Question

in the figure below, \\(\overline{ad}\\) and \\(\overline{ce}\\) are diameters of circle \\(p\\). what is the arc measure of minor arc \\(\overset{\frown}{ab}\\) in degrees?

Explanation:

Step1: Recall that a diameter divides a circle into two 180° arcs. So, ∠APE (or the angle related to the straight line) is 180°. Wait, actually, AD and CE are diameters, so they intersect at the center P, forming vertical angles and supplementary angles. Let's look at the given angles. The angle between CE and the other radius is 38°, and the angle between the other diameter and something is 95°? Wait, maybe first, since AD is a diameter, the straight line AD means that the sum of angles around point P on one side should be 180°. Wait, let's see: the angle between BP and CP is 38°, and the angle between DP and EP is 95°? Wait, maybe the key is that vertical angles are equal, and linear pairs sum to 180°. Wait, actually, since AD and CE are diameters, ∠APC and ∠DPE are vertical angles, and ∠AP C + ∠CPE + ∠EPD + ∠DPA? No, better: the total around a point is 360°, but diameters make 180° each. Wait, let's find the measure of ∠APB. Wait, first, let's note that CE is a diameter, so ∠CPE is a straight line? No, CE is a diameter, so from C to E through P is 180°. Wait, maybe the angle given as 95°: let's see, the angle between DP and EP is 95°, so the angle opposite to it (vertical angle) would be equal. Wait, maybe the angle ∠APB: let's calculate. Since AD is a diameter, the angle from A to D through P is 180°. Let's see the angles around P: we have 38°, 95°, and we need to find ∠APB. Wait, maybe the sum of angles on a straight line (AD) is 180°. So, the angle from A to B to C to E? Wait, no, let's re-express. Let's denote the angles: ∠CPB = 38°, ∠DPE = 95° (since it's given). Since CE and AD are diameters, ∠APC and ∠DPE are vertical angles? Wait, no, CE and AD intersect at P, so ∠APC and ∠DPE are vertical angles, so they should be equal? Wait, no, maybe I misread. Wait, the figure: points A, B, C on the left half, D, E on the right? Wait, the angle between BP and CP is 38°, the angle between DP and EP is 95°, and AD is a diameter, so from A to D is 180°. So, the angle from A to B to C to P to D? Wait, maybe the angle ∠APB is equal to 180° - 38° - 95°? Wait, no, let's think again. The total around point P is 360°, but since AD and CE are diameters, they divide the circle into four angles? Wait, no, two diameters intersecting form four angles, but here maybe there are more radii. Wait, the minor arc AB's measure is equal to the measure of its central angle ∠APB. So we need to find ∠APB. Let's see: since CE is a diameter, the angle from C to E through P is 180°, so ∠CPE = 180°. Wait, no, CE is a diameter, so the arc CE is 180°, so the central angle ∠CPE is 180°. But we have ∠CPB = 38°, ∠BPA =?, and ∠APE =? Wait, maybe the angle ∠DPE is 95°, so its vertical angle ∠APC is also 95°? Wait, no, vertical angles are equal. Wait, if AD and CE are diameters, then ∠APC and ∠DPE are vertical angles, so they should be equal. Wait, but 95°: if ∠DPE is 95°, then ∠APC is 95°. Then, since CE is a diameter, ∠CPE is 180°, so ∠CPB + ∠BPA + ∠APC = 180°? Wait, no, ∠CPB is 38°, ∠BPA is what we need, and ∠APC is 95°? Wait, that can't be, because 38 + 95 = 133, 180 - 133 = 47? Wait, no, maybe I got the angles wrong. Wait, let's start over. The central angle for minor arc AB is equal to the measure of ∠APB. Since AD is a diameter, the straight line AD means that the sum of angles from A to B to C to E (along the diameter CE? No, AD is the diameter. Wait, maybe the angle ∠APB: let's calculate the sum of angles on a straight line (AD) is 180°. So, the angle from A to P to D is 180°. The angles on one side of AD (say, the upper half) would be ∠APB + ∠BPC + ∠CPD? No, CE is another diameter, so CE is perpendicular? No, not necessarily. Wait, the given angle is 38° (∠CPB) and 95° (∠DPE). Since ∠DPE and ∠APC are vertical angles, ∠APC = 95°. Then, since CE is a diameter, ∠CPE = 180°, so ∠CPB + ∠BPA + ∠APC = 180°? Wait, ∠CPB is 38°, ∠APC is 95°, so ∠BPA = 180° - 38° - 95° = 47°? Wait, that makes sense. So the central angle ∠APB is 47°, so the minor arc AB is 47 degrees. Wait, let's check: 38 + 95 + 47 = 180, which is the straight line AD. Yes, because AD is a diameter, so the sum of angles from A to B to C to P to D should be 180°? Wait, no, from A to P to D is 180°, so the angles at P on that line: ∠APB + ∠BPC + ∠CPD? No, ∠APB, ∠BPC, and then ∠CPD? Wait, no, CE is a diameter, so C to E through P is 180°, so ∠CPE is 180°, but AD is also a diameter, so A to D through P is 180°. So the intersection of the two diameters at P creates angles: ∠APB, ∠BPC = 38°, ∠CPD, ∠DPE = 95°, ∠EPA, and ∠APB? Wait, no, around a point, the sum is 360°, but two diameters divide it into four angles, but here there are more radii. Wait, maybe the key is that ∠APB = 180° - 38° - 95° = 47°, because along the diameter AD, the sum of angles on one side (from A to D) is 180°, so ∠APB (angle at P between A and B) + ∠BPC (38°) + ∠CPD (which is equal to ∠DPE? No, ∠DPE is 95°, so ∠CPD is supplementary to ∠DPE? Wait, no, ∠CPD + ∠DPE = 180° because CE is a diameter. Wait, ∠CPD + 95° = 180°, so ∠CPD = 85°? No, that contradicts. Wait, I think I made a mistake. Let's try again. The measure of a minor arc is equal to the measure of its central angle. So we need to find ∠APB. Since AD and CE are diameters, they intersect at P, so ∠APC and ∠DPE are vertical angles, so ∠APC = ∠DPE = 95° (given). Also, ∠BPC = 38° (given). Now, since AD is a diameter, the straight line AD means that ∠APB + ∠BPC + ∠APC = 180° (because they are on a straight line). So ∠APB = 180° - ∠BPC - ∠APC = 180° - 38° - 95° = 47°. Yes, that works. So the central angle ∠APB is 47°, so the minor arc AB is 47 degrees.

Step1: Identify that AD is a diameter, so the sum of angles on line AD (at center P) is 180°.

Step2: Given ∠BPC = 38° and ∠APC = 95° (vertical angle to ∠DPE = 95°), calculate ∠APB = 180° - 38° - 95°.

$180 - 38 - 95 = 47$

Answer:

47