Abcd is cyclic if angle b 110 then find /_d

  1. [Solved] In this given figure ABCD is a cyclic quadrilateral. &a
  2. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC =70∘ and ∠BAC =30∘, then ∠BCD =A. 120∘B. 110∘C. 100∘D. 80∘
  3. In a kite ABCD, AB=AD and BC =DC ,if angle A= 80, angle B =40 then find angle C and angle D ?
  4. ABCD is a cyclic quadrilateral PQ is a tangent at B. If ∠DBQ =65∘, then ∠BCD isA. 35∘B. 85∘C. 115∘D. 90∘
  5. Quadrilateral ABCD is inscribed in a circle. If angle A measures 85 degrees and angle D measures 80 degrees, what is the measure of: Angle B? Arc ADC?


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[Solved] In this given figure ABCD is a cyclic quadrilateral. &a

Given: ABCD is acyclic quadrilateral ∠CDA = 110° ∠a ∶ ∠b = 2∶ 3 Concept: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. The sum of angles on a line is 180°, called linear pair of angles. Calculation: ∠CDA +∠CBA = 180° (The sum of either pair of opposite angles of a cyclic quadrilateral is 180°) ⇒ 110° +∠CBA = 180° ⇒∠CBA = 70° ∠CBA +∠CBE = 180° (The sum of angles on a line is 180°, called linear pair of angles.) ⇒ 70° +∠CBE = 180° ⇒∠CBE = 110° ∠CBE =∠a +∠b ⇒ 110° = 2x + 3x ⇒ x = 22° ∠b = 3× 22° = 66° ∠FBA =∠FBC +∠CBA ⇒∠FBA = 66° + 70° ∴∠FBA = 136° The correct option is 3 i.e.136°

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC =70∘ and ∠BAC =30∘, then ∠BCD =A. 120∘B. 110∘C. 100∘D. 80∘

The correct option is D 80 ∘ Given that ∠ D B C = 70 ∘ and ∠ B A C = 30 ∘ . Since the angles in the same segment are equal, ∠ D B C = ∠ D A C = 70 ∘ . Now, ∠ D A C + ∠ C A B = ∠ D A B . ⟹ ∠ D A B = 30 ∘ + 70 ∘ = 100 ∘ Now since opposite angles are supplementary in a cyclic quadrilateral, we have ∠ D A B + ∠ B C D = 180 ∘ . ⟹ ∠ B C D = 180 ∘ − ∠ D A B = 180 ∘ − 100 ∘ = 80 ∘

In a kite ABCD, AB=AD and BC =DC ,if angle A= 80, angle B =40 then find angle C and angle D ?

A Kite is a flat shaped with straight sides. It has 2 pairs of equal adjacent sides. #AB=AD and BC= DC# There is one set of congruent angles. These are opposite of each other and are between sides that are different lengths. The angles #/_B = /_D =40^0# as at #B and D# two pairs meet. Angle sum of a quadrilateral's interior angles must be 360°. #/_A + /_B +/_C + /_D=360^0 # #:. /_C=360-(80+2*40)= 200^0 # #:. /_C=200^0 and /_D=40^0# [Ans]

ABCD is a cyclic quadrilateral PQ is a tangent at B. If ∠DBQ =65∘, then ∠BCD isA. 35∘B. 85∘C. 115∘D. 90∘

The correct option is C 115° Join OB and OD We know that OB is perpendicular to PQ ∠OBD = ∠OBQ - ∠DBQ ∠OBD = 90 ∘– 65 ∘ ∠OBD = 25 ∘ OB = OD (radius) ∠OBD = ∠ODB = 25 ∘ In △ODB ∠OBD + ∠ODB + ∠BOD = 180 ∘ 25 ∘ + 25 ∘ + ∠BOD = 180 ∘ ∠BOD = 130 ∘ ∠BAD = 1 2 ∠BOD (Angle subtended by a chord on the center is double the angle subtended on the circle) ∠BAD = 1 2 ( 130 ∘) ∠BAD = 65 ∘ ABCD is a cyclic quadrilateral ∠BCD + ∠BAD = 180 ∘ ∠BCD + 65 ∘ = 180 ∘ ∠BCD = 115 ∘

Quadrilateral ABCD is inscribed in a circle. If angle A measures 85 degrees and angle D measures 80 degrees, what is the measure of: Angle B? Arc ADC?

In a cyclic quadrilateral, opposite angles add to 180 i.e. #/_A + /_C = 180, /_B + /_D = 180# 1) #:. /_B = 180 - /_D = 180 - 80 = color(blue)(100^0)# Angle at the center is twice the angle at the circumference. 2) ARC length ADC will have angle #color(blue)(200^0)# at the center. #:. Arc ADC = (X / 360) 2 pi r# where X is the angle subtended by the arc at the center and r the radius of the circle. #Arc ADC = (200 / 360) * 2 * pi r = (10/9) pi r ~~ color(blue)(3.49r)#

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