Question

Can \(A\), \(B\), and \(C\) be the angles of a triangle? Statement (I): \(\angle A+\angle B<90^\circ\) Statement (II): \(\angle B+\angle C<120^\circ\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

To verify whether three angles form a triangle, the crucial condition is \(A+B+C=180^\circ\). Inequalities alone are generally not enough.

Answer 1

The Correct Option is D

Concept: Three angles can form a triangle if and only if: \[ A+B+C=180^\circ \] and each angle is positive.

Step 1: Analyze Statement (I). Given: \[ A+B<90^\circ. \] No information about \(C\). For example, \[ A+B=80^\circ,\quad C=100^\circ \] could form a triangle. But \[ A+B=80^\circ,\quad C=50^\circ \] would not. Hence Statement (I) alone is insufficient.

Step 2: Analyze Statement (II). Given: \[ B+C<120^\circ. \] Again, nothing about \(A\). A triangle may or may not be formed. Therefore Statement (II) alone is insufficient.

Step 3: Combine both statements. We know: \[ A+B<90^\circ \] and \[ B+C<120^\circ. \] These inequalities do not imply \[ A+B+C=180^\circ. \] Example: \[ A=40^\circ,\;B=30^\circ,\;C=50^\circ. \] Both statements are true, but \[ A+B+C=120^\circ. \] Not a triangle. Another example: \[ A=50^\circ,\;B=30^\circ,\;C=100^\circ. \] Both statements remain true and \[ A+B+C=180^\circ. \] A triangle is formed. Since both possibilities exist, the answer cannot be determined uniquely. Therefore even together the statements are insufficient.