Respuesta :
Answer:
40°
Step-by-step explanation:
Opposite angles in a cyclic quadrilateral are equal
3x + 10 = 2x + 20
Take away 2x from both sides
x + 10 = 20
Take away 10 from both sides
x = 10
Now that we know the value of x, substitute it back into the equations
3x + 10 = 3 x 10 + 10 = 30 + 10 = 40
2x + 20 = 2 x 10 + 20 = 20 + 20 = 40
Both angles are equal to 40°
Step-by-step explanation:
Need to FinD :
- We have to find the angles.
[tex] \red{\frak{Given}} \begin{cases} & \sf {We\ are\ given\ a\ cyclic\ quadrilateral.} \\ & \sf {The\ measures\ of\ the\ opposite\ angles\ of\ the\ quadrilateral\ are\ {\pmb{\sf{(3x\ +\ 10)^{\circ}}}}\ and\ {\pmb{\sf{(2x\ +\ 20)^{\circ}}}}.} \end{cases}[/tex]
We know that,
- The opposite angles of the cyclic quadrilateral are (3x + 10)° and (2x + 20)°. And we have to find the angles.
In a cyclic quadrilateral, all the four vertices of the quadrilateral lie on the circumference of the circle. The opposite angles in a cyclic quadrilateral add up to 180°. So simply, we just simplify the measures and find the value of x. And at last, we'll find the angles of the quadrilateral.
[tex]\rule{200}{3}[/tex]
[tex] \sf \dashrightarrow {(3x\ +\ 10)\ +\ (2x\ +\ 20)\ =\ 180} \\ \\ \\ \sf \dashrightarrow {3x\ +\ 10\ +\ 2x\ +\ 20\ =\ 180} \\ \\ \\ \sf \dashrightarrow {5x\ +\ 30\ =\ 180} \\ \\ \\ \sf \dashrightarrow {5x\ =\ 180\ -\ 30} \\ \\ \\ \sf \dashrightarrow {5x\ =\ 150} \\ \\ \\ \sf \dashrightarrow {x\ =\ \dfrac{\cancel{150}}{\cancel{5}}} \\ \\ \\ \dashrightarrow {\underbrace{\boxed{\pink{\frak{x\ =\ 30.}}}}_{\sf \blue{\tiny{Value\ of\ x}}}} [/tex]
∴ Hence, the required value of x is 30. Now, let us find out the angles.
[tex]\rule{200}{3}[/tex]
First AnglE :
- 3x + 10
- 3(30) + 10
- 90 + 10
- 100
∴ Hence, the measure of the first angle of the quadrilateral is 100°.
[tex]\rule{200}{3}[/tex]
Second AnglE :
- 2x + 20
- 2(30) + 20
- 60 + 20
- 80
∴ Hence, the measure of the second angle of the quadrilateral is 80°.
