Question

what is the measure of ∠b?
○ 98°
○ 108°
○ 118°
○ 128°
(diagram shows a polygon with angles: ∠c=133°, ∠d=102°, ∠e=117°, ∠a=170°, ∠f=90° (right angle), and ∠b unknown)

Explanation:

Step1: Recall polygon angle sum formula

For a hexagon? Wait, no, the figure here: Wait, the angle at A is 170°, but actually, the angle at A: Wait, the figure is a hexagon? Wait, no, let's count the vertices: B, C, D, E, F, A? Wait, no, the points are B, C, D, E, F, A? Wait, no, the angle at A is adjacent to B and F. Wait, actually, the figure is a hexagon? Wait, no, the sum of interior angles of a hexagon is \((6 - 2)\times180^\circ= 720^\circ\). Wait, but angle at A: the angle given is 170°, but is that an interior angle? Wait, no, the angle at A: if it's a reflex angle? Wait, no, maybe it's a hexagon with one angle being a reflex angle? Wait, no, let's check the given angles: ∠C = 133°, ∠D = 102°, ∠E = 117°, ∠A: wait, the angle at A is 170°, but maybe it's a straight angle? No, wait, the problem: let's list the angles. Wait, the figure is a hexagon? Wait, no, the vertices are B, C, D, E, F, A. Wait, but angle at A: the angle marked 170°—maybe it's a convex hexagon? Wait, no, the sum of interior angles of a hexagon is \((n - 2)\times180^\circ\) where \(n = 6\), so \(4\times180 = 720^\circ\). Wait, but angle at F: is there a right angle? Wait, the figure shows a right angle at F? Wait, the original image: maybe F has a right angle (90°). Let's assume the angles are: ∠B (unknown), ∠C = 133°, ∠D = 102°, ∠E = 117°, ∠F = 90° (right angle), ∠A: wait, the angle at A is 170°? No, wait, maybe the angle at A is supplementary? Wait, no, the angle marked 170° is adjacent to B and F, maybe it's an exterior angle? No, let's re-express.

Wait, maybe the figure is a hexagon with angles: ∠B (x), ∠C = 133°, ∠D = 102°, ∠E = 117°, ∠F = 90° (right angle), and the angle at A: wait, the angle at A is 170°, but maybe it's a reflex angle? No, that can't be. Wait, maybe the sum of interior angles: let's count the number of sides. The points are B, C, D, E, F, A—so 6 sides, hexagon. Sum of interior angles: \((6 - 2)\times180 = 720^\circ\). Now, let's list the known angles: ∠C = 133°, ∠D = 102°, ∠E = 117°, ∠F = 90° (assuming the right angle), ∠A: wait, the angle at A is 170°? No, maybe the angle at A is 180° - 170° = 10°? No, that doesn't make sense. Wait, maybe the figure is a pentagon? Wait, B, C, D, E, F—no, A is also there. Wait, maybe the angle at A is 170°, but it's an interior angle. Wait, let's check the options. Let's suppose the sum is 720°, and we have angles: ∠B = x, ∠C = 133, ∠D = 102, ∠E = 117, ∠F = 90, ∠A = 170. Then sum: x + 133 + 102 + 117 + 90 + 170 = 720. Let's calculate: 133 + 102 = 235; 235 + 117 = 352; 352 + 90 = 442; 442 + 170 = 612. Then x = 720 - 612 = 108? No, 720 - 612 = 108? Wait, 612 + 108 = 720. But the options include 108. Wait, but maybe the angle at F is not 90°? Wait, maybe the figure is a hexagon with angles: ∠B, ∠C=133, ∠D=102, ∠E=117, ∠F (right angle, 90), and ∠A: wait, the angle at A is 170, but maybe it's a different approach. Wait, maybe the angle at A is 180 - 170 = 10? No, that's not right. Wait, let's re-express.

Wait, maybe the figure is a pentagon? Wait, B, C, D, E, F—no, A is connected. Wait, maybe the angle at A is 170°, but it's an exterior angle. No, interior angles. Wait, let's check the sum again. Let's assume the correct sum: for a hexagon, sum is 720. Let's list the angles:

∠B = x

∠C = 133°

∠D = 102°

∠E = 117°

∠F = 90° (right angle)

∠A = 170°

Sum: x + 133 + 102 + 117 + 90 + 170 = x + (133+102) + (117+90) + 170 = x + 235 + 207 + 170 = x + 612

Set equal to 720: x = 720 - 612 = 108°

Yes, that matches one of the options (108°). So the measure of ∠B is 108°.

Step2: Verify the sum

Sum of angles: 108 + 133 + 102 + 117 + 90 + 170 = 108+133=241; 241+102=343; 343+117=460; 460+90=550; 550+170=720. Which is the sum of interior angles of a hexagon (\((6-2)\times180=720\)). So correct.

Answer:

108° (Option: 108°)