Question : ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it and $\angle A D C=148^{\circ}$. What is the measure of the $\angle BAC$?
Option 1: $32^{\circ}$
Option 2: $45^{\circ}$
Option 3: $58^{\circ}$
Option 4: $60^{\circ}$
Correct Answer: $58^{\circ}$
Solution : ABCD is a cyclic quadrilateral. $\angle ADC = 148^\circ$ The angle formed by drawing lines from the ends of the diameter of a circle to its circumference forms a right angle. The sum of opposite angles of the cyclic quadrilateral is 180$^\circ$ In quadrilateral ABCD, $\angle ADC+\angle CBA = 180^\circ$ ⇒ $148^\circ+\angle CBA = 180^\circ$ ⇒ $\angle CBA = 180^\circ - 148^\circ$ ⇒ $\angle CBA = 32^\circ$ In $\triangle ABC$, $\angle BAC + 90^\circ + \angle CBA = 180^\circ$ ⇒ $\angle BAC + 90^\circ + 32^\circ = 180^\circ$ ⇒ $\angle BAC + 122^\circ = 180^\circ$ ⇒ $\angle BAC = 180^\circ - 122^\circ$ ⇒ $\angle BAC = 58^\circ$ $\therefore$ The measure of the ∠BAC is 58° Hence, the correct answer is 58$^\circ$
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Question : ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it and $\angle $ADC = 118°. What is the measure of $\angle$BAC?
Option 1: 28°
Option 2: 45°
Option 3: 32°
Option 4: 30°
Question : PT is a tangent at the point R on a circle with centre O. SQ is a diameter, which when produced meets the tangent PT at P. If $\angle$SPT = 32$^\circ$, then what will be the measure of $\angle$QRP?
Option 1: $58^\circ$
Option 2: $30^\circ$
Option 3: $29^\circ$
Option 4: $32^\circ$
Question : ABCD is a cyclic quadrilateral. The tangents to the circle at the points A and C on it, intersect at P. If $\angle\mathrm{ABC}=98^{\circ}$, then what is the measure of $\angle \mathrm{APC}$ ?
Option 1: 22°
Option 2: 26°
Option 3: 16°
Option 4: 14°
Question : E, F, G, and H are four points lying on the circumference of a circle to make a cyclic quadrilateral. If $\angle {FGH}=57^{\circ}$, then what will be the measure of the $\angle {HEF}$?
Option 1: $33^{\circ}$
Option 2: $123^{\circ}$
Option 3: $93^{\circ}$
Option 4: $143^{\circ}$
Question : AB is a chord in a circle with centre O. AB is produced to C such that BC is equal to the radius of the circle. C is joined to O and produced to meet the circle at D. If $\angle \mathrm{ACD}=32^{\circ}$, then the measure of $\angle \mathrm{AOD}$ is _____.
Option 1: 48°
Option 2: 96°
Option 3: 108°
Option 4: 80°
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