Find all unknown angles and arcs for the Cyclic Quadrilateral | Important Geometry skills explained

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Find all unknown angles and arcs for the Cyclic Quadrilateral | Important Geometry skills explained

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[misterenter-izrz]

First BCD=180-114=66, secondly angle COD=142, thus angle ODC=angle OCD=(180-142)/2=19, likewise angle OAB=angle OBA=40, thus angle OAD=angle ODA=114-40=74, so angle ADC=19+74=93, so angle ABC=180-93=87, lastly angle OCB=66-19=47, thus m BC=180-2x47=180-94=86, m AD=180-2x74=180-148=32.😃

[alinayfeh]

The angle subtended by an arc of a circle it center divide half "is twice the angle
<ABC=87
<BCD=66
<ADC= 93
mBC=<86
mAD=<32

[JSSTyger]

I'll just say I got the same answer! Excellent as usual.

[marioalb]

Angle A = 114° (Data)
Angle C = 180°-114° = 66°
Angle (B+D) = 180°
Angle BOD = 66°x2 = 132°
Angle AOB = 100° (Data)
Angle AOD = 132-100=32°
Angle DOC = 142° (Data)
Angle BOC = 360-132-142=86°
Angle B = (142+32)/2=87°
Angle D = (100+86)/2=93°
All central angles=
142°+100°+86°+32°=360°
All trapezoid angles=
114°+93°+66°+87°=360°

[theoyanto]

Nice little tour of cyclic quadrilaterals and circle stuff, no maths needed.... Nice

[ybodoN]

Alternative approach:
Since two of their legs are radii of the circle, the triangles AOB and AOD are isosceles (also true for the triangles COD and BOC).
Thus, mAB = 100° = ∠AOB ⇒ ∠OAB = (180° − 100°)/2 = 40° ⇒ ∠OAD = 114° − 40° = 74° ⇒ ∠AOD = 180° − 2 × 74° = 32° = mAD.
Therefore, mBC = 360° − 100° − 32° − 142° = 86°. We can then find all the pairs in each triangle and add the neighboring angles.

[vidyadharjoshi]

BCD = 180 - 114 = 66;
Facing 114 angle DOB = 2*66 = 132. AOB = 100 so mAD = 32.
Facing C angel DOB = 2*114 = 228. DOC = 142 so mBC = 86.

In DOC angle DOC = 142 so other angles are same = 19. Since DCB = 66 so OCB = 66 - 19 = 47 = OBC
Similarly in AOB other angles each = 40. Since AOD = 32 so ODA = OAD = 74.
ADC = 74 + 19 = 93
ABC = 47 + 40 = 87

[msafasharhan]

Hello sir, how you know that arc AD is32

[bigm]

Arcs and angles are no match for the Professor’s numerical acumen!🥂📐❤