SOLUTION: A circle has a cyclic quadrilateral inside it marked A B C D there is a line across the middle D to B and also from the top A to C. The letters are clockwise in order if angle

Algebra ->  Circles -> SOLUTION: A circle has a cyclic quadrilateral inside it marked A B C D there is a line across the middle D to B and also from the top A to C. The letters are clockwise in order if angle       Log On


   

Question 1086563: A circle has a cyclic quadrilateral inside it marked A B C D there is a line across the middle D to B and also from the top A to C. The letters are clockwise in order if angle ADC is 89 and ACD is 42 find angle DBC giving reasons for your answer
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Angle ADC = 89 degrees (given)
Angle ACD = 42 degrees (given)

Focus on triangle ADC. Two angles are given.
The third angle, angle DAC, is unknown.
We can use the idea that three angles of a triangle add to 180 degrees to find angle DAC.

(Angle DAC) + (Angle ADC) + (Angle ACD) = 180
(Angle DAC) + (89) + (42) = 180
(Angle DAC) + 131 = 180
(Angle DAC) + 131 - 131 = 180 - 131
Angle DAC = 49

The angle DAC is an inscribed angle that cuts off minor arc CD.
The inscribed angle DBC also cuts off minor arc CD.
Therefore, these two inscribed angles are congruent

angle DBC = angle DAC
angle DBC = 49

Answer: 49 degrees