Question

apply: finding unknown angle measures
the five angles shown are formed by two lines and a ray.
find the measure of each angle.
click or tap the correct measure from each drop - down menu.
the measure of ∠1 is choose
the measure of ∠2 is choose
the measure of ∠3 is choose

Explanation:

Step1: Find ∠1

We know that a right angle is \(90^\circ\), and the angle of \(72^\circ\) and \(\angle1\) form a right angle. So, \(\angle1 = 90^\circ - 72^\circ = 18^\circ\).

Step2: Find ∠3

From the diagram, \(\angle3\) is a right angle, so \(\angle3 = 90^\circ\).

Step3: Find ∠2

We know that \(\angle2\) and the \(72^\circ\) angle are vertical angles? Wait, no. Wait, \(\angle2\) and the angle adjacent to \(72^\circ\) (the one with \(\angle3\)): Wait, actually, \(\angle2\) and the angle that is \(72^\circ\) plus \(\angle1\)? No, wait, looking at the straight line. Wait, \(\angle2\) and the angle that is \(72^\circ\) (since vertical angles or supplementary? Wait, no, let's re - examine. The angle with \(72^\circ\) and \(\angle3\) (which is \(90^\circ\)) and \(\angle2\): Wait, actually, \(\angle2\) and the \(72^\circ\) angle are equal? No, wait, \(\angle2\) and the angle that is \(72^\circ\) are vertical angles? Wait, no, the line with \(\angle2\) and the line with the \(72^\circ\) angle: Wait, maybe \(\angle2 = 72^\circ\)? Wait, no, let's use the fact that \(\angle3\) is \(90^\circ\), and the angle between \(\angle3\) and \(\angle2\) and the \(72^\circ\) angle: Wait, actually, \(\angle2\) and the angle of \(72^\circ\) are equal because they are vertical angles? Wait, no, let's do it step by step.

First, for \(\angle1\): The angle between the right angle (\(90^\circ\)) and the \(72^\circ\) angle. So \(\angle1=90 - 72=18^\circ\).

For \(\angle3\): We can see from the diagram that \(\angle3\) is a right angle, so \(\angle3 = 90^\circ\).

For \(\angle2\): The angle \(\angle2\) and the \(72^\circ\) angle are vertical angles? Wait, no, the line that forms \(\angle2\) and the line that forms the \(72^\circ\) angle: Wait, actually, \(\angle2\) and the angle of \(72^\circ\) are equal because they are vertical angles? Wait, no, let's look at the straight line. The sum of angles on a straight line is \(180^\circ\). But \(\angle3\) is \(90^\circ\), and the angle adjacent to \(\angle2\) and \(\angle3\): Wait, maybe \(\angle2 = 72^\circ\)? Wait, no, let's correct.

Wait, the angle with \(72^\circ\) and \(\angle1\) and the right angle: So \(72+\angle1 + 90=180\)? No, that's not right. Wait, the right angle is between two lines, so the angle between the \(72^\circ\) angle and \(\angle1\) is \(90^\circ\), so \(\angle1=90 - 72 = 18^\circ\).

For \(\angle3\): We can see from the diagram that there is a right - angle symbol, so \(\angle3 = 90^\circ\).

For \(\angle2\): The angle \(\angle2\) and the \(72^\circ\) angle are equal because they are vertical angles (the lines that form them are intersecting lines, so vertical angles are equal). So \(\angle2 = 72^\circ\).

Step1: Calculate \(\angle1\)

The angle consisting of \(\angle1\), the \(72^{\circ}\) angle, and the right angle (\(90^{\circ}\))? No, the \(72^{\circ}\) angle and \(\angle1\) form a right angle. So we use the formula \(\angle1=90^{\circ}-72^{\circ}\)
\(\angle1 = 18^{\circ}\)

Step2: Determine \(\angle3\)

From the diagram, we can see that \(\angle3\) is a right angle. So \(\angle3 = 90^{\circ}\)

Step3: Determine \(\angle2\)

\(\angle2\) and the \(72^{\circ}\) angle are vertical angles (vertical angles are equal). So \(\angle2 = 72^{\circ}\)

Answer:

\(\angle1 = 18^{\circ}\), \(\angle2 = 72^{\circ}\), \(\angle3 = 90^{\circ}\)