Question

in the figure below, the measure of $\angle 4=90^\circ$, the measure of $\angle 5=54^\circ$, and the measure of $\angle 6=36^\circ$. what are the measures of $\angle 1$, $\angle 2$, and $\angle 3$? $m\angle 1=\square^\circ$

Explanation:

Step1: Identify linear pair or vertical angles

We know that \(\angle 1\), \(\angle 5\), and \(\angle 6\) form a straight line (180°), but also, looking at the right angle \(\angle 4 = 90^\circ\), and vertical angles or supplementary angles. Wait, actually, \(\angle 1\), \(\angle 5\), and \(\angle 6\) – no, let's check the angles around the point. Wait, \(\angle 4 = 90^\circ\), so \(\angle 3\) and \(\angle 4\) – no, let's see: \(\angle 1\) and the angle opposite? Wait, maybe \(\angle 1\) is supplementary to \(\angle 5 + \angle 6\)? Wait, no, \(\angle 4 = 90^\circ\), so \(\angle 3 = \angle 5\) (vertical angles)? Wait, no, let's start with \(\angle 1\).

Wait, the sum of angles around a point is 360°, but also, straight lines are 180°. Let's look at the angles: \(\angle 4 = 90^\circ\), so \(\angle 3\) and \(\angle 4\) – no, \(\angle 3\) and \(\angle 5\) are vertical angles? Wait, \(\angle 5 = 54^\circ\), so \(\angle 3 = 54^\circ\)? Wait, no, maybe \(\angle 1\) is equal to \(\angle 4\)? No, wait, \(\angle 1\), \(\angle 6\), \(\angle 5\) – wait, \(\angle 6 = 36^\circ\), \(\angle 5 = 54^\circ\), so \(\angle 1 + \angle 6 + \angle 5 = 180^\circ\)? Wait, no, because \(\angle 4 = 90^\circ\), so the vertical angle of \(\angle 4\) is also 90°, so \(\angle 1 + \angle 6 + \angle 5 = 90^\circ\)? Wait, no, maybe I'm overcomplicating. Wait, the problem is about angles around a point, with some right angles.

Wait, let's see: \(\angle 4 = 90^\circ\), so the angle opposite to \(\angle 4\) (let's say \(\angle 1 + \angle 6 + \angle 5\))? No, wait, \(\angle 1\) and \(\angle 4\) – no, maybe \(\angle 1\) is equal to \(\angle 4\)? No, \(\angle 4\) is 90°, but \(\angle 5 = 54^\circ\), \(\angle 6 = 36^\circ\), so \(\angle 5 + \angle 6 = 54 + 36 = 90^\circ\). Then, since \(\angle 1\) is adjacent to \(\angle 5\) and \(\angle 6\) and forms a right angle? Wait, no, \(\angle 1\), \(\angle 5\), \(\angle 6\) – wait, if \(\angle 4 = 90^\circ\), then the angle opposite to \(\angle 4\) (let's call it \(\angle 1 + \angle 6 + \angle 5\)) should also be 90°? Wait, no, maybe \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles? Wait, no, \(\angle 4\) is 90°, so if \(\angle 1\) is vertical to the angle that's 90°, then \(\angle 1 = 90^\circ\)? Wait, no, let's check the sum: \(\angle 5 + \angle 6 = 54 + 36 = 90^\circ\), and \(\angle 1\) is adjacent to them, forming a straight line? No, maybe \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles. Wait, the diagram shows intersecting lines, so vertical angles are equal. \(\angle 4 = 90^\circ\), so the angle opposite to \(\angle 4\) (which is \(\angle 1 + \angle 6 + \angle 5\)?) No, maybe \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles. Wait, maybe I made a mistake. Let's re-express:

Wait, \(\angle 5 = 54^\circ\), \(\angle 6 = 36^\circ\), so \(\angle 5 + \angle 6 = 90^\circ\). Then, since \(\angle 4 = 90^\circ\), the angle \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles? Wait, no, vertical angles are opposite each other. Wait, maybe \(\angle 1\) is supplementary to \(\angle 5 + \angle 6\)? No, \(\angle 5 + \angle 6 = 90^\circ\), so \(\angle 1 = 90^\circ\) because \(\angle 1 + \angle 5 + \angle 6 = 180^\circ\)? Wait, no, 90 + 54 + 36 = 180? 90 + 90 = 180, yes! So \(\angle 1 + \angle 5 + \angle 6 = 180^\circ\)? Wait, no, 54 + 36 = 90, so \(\angle 1 + 90 = 180\)? No, that would be 90, but \(\angle 4\) is 90. Wait, maybe the lines are perpendicular, so \(\angle 4 = 90^\circ\), and \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles. Wait, the answer is 90? Wait, let's check:

If \(\angle 4 = 90^\circ\), and \(\angle 1\) is vertical to the angle that's 90°, then \(\angle 1 = 90^\circ\). Alternatively, \(\angle 5 + \angle 6 = 54 + 36 = 90^\circ\), and since \(\angle 1\) is adjacent to them and forms a right angle? Wait, maybe the diagram has two intersecting lines, one horizontal and one vertical, making \(\angle 4 = 90^\circ\), so \(\angle 1\) is also 90° because it's vertical to \(\angle 4\). So \(\angle 1 = 90^\circ\).

Step2: Confirm with angle sum

Wait, \(\angle 5 = 54^\circ\), \(\angle 6 = 36^\circ\), so \(\angle 5 + \angle 6 = 90^\circ\). Then, since \(\angle 1\) is adjacent to them and forms a straight line with \(\angle 4\)'s vertical angle? No, maybe the key is that \(\angle 1\) is equal to \(\angle 4\) because they are vertical angles. So \(\angle 1 = 90^\circ\).

Answer:

\(90\)