES" and a "NO": choose (E), not enough information.

Staurt,

Didn't mean to question your solution.

I would like to go with B as the ans.

Stmt 2 states - The degree measure of ABC is twice the measure of BCD.

It is clearly stating the sum of the angles on the straight line on one side of the quadrilateral.

This gives x + 2x = 180 ==> x = 60.

This is sufficient to say that one angle of the quadrilateral is 60.

Hi! Questioning is good!

You have made the assumption that our shape is a parallelogram - that each adjoining pair of angles must add up to 180 degrees. However, nowhere in the question are we given that information. In data sufficiency, if we're not told something explicitly, we cannot make any assumptions.

A quadrilateral can be any enclosed 4 sided shape (with straight sides, of course, in 2-dimensional space). So, your equation does not have to apply to this question.

Based on statement (2) alone, the angles in our shape could be 60/60/120/120, but they could also easily be 1/2/3/354 or 45/90/90/135.