Triangle

Hey there, knowing the angle ABD is 60, doesn't mean that the triangle is iscoseles or equilateral. It just means that 1 angle is 60, others can be anything that sum up to 180. SO, it is insufficient, because we can't figure out the what is DB. A vertice may be even to the left of the C.

+ stmt one, we know that the triangle is equilateral, because the 2 sides and two angles are the same.
Hence , 1+2 is sufficient.

NikolayZ wrote:Hey there, knowing the angle ABD is 60, doesn't mean that the triangle is iscoseles or equilateral. It just means that 1 angle is 60, others can be anything that sum up to 180. SO, it is insufficient, because we can't figure out the what is DB. A vertice may be even to the left of the C.

+ stmt one, we know that the triangle is equilateral, because the 2 sides and two angles are the same.
Hence , 1+2 is sufficient.

But why can't it be that AD divides CB into equal halves ? Do I have to know first that the triangle is isosceles or equilateral first ?

You should always look at the graph in abstract. They are all were drawn NOT INTO A SCALE.
All you must see is the information that is GIVEN. In the question stem it is the base.
In stmt 1) it is the length of the right side. This could only lead you to the conclusion that given triangle is iscoseles. But you don't know any angles, and the length of the base.
And that is it. you don't know anything more about the triangle.

stmt 2) you know the angle between BC and AB. Still no key-onfo about triangle. it may be equilateral, iscoseles or right.

Try to move the "A" vertice horisontaly from right to left, and you will get that we can't find the length of DB. From the problem's "given" combined with 1 of the statements we can do that freely. We need to know more to find how height divides base.

NikolayZ wrote:You should always look at the graph in abstract. They are all were drawn NOT INTO A SCALE.
All you must see is the information that is GIVEN. In the question stem it is the base.
In stmt 1) it is the length of the right side. This could only lead you to the conclusion that given triangle is iscoseles. But you don't know any angles, and the length of the base.
And that is it. you don't know anything more about the triangle.

stmt 2) you know the angle between BC and AB. Still no key-onfo about triangle. it may be equilateral, iscoseles or right.

Try to move the "A" vertice horisontaly from right to left, and you will get that we can't find the length of DB. From the problem's "given" combined with 1 of the statements we can do that freely. We need to know more to find how height divides base.

Last thing. The OE says this is an equilateral triangle, why not an isosceles ?

I assume that you got the idea why 1 and 2 both alone are insufficient.
Let's figure out why it is an equilateral triangle.
You know that 2 sides of a triangle have exactly the same length. So, this triangle must be iscoseles. THEN, because this triangle is iscoseles, the angles A and C must be equal (rule of iscoseles triangle: if sides are equal, then the opposite angles are also equal). We can find those angles this way ( let angle A and C be x)==>
180=x+x+B, or 2x+B=180, x=(180-B)/2.
Stmt two gives you the measure of angle B. hence X=120/2=60. Oooops, all angles equal to 60. All angles are the same, hence all sides are the same also. => ABC is equilateral.

heshamelaziry wrote:Pleas help me understand why B alone is insufficient ? If x = 60 then this is a 30-60-90 triangle, then can find height and the base is 4/2 = 2 and though, can find area

Isn't it true that a line coming down from the highest point of an isosceles or equilateral triangle and form a 90 angle, will divide the base into 2 equal halves ?

Hi heshamelaziry,

You are asking about why statement 2 alone is not sufficient. But you are pointing out that the "line coming down from the highest point...will divide the base into 2 equal halves".

But from statement 2 we do not know ANYTHING about line segment EG. Remember, when you are looking at statement 2 by itself, you have to pretend as though statement 1 does not exist. If statement 1 does not exist, then you do not know that the line coming down from the top bisects (ie, cuts in half) line DF!