The angles of a cyclic quadrilateral are the consecutive terms of an arithmetic sequence and one of the angles is 45 degree. What is the measurement of angle opposite to angle 45? Find the measure of other two angles of the quadrilateral.
Answers
Cyclic quadrilateral is a 4 sided polygon whose vertices lie on the circumference of a single circle.
⇒ opposite angles add up to 180º
Given that one of the angles is 45º:
⇒ Opposite angle = 180 - 45 = 135º
Given that the angles are consecutive terms of an arithmetic sequence:
⇒ There is a common difference between one angle and the next
Define the difference:
Let the difference be d.
First angle = 45º (Given)
Second angle = 45 + d
Third angle = 45 + 2d = 135 (We found it earlier)
Fourth angle = 45 + 3d
Solve d:
Since we also know that all the angles in a quadrilateral add up to 360º
45 + (45 + d) + 135 + (45 + 3d) = 360
45 + 45 + d + 135 + 45 + 3d = 360
4d + 370 = 360
4d = 90
d = 22.5
Find the angles:
Second angle = 45 + d = 45 + 22.5 = 67.5º
Fourth angle = 45 + 3d = 45 + 3(22.5) = 112.5º
Answer: The opposite angle is 135º and the 2 other angles are 67.5º and 112.5º
Answer:
A) 180°-45° = 135°
B) 67.5 and 112.5