Question

Quadrilateral ABCD ​ is inscribed in this circle.

What is the measure of angle C?

Enter your answer in the box.


°




A quadrilateral inscribed in a circle. The vertices of the quadrilateral lie on the edge of the circle and are labeled as A, B, C, D. The interior angle A is labeled as left parenthesis 2 x plus 3 right parenthesis degrees. The angle B is labeled as left parenthesis 2 x minus 4 right parenthesis degrees. The angle D is labeled as left parenthesis 3 x plus 9 right parenthesis degrees.

Answer 1

The measure of angle C is 107°

Step-by-step explanation:

Given that ABCD is a quadrilateral.

The vertices A, B, C, D lie on the edge of the circle.

The measure of angle A is
`\angle A=(2x+3)^{\circ`

The measure of angle B is
`\angle B=(2x-4)^(\circ)`

The measure of angle D is
`\angle D=(3x+9)^(\circ)`

We need to determine the measure of angle C

Since, we know that the opposite angles B and D are supplementary.

Thus, we have,

`\angle B+\angle D=180^(\circ)`

Substituting the values, we get,

`2x-4+3x+9=180`

`5x+5=180`

`5x=175`

`x=35`

Thus, the value of x is 35

Substituting the value of x in the measures of angles A, B and D, we get,

`\angle A=(2(35)+3)^{\circ`

`\angle A=(70+3)^{\circ`

`\angle A=73^{\circ`

Similarly,
`\angle B=(2x-4)^(\circ)`

`\angle B=(2(35)-4)^(\circ)`

`\angle B=(70-4)^(\circ)`

`\angle B=66^(\circ)`

For angle D, we have,

`\angle D=(3(35)+9)^(\circ)`

`\angle D=(105+9)^(\circ)`

`\angle D=114^(\circ)`

Now, we shall find the measure of angle C

Since, we know that all the angles in a quadrilateral add up to 360°

Thus, we have,

`\angle A+\angle B+\angle C+\angle D=360^{\circ`

Substituting the values of A, B and D, we get,

`73^(\circ)+66^(\circ)+\angle C+114^(\circ)=360^(\circ)`

`253^(\circ)+\angle C=360^(\circ)`

`\angle C=107^(\circ)`

Thus, the measure of angle C is 107°