Question

1. choose the best answer. which angle is acute? ∠1 ∠6 ∠8 ∠3

Explanation:

Step1: Recall acute angle definition

An acute angle is an angle whose measure is less than \(90^\circ\).

Step2: Analyze each angle

• For \(\angle1\): From the diagram, \(\angle1 = 60^\circ\), and \(60^\circ< 90^\circ\), so it could be acute.
• For \(\angle6\): \(\angle6\) is adjacent to a right angle and forms a straight line? Wait, actually, looking at the diagram, \(\angle6\) is supplementary to the angle with the transversal? Wait, no, more simply, \(\angle6\) is part of a linear pair? Wait, actually, from the diagram, \(\angle6\) is adjacent to the right angle area? Wait, no, let's check the other angles. \(\angle3\) is a right angle (\(90^\circ\)), so not acute. \(\angle8\): \(\angle8\) and the angle with the transversal, but actually, \(\angle8\) is equal to the angle opposite? Wait, no, \(\angle6\) and \(\angle8\) are vertical angles? Wait, no, the transversal crosses the horizontal line. So \(\angle6\) and \(\angle8\) are vertical angles, and the angle adjacent to them (like \(\angle7\) and \(\angle9\)): but actually, \(\angle6\) is a straight line adjacent? Wait, no, the horizontal line is a straight line (\(180^\circ\)). The angle \(\angle6\) is adjacent to the right angle (\(90^\circ\))? Wait, no, the diagram has a right angle (\(\angle3 = 90^\circ\)), and the horizontal line. So \(\angle6\) is between the transversal and the horizontal line, but actually, let's check the measures. \(\angle1 = 60^\circ\) (given as \(60^\circ\) in the diagram), \(\angle3 = 90^\circ\) (right angle), \(\angle6\): since the horizontal line is straight, and the transversal makes angles, but actually, \(\angle6\) is supplementary to the angle with the transversal? Wait, no, maybe \(\angle6\) is a straight angle? No, no. Wait, the key is: acute angle is less than \(90^\circ\). \(\angle1 = 60^\circ\) (acute), \(\angle6\): let's see, the angle next to the right angle (\(90^\circ\)) and the transversal. Wait, maybe \(\angle6\) is a right angle? No, the right angle is \(\angle3\). Wait, no, the diagram shows \(\angle1 = 60^\circ\), \(\angle2 = 30^\circ\), \(\angle3 = 90^\circ\), \(\angle4 = 150^\circ\), \(\angle5 = 30^\circ\). Then \(\angle6\): since the horizontal line is straight, the angle \(\angle6\) and the angle with the transversal (like \(\angle7\)): but actually, \(\angle6\) is equal to the angle opposite? Wait, no, the transversal crosses the horizontal line, so \(\angle6\) and \(\angle8\) are vertical angles, and \(\angle7\) and \(\angle9\) are vertical angles. But from the diagram, the angle with the transversal: maybe \(\angle6\) is a straight angle? No, no. Wait, the correct approach: check each option. \(\angle1 = 60^\circ\) (acute), \(\angle6\): let's assume that \(\angle6\) is a straight angle? No, the horizontal line is \(180^\circ\), so \(\angle6\) plus the angle adjacent (like the angle with the transversal) would be \(180^\circ\), but if \(\angle1 = 60^\circ\), \(\angle3 = 90^\circ\), then the angle between the transversal and the vertical line: maybe \(\angle6\) is \(90^\circ\)? No, \(\angle3\) is \(90^\circ\). Wait, no, the answer is \(\angle1\) because \(60^\circ < 90^\circ\), \(\angle3 = 90^\circ\) (right, not acute), \(\angle6\) and \(\angle8\): if the transversal makes an angle, but from the diagram, \(\angle1\) is \(60^\circ\) (acute), so \(\angle1\) is acute.

Answer:

\(\boldsymbol{\angle1}\) (Option: \(\boldsymbol{\angle1}\))