Question

In circle D, angle ADC measures (7x + 2)°. Arc AC

Answer 1

Answer:

The measure of angle ABC is 36° ⇒ 1st answer

Step-by-step explanation:

Let us revise some important facts in the circle

The measure of the center angle is equal to the measure of its subtended arc

The measure of the inscribed angle is equal to half the measure of the central angle subtended by the same arc

The vertex of a central angle is the center of the circle and its sides are radii in the circle

The vertex of an inscribed angle is a point on the circle, and its sides are chords in the circle

In circle D

∵ D is the center of the circle

∵ A and C lie on the circle

- DA and DC are radii

∴ ∠ADC is a central angle subtended by arc AC

∴ m∠ADC = m of arc AC

∵ m∠ADC = (7x + 2)°

∵ m of arc AC = (8x - 8)°

- Equate them to find x

∴ 8x - 8 = 7x + 2

- subtract 7x from both sides

∴ x - 8 = 2

- Add 8 to both sides

∴ x = 10

Substitute the value of x in the measure of ∠ADC

∵ m∠ADC = 7(10) + 2 = 70 + 2

∴ m∠ADC = 72°

∵ AB and BC are two chords in circle D

∴ ∠ABC is an inscribed angle subtended by arc AC

∵ ∠ADC is a central angle subtended by arc AC

- By using the 2nd fact above

∴ m∠ABC =  m∠ADC

∴ m∠ABC =  × 72

∴ m∠ABC = 36°