Question

in △jkl, if m∠j < 90°, then ∠k and ∠l are ______. a) congruent b) complementary c) both obtuse d) one obtuse and one acute

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So in \(\triangle JKL\), \(m\angle J + m\angle K + m\angle L = 180^\circ\).

Step2: Analyze given angle

Given \(m\angle J < 90^\circ\) (acute angle). Then \(m\angle K + m\angle L = 180^\circ - m\angle J\). Since \(m\angle J < 90^\circ\), \(180^\circ - m\angle J > 90^\circ\).

Step3: Evaluate options

• Option A: Congruent? No info for equality, eliminate.
• Option B: Complementary? Sum of two angles would be \(90^\circ\), but we have \(m\angle K + m\angle L > 90^\circ\), eliminate.
• Option C: Both obtuse? Obtuse is \(> 90^\circ\), sum of two obtuse angles \(> 180^\circ\), but \(m\angle K + m\angle L = 180^\circ - m\angle J < 180^\circ\) (since \(m\angle J > 0\)), so sum of two angles \(< 180^\circ\), can't both be obtuse, eliminate.
• Option D: One obtuse (\(> 90^\circ\)) and one acute (\(< 90^\circ\))? Let's check: If one is obtuse (\(> 90^\circ\)), the other would be \(180^\circ - m\angle J - \text{obtuse angle}\). Since \(180^\circ - m\angle J > 90^\circ\), if one angle is \(> 90^\circ\) (obtuse), the remaining angle is \(180^\circ - m\angle J - \text{obtuse angle} < 90^\circ\) (acute). This fits.

Answer:

D) one obtuse and one acute