The measure of the angles are ∠A = 91°, ∠C = 89° and ∠D = 34°
Explanation:
Given that the quadrilateral ABCD is inscribed in a circle.
The given angles are ∠A = (2x + 3), ∠C = (2x + 1) and ∠D = (x - 10)
We need to determine the measures of the angles A, C and D
The value of x:
We know that, the opposite angles of a cyclic quadrilateral add up to 180°
Thus, we have,
[tex]\angle A+\angle C=180^{\circ}[/tex]
Substituting the values, we have,
[tex]2x+3+2x+1=180[/tex]
[tex]4x+4=180[/tex]
[tex]4x=176[/tex]
[tex]x=44[/tex]
Thus, the value of x is 44.
Measure of ∠A:
Substituting [tex]x=44[/tex] in ∠A = (2x + 3)°, we get,
[tex]\angle A=(2(44)+3)^{\circ}[/tex]
[tex]=(88+3)^{\circ}[/tex]
[tex]\angle A=91^{\circ}[/tex]
Thus, the measure of angle A is 91°.
Measure of ∠C :
Substituting [tex]x=44[/tex] in ∠C = (2x + 1)°, we get,
[tex]\angle C=(2(44)+1)^{\circ}[/tex]
[tex]=(88+1)^{\circ}[/tex]
[tex]\angle C=89^{\circ}[/tex]
Thus, the measure of angle C is 89°.
Measure of ∠D :
Substituting [tex]x=44[/tex] in ∠D = (x - 10)°, we get,
[tex]\angle D=(44-10)^{\circ}[/tex]
[tex]\angle D=34^{\circ}[/tex]
Thus, the measure of angle D is 34°.
Hence, the measure of the angles are ∠A = 91°, ∠C = 89° and ∠D = 34°
