Are all the angles of triangle ABC less than 90 degrees? (1) AB^2 + B

Question

Are all the angles of triangle ABC less than 90 degrees?

(1) AB^2 + BC^2 > AC^2

(2) BC^2 + AC^2 > AB^2

rahul16singh28 · Answer

From Statement I: B < 90

From Statement II: C < 90

Combining I & II:

We cannot decide if Angle A < or > 90. Hence, E.

GMATBusters · Answer

CONCEPT: How to find the nature of angles in a triangle.

If \(a^2+b^2 < c^2\), it means the angle between side a & b is obtuse, the triangle is obtuse.
If \(a^2+b^2 > c^2\), it means the angle between a & b is acute, but since other angle can be obtuse, the triangle cant be taken as acute.
If \(a^2+b^2 =c^2\), it means the angle between a & b is 90 deg, hence the triangle is right.

Solution:
Statement 1) AB^2 + BC^2 > AC^2
it means angle B is acute, No other information.
INSUFFICIENT.

Statement 2) BC^2 + AC^2 > AB^2
it means angle C is acute, No other information.
INSUFFICIENT.

Even Combing Statement 1 & 2, we get angle B & C are acute, but angle A can be Acute, 90 deg or Obtuse.
Angles (A, B, C) can be (60,60,60) or (30,60, 90) or (30,50,100)
Hence INSUFFICIENT.

Answer - E

Attachment:

gmatbusters.jpg [ 120.91 KiB | Viewed 1255 times ]

Must Do similar questions based on same logic:
1) https://gmatclub.com/forum/is-triangle-abc-with-sides-a-b-and-c-acute-angled-263435.html?hilit=acute
2) https://gmatclub.com/forum/is-triangle-abc-obtuse-angled-203020.html?hilit=acute
3) https://gmatclub.com/forum/are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html?hilit=acute

Are all the angles of triangle ABC less than 90 degrees? (1) AB^2 + B

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