Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. find the measure of angle $\angle aef$ in the figure below. enter only the number. figure with lines and angles, including a 120° angle and points a, b, c, d, e, f, g

Explanation:

Step1: Identify straight line angles

AB and CD are straight lines, so ∠AEB and ∠CED are straight angles (180°). The angle between EB and EF (with the 120°) and ∠AEF: since AB is a straight line (180°), and the angle between EB and the other side (let's see, the angle given is 120° between EB and... Wait, actually, AB is horizontal, CD is vertical, so they are perpendicular? Wait, no, AB and CD intersect at E, forming vertical and horizontal lines, so they are perpendicular (90° each quadrant? Wait, no, the angle between EB (right horizontal) and the angle marked 120°: wait, maybe AB and CD are perpendicular? Wait, no, the angle between EB (right) and the angle with EF: let's think. The angle between EB (right) and EF's adjacent angle is 120°, but AB is a straight line (180°), and CD is a straight line (180°). Wait, maybe the angle between EB and ED? No, CD is vertical. Wait, the key is that AB and CD are perpendicular? Wait, no, the angle between EB (horizontal right) and the angle marked 120°: actually, the angle between EB (right) and EF is such that the angle between EB and the vertical line (CD) is 90°, but the given angle is 120°. Wait, maybe the angle between EB and EF is 120°, but AB and CD are perpendicular? No, let's re-examine.

Wait, AB is a horizontal line (A to E to B), CD is a vertical line (C to E to D). So ∠AEB is 180° (straight line), ∠CED is 180° (straight line). The angle between EB (right) and EF is 120°? Wait, no, the angle marked 120° is between EB (right) and... maybe ED? No, CD is vertical. Wait, maybe the angle between EB (right) and EF is 120°, but since AB is a straight line (180°), and CD is a straight line (180°), and they intersect at E, so ∠AEB = 180°, ∠CED = 180°, and ∠BEC = 90°? Wait, no, maybe AB and CD are perpendicular, so ∠BEC = 90°, but the angle given is 120°, which is more than 90°, so maybe AB and CD are not perpendicular. Wait, no, the figure: A is left, B is right (horizontal), C is top, D is bottom (vertical). So AB and CD are perpendicular, so ∠BEC = 90°, ∠AEC = 90°, etc. But the angle marked 120°: maybe the angle between EB (right) and EG is... No, the question is ∠AEF. Let's see: AB is horizontal (180°), so ∠AEB = 180°. The angle between EB (right) and EF: the angle given is 120°, but since AB and CD are perpendicular (90° between EB and ED), wait, no. Wait, the sum of angles on a straight line is 180°. The angle between EB (right) and EF is 120°, so the angle between EA (left) and EF would be 180° - 120°? No, that can't be. Wait, maybe CD is vertical, so ∠AED is 90° (since AB is horizontal and CD is vertical). Wait, the angle between EB (right) and EF is 120°, so the angle between EF and ED (down vertical) is 120° - 90° = 30°? No, that's not right. Wait, let's start over.

AB is a straight line, so ∠AEB = 180°. The angle between EB and EF is 120°, so the angle between EA and EF is 180° - 120° = 60°? Wait, no, because CD is vertical, so ∠AED is 90° (since AB is horizontal and CD is vertical, they are perpendicular). Wait, maybe the angle between EB and CD (vertical) is 90°, so the angle between EB and EF is 120°, which is 90° + 30°, so the angle between EF and CD (down) is 30°, then ∠AEF is 90° - 30° = 60°? Wait, no. Wait, AB is horizontal (left to right), CD is vertical (top to bottom), intersecting at E. So ∠AEC = 90°, ∠CEB = 90°, ∠BED = 90°, ∠DEA = 90°. The angle marked 120° is between EB (right) and EF. So ∠BEF = 120°, but ∠BED is 90°, so that would mean EF is below the horizontal? Wait, no, the figure shows F below E, towards the left. Wait, maybe the angle between EB (right) and EF is 120°, so the angle between EA (left) and EF is 180° - 120° = 60°? Wait, let's check: if AB is a straight line (180°), and ∠BEF = 120°, then ∠AEF = 180° - 120° = 60°? Wait, but CD is vertical, so ∠AED is 90°, so maybe that's conflicting. Wait, no, maybe the 120° is between EB and CD? No, CD is vertical. Wait, the key is that AB and CD are perpendicular, so ∠BED = 90°, but the angle given is 120°, which is ∠BEF = 120°, so ∠DEF = 120° - 90° = 30°, then ∠AEF = 90° - 30° = 60°? Yes, that makes sense. Because ∠AED is 90° (AB horizontal, CD vertical), ∠BEF is 120°, so ∠DEF = ∠BEF - ∠BED = 120° - 90° = 30°, then ∠AEF = ∠AED - ∠DEF = 90° - 30° = 60°. So ∠AEF is 60 degrees.

Step2: Calculate ∠AEF

Since AB and CD are perpendicular (forming 90° angles at E), ∠BED = 90°. The angle ∠BEF = 120°, so ∠DEF = ∠BEF - ∠BED = 120° - 90° = 30°. Then, since ∠AED = 90° (perpendicular), ∠AEF = ∠AED - ∠DEF = 90° - 30° = 60°. Alternatively, since AB is a straight line (180°), ∠AEF + ∠BEF = 180°? Wait, no, ∠BEF is 120°, so ∠AEF = 180° - 120° = 60°. Yes, that's simpler. AB is a straight line, so the sum of ∠AEF and ∠BEF is 180°. So ∠AEF = 180° - 120° = 60°.

Answer:

60