Question

​ Quadrilateral ABCD ​ is inscribed in this circle.

What is the measure of angle A?


Enter your answer in the box.

°

Answer 1

a quadrilateral, has 4 sides and its internal angles sum, add up to 360, now... you have 3 angles give.. .but we don't have C

so.. C is the difference of all the three angles from 360 or
`\bf \measuredangle C=360-x-(2x+1)-148\implies \measuredangle C=360-x-2x-1-148` whatever that is, now, you'll get some value in x-terms

so.... now once we know what C is

you can if you want, do a search in google for "inscribed quadrilateral conjecture", I can do a quick proof if you need one

but in short, for a quadrilateral inscribed in a circle, each pair of opposites angles are "supplementary angles", namely they add up to 180°

so.. what the dickens does all that mean? well D+B=180 and A+C = 180

now. we know what A is, 2x+1
and by now, you'd know what C is from 360-x-2x-1-148

so... add them together then and


`\bf \begin{array}{cccclllll} A&+&C&=&180\\ \uparrow &&\uparrow \\ (2x+1)&+&(360-x-2x-1-148)&=&180 \end{array}`

solve for "x"

Answer 2

The measure of ∠A=65°

Explanation:

From the given figure, it can be seen that quadrilateral ABCD is inscribed in a circle and

Since, we know that opposite angles of the quadrilateral inscribed in circle is supplementary, therefore

∠D+∠B=180°

⇒x+148=180

⇒x=180-148

⇒x=32°

therefore, the measure of ∠A=
`2x+1=2(32)+1=65^{{\circ}}`