Question

which statement about the angles in the figure is true?
m∠3 + m∠5 = 180°
m∠7 + m∠10 = 180°
m∠8 + m∠9 = 90°
m∠1 + m∠8 = 90°

Explanation:

Step1: Analyze \( m\angle3 + m\angle5 \)

\(\angle3\) and \(\angle5\) are same - side interior angles? No, wait, looking at the figure, the lines: the two non - vertical lines (the sides of the triangle) and the transversal. Wait, actually, \(\angle3\) and \(\angle8\) are alternate interior angles? No, let's check the right angles. There is a right angle in the triangle, and also a right angle at the intersection of the two crossing lines at the bottom.

Wait, let's check each option:

Option 1: \( m\angle3 + m\angle5 \). Are they supplementary? Let's see, \(\angle3\) and \(\angle2\) are supplementary (linear pair), \(\angle5\) and \(\angle8\) are supplementary (linear pair). But \(\angle3\) and \(\angle5\): no, they are not same - side interior angles in a way that they sum to 180.

Option 2: \( m\angle7 + m\angle10 \). \(\angle7\) and \(\angle8\) are supplementary (linear pair), \(\angle10\) is part of a right - angled triangle? No.

Option 3: \( m\angle8 + m\angle9 = 90^{\circ}\). Let's see, in the triangle, there is a right angle, and the angles in a triangle sum to \(180^{\circ}\). Also, the vertical angles and linear pairs. Wait, the two lines that cross at the bottom have a right angle, so the triangle formed has a right angle. Also, \(\angle3\) and \(\angle1\) are vertical angles, \(\angle8\) and \(\angle6\) are vertical angles. Wait, let's think about the angles in the right - angled triangle and the transversals.

Wait, the triangle has a right angle, so the other two angles (non - right) sum to \(90^{\circ}\). Also, \(\angle8\) and \(\angle3\) are alternate interior angles (if we consider the two parallel? No, wait, the lines: the top line and the base of the triangle. Wait, maybe \(\angle8\) and \(\angle3\) are equal (alternate interior angles), and \(\angle9\) and the other angle. Wait, let's look at the right angle in the triangle. The segment inside the triangle is perpendicular to the base, so it forms two right angles. Then, in the triangle, the angles \(\angle8\) and \(\angle9\) (along with the right angle? No, wait, the angle at the bottom vertex: the two lines cross at a right angle, so \(\angle9+\angle10 = 90^{\circ}\), and \(\angle8\) and \(\angle3\) are equal (alternate interior angles), \(\angle1\) and \(\angle3\) are vertical angles. Wait, maybe the key is that in the right - angled triangle, the sum of \(\angle8\) and \(\angle9\) is \(90^{\circ}\). Let's verify:

In the triangle, one angle is \(90^{\circ}\), so the sum of the other two non - right angles is \(90^{\circ}\). \(\angle8\) and \(\angle9\) are those two non - right angles? Wait, maybe. Let's check the other options:

\(m\angle1 + m\angle8\): \(\angle1=\angle3\) (vertical angles), and \(\angle3\) and \(\angle8\) are alternate interior angles, so \(\angle1 = \angle8\)? No, that would mean \(m\angle1 + m\angle8 = 2m\angle1\), not \(90^{\circ}\) unless \(\angle1 = 45^{\circ}\), which we don't know.

\(m\angle3 + m\angle5\): \(\angle3\) and \(\angle5\) are same - side interior angles? If the lines are parallel, but we don't know if they are parallel. Wait, no, the figure has a triangle, so the lines are not parallel. So \(m\angle3 + m\angle5
eq180^{\circ}\).

\(m\angle7 + m\angle10\): \(\angle7\) is supplementary to \(\angle8\), \(\angle10\) is part of a right angle, so \(m\angle7 + m\angle10\) is \(180 - m\angle8+m\angle10\), which is not \(180^{\circ}\).

So the correct option is \(m\angle8 + m\angle9 = 90^{\circ}\).

Answer:

\(m\angle8 + m\angle9 = 90^{\circ}\) (the option with this statement)