Triangle ABC is isosceles. What is the measure of angle BAC?

OFFICIAL SOLUTION:

We use the fact that the sum of angles in any triangle is 180°. We also use the fact that an isosceles triangle has two equal angles. Let’s denote each angle by its vertex, i.e. angle BAC is angle A, angle ACB is angle C and angle ABC is angle B.

If we use statement (1) it defines angle C = 40°. If it is NOT one of the equal angles, then the sum of the equal angles is 180° – 40° = 140°. So each one of the equal angles is 140°/2 = 70°. In this case angle A is 70°.

However, if the given angle (angle C) is one of the equal angles, then we do NOT know which one is the other equal angle. If it is angle A, then it equals 40°. If not – then angle A equals 180° – (40° + 40°) = 100°. Therefore we do NOT have a definite value of angle A. So statement (1) by itself is NOT sufficient.

Statement (2) defines angle B = 100°. It looks very similar to statement (1), but there is a major difference. Angle B can NOT be one of the equal angles. If it had been so, the sum of the angles in the triangle would have exceeded 180° (100° + 100° + one more angle). That is impossible because the sum of the angles in any triangle is always 180°.Therefore angle B is NOT one of the equal angles, but angle A is. So angle A = (180° – 100°)/2 = 40°. Statement (2) by itself is sufficient to answer the question.Statement (2) by itself is sufficient to answer the question, but statement (1) by itself is not.

The correct answer is B.

Similar question to practice: in-isosceles-triangle-rst-what-is-the-measure-of-angle-r-129378.html

Are we sure this is 600-700 level?

The triangle is isosceles, but we do not know which pair of sides or angles is congruent.

(1) BAC could be 40 or (180-40)/2 = 70. NOT sufficient.

Strike A/D.

(2) BAC cannot be 100, because the third angle would have to be negative. There is an implicit constraint that each angle is non-negative. BAC must therefore be (180-100)/2 = 40. Sufficient.

The answer is B.

Hi Bunuel, I have an issue with the explanation that the answer is B.

In 1) BCA is 40 degrees. THE ONLY OPTION for the rest of the triangles is 70 degrees.
My reasoning is that, if you take into account your answer where another angle could be 40, so 40/40/100, then you are breaking the main rule of inner triangle angles. The rule is that the sum of 2 angles may always be higher that the third angle. In this case (40+40) < 100, which means this is NOT A POSSIBLE triangle.

Does this make sense??


Bunuel

There is no such rule.

I think you meant the following rule: The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

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