EH is a diameter of CircleD. The measure of Arc E F is (10x + 8)° and the measure of Arc G H is (11x)°. Circle D is shown. Line

Answer:

The value of x equals 5

The measurement of arc EF is 58°

The measurement of arc GH is 55°

Step-by-step explanation:

Let’s go over some essential concepts related to circles

• The total degrees in a circle is 360°
• A diameter splits a circle into two equal arcs, each measuring 180°
• An arc's measure corresponds to the central angle that subtends it

In circle D

∵ FD and GD serve as radii

- An angle formed at the center of a circle, with sides along the radii, is termed a central angle

∴ ∠FDG functions as a central angle corresponding to arc FG

- The measure of a central angle matches the measure of the arc it subtends

∴ m∠FDG = measurement of arc FG

∵ m∠FDG = 67°

∴ Measurement of arc FG = 67°

∵ EH acts as a diameter of circle D

- This divides the circle into two equal arcs

∴ The measurement of arc EH = 180°

∵ Arc EH includes arcs EF, FG, and GH

∴ m arc EH = m arc EF + m arc FG + m arc GH

∵ m arc EF = (10 x + 8)°

∵ m arc FG = 67°

∵ m arc GH = (11 x)°

- Plugging them into the equation above

∴ 180 = (10 x + 8) + 67 + 11 x

- Combine like terms on the right side

∴ 180 = 21 x + 75

- Deduct 75 from each side

∴ 105 = 21 x

- Divide both sides by 21

∴ 5 = x

The value of x is 5

∵ m arc EF = (10 x + 8)°

- Replace x with 5

∴ m arc EF = 10(5) + 8 = 50 + 8 = 58°

The measurement of arc EF is 58°

∵ m arc GH = (11 x)°

- Replace x with 5

∴ m arc GH = 11(5) = 55°

The measurement of arc GH is 55°

Answer:

The value of x is

5

.

The measurement of Arc E F is

58

degrees.

The measurement of Arc G H is

55

degrees.