Question

Given below is a question followed by 2 statements.
Which of the following statement(s) is/are sufficient to answer the question?
Question: What is the measure of angle A in triangle ABC?
Statements:
I. Triangle ABC is isosceles
II. The measure of the angle B in triangle ABC is 45 degrees

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Both statements I and II together are sufficient
• D. Neither statement I nor statement II are sufficient

Answer 1

The Correct Option is D

We are asked to determine the measure of angle A in triangle ABC.

Statement I alone: Knowing that triangle ABC is isosceles is insufficient. We do not know which sides (and therefore which angles) are equal. Angle A could be the unique angle, or one of the two equal angles.

Statement II alone: Knowing that angle B is 45 degrees is insufficient. We know nothing about the other angles.

Statements I and II together: Since the triangle is isosceles (Statement I), two angles are equal. Since angle B is 45 degrees (Statement II), there are two possibilities:

• Case 1: Angle B is the unique angle. Then angles A and C are equal. So, A+C+B=180◦ ⇒2A+45◦ =180◦ ⇒2A=135◦ ⇒A=67.5◦.
• Case 2: Angle B is one of the two equal angles. Then angle C is also 45◦. So, A+B+C=180◦ ⇒A+45◦+45◦ =180◦ ⇒A=90◦.

Since the two cases lead to different answers, the statements together are not sufficient.

Conclusion: Therefore, the correct answer is (D) Neither statement I nor statement II are sufficient.