Question

triangle abc is an obtuse triangle with the obtuse angle at vertex b. angle a must be
○ less than 90°.
○ greater than 90°.
○ congruent to angle b.
○ congruent to angle c.

Explanation:

Step1: Recall triangle angle sum and obtuse angle definition

In a triangle, the sum of interior angles is \(180^\circ\). An obtuse angle is greater than \(90^\circ\) and less than \(180^\circ\). Since \(\triangle ABC\) has an obtuse angle at \(B\), \(\angle B>90^\circ\).

Step2: Analyze \(\angle A\)

Let \(\angle A = x\), \(\angle B = y\) (where \(y>90^\circ\)), \(\angle C = z\). We know \(x + y+ z=180^\circ\). So \(x + z=180^\circ - y\). Since \(y>90^\circ\), \(180^\circ - y<90^\circ\). So \(x\) (and \(z\)) must be less than \(90^\circ\) (because \(x\) and \(z\) are positive angles and their sum is less than \(90^\circ\), so each is less than \(90^\circ\)). Also, \(\angle A\) can't be congruent to \(\angle B\) (as \(\angle B>90^\circ\) and \(\angle A<90^\circ\)) and there's no info to say \(\angle A\cong\angle C\).

Answer:

less than \(90^\circ\) (the option: less than \(90^\circ\))