Question

the angles of a cyclic quadrilateral are the consecutive terms of an arithmetic sequence and one of the angles is 45 degree find the measures of other 3 angles of the quadrilateral

Answer 1

l think so the answer is 45

Answer 2

Answer: The other three angles are 75°, 105° and 135°.

Step-by-step explanation:  Given that the quadrilateral is cyclic, so the sum of all the angles of the quadrilateral will be 360°.

Also, the measures of the four angles are consecutive terms of an arithmetic progression. So, let the measure of the angles be a, a+d, a+2d and a+3d.

Let us consider that  a=45°.

Now, we can write

$a+(a+d)+(a+2d)+(a+3d) = 360^{\circ}\\\\\Rightarrow 4a+6d=360^\circ\\\\\Rightarrow 4\times 45^\circ +6d=360^\circ\\\\\Rightarrow 180^\circ+6d=360^\circ\\\\\Rightarrow 6d=360^\circ-180^\circ\\\\\Rightarrow 6d=180^\circ\\\\\Rightarrow d=30^\circ.$

So, the other three measures are

a+d, a+2d and a+3d

=45°+30°, 45°+60° and 45°+90°

=75°, 105° and 135°.

Hence the answer is 75°, 105° and 135°.