Question

in the figure below, the measure of ∠4=90°, the measure of ∠5=54°, and the measure of ∠6=36°. what are the measures of ∠1, ∠2, and ∠3? m∠1=90° m∠2=□°

Explanation:

Step1: Identify vertical angles or linear pairs

∠2 and ∠5 are vertical angles? Wait, no, let's check the lines. Wait, ∠2, ∠5, and the right angle? Wait, actually, ∠2 and ∠5: Wait, no, let's see the straight line. Wait, ∠2, ∠3, ∠4: but ∠4 is 90°. Wait, maybe ∠2 and ∠5 are vertical angles? Wait, no, vertical angles are opposite. Wait, ∠2 and ∠5: Wait, the lines intersect, so vertical angles are equal. Wait, ∠5 is 54°, so ∠2 should be equal to ∠5? Wait, no, maybe not. Wait, let's look at the angles around the point. The sum of angles on a straight line is 180°, and vertical angles are equal.

Wait, ∠2 and ∠5: Wait, actually, ∠2 and ∠5 are vertical angles? Wait, no, let's see the diagram. The angles ∠2, ∠3, ∠4: ∠4 is 90°, ∠3 and ∠6? Wait, maybe I made a mistake. Wait, the problem says ∠4 is 90°, ∠5 is 54°, ∠6 is 36°. Let's find ∠2 first.

Wait, ∠2 and ∠5: Are they vertical angles? Wait, vertical angles are opposite each other when two lines intersect. So if two lines intersect, the opposite angles are equal. So ∠2 and ∠5: Let's check. If we have two intersecting lines, then ∠2 and ∠5 are vertical angles? Wait, no, maybe ∠2 and ∠5 are equal? Wait, ∠5 is 54°, so ∠2 should be 54°? Wait, no, maybe not. Wait, let's think again.

Wait, the sum of angles around a point is 360°, but also, linear pairs sum to 180°. Wait, ∠1 is 90°, as given. Let's check ∠1, ∠6, ∠5, ∠4: Wait, ∠1 + ∠6 + ∠5 + ∠4 = 90 + 36 + 54 + 90 = 270? No, that can't be. Wait, maybe the lines are intersecting, so ∠2 is equal to ∠5 because they are vertical angles. Wait, vertical angles are equal, so if ∠5 is 54°, then ∠2 is 54°? Wait, but let's confirm.

Wait, another approach: ∠3, ∠4, ∠2: Wait, ∠3 + ∠4 + ∠2 = 180°? No, because ∠4 is 90°, and ∠3 and ∠6 are vertical angles? Wait, ∠3 and ∠6: ∠6 is 36°, so ∠3 is 36°? Then ∠2 would be 180° - ∠3 - ∠4 = 180 - 36 - 90 = 54°? Yes, that makes sense. So ∠2 is 54°, because ∠3 is 36° (vertical angle with ∠6), ∠4 is 90°, so ∠2 = 180 - 36 - 90 = 54°.

Step2: Calculate ∠2

Since ∠3 and ∠6 are vertical angles, m∠3 = m∠6 = 36°. Then, in the straight line containing ∠2, ∠3, and ∠4, the sum of angles is 180°. So:

m∠2 + m∠3 + m∠4 = 180°

Substitute m∠3 = 36° and m∠4 = 90°:

m∠2 + 36° + 90° = 180°

m∠2 + 126° = 180°

m∠2 = 180° - 126° = 54°

Answer:

54