The different angle measures are
1. [tex]m\angle EBD = 34^\circ[/tex]
2. [tex]m\angle ACE = 52^{\circ}[/tex]
3. [tex]m \overset{\LARGE\frown}{GB} = 76^\circ[/tex]
4. [tex]m \overset{\LARGE\frown}{GBD} = 256^\circ[/tex]
5. [tex]m \angle DBA= 90^\circ[/tex]
6. [tex]m \overset{\LARGE\frown}{GD} = 104^\circ[/tex]
7. [tex]m\angle DFC= 56^\circ[/tex]
Calculating measures of angles
From the question, we are to determine the measures of the different angles
1.
From the given information,
[tex]m \overset{\LARGE\frown}{HD} = 68^\circ[/tex]
This implies that the measure of the central angle is 68°
Then,
[tex]m\angle EBD = \frac{1}{2} \times 68^\circ[/tex] (Angle at the center is twice the angle at the circumference)
∴ [tex]m\angle EBD = 34^\circ[/tex]
2.
From the diagram
[tex]m\angle ACE + m\angle BED + 90^\circ = 180^\circ[/tex]
From the given information
[tex]m\angle BED = 38^\circ[/tex]
Then,
[tex]m\angle ACE + 38^\circ + 90^\circ = 180^\circ[/tex]
[tex]m\angle ACE + 128^{\circ} = 180^\circ[/tex]
[tex]m\angle ACE = 180^\circ-128^{\circ}[/tex]
[tex]m\angle ACE = 52^{\circ}[/tex]
3.
First, we will determine [tex]m \overset{\LARGE\frown}{GD}[/tex]
[tex]m \overset{\LARGE\frown}{GD} = 2 \times m\angle ACE[/tex] (Angle at the center is twice the angle at the circumference)
[tex]m \overset{\LARGE\frown}{GD} = 2 \times 52^\circ[/tex]
[tex]m \overset{\LARGE\frown}{GD} = 104^\circ[/tex]
But,
[tex]m \overset{\LARGE\frown}{GD} + m \overset{\LARGE\frown}{GB} = 180^\circ[/tex]
[tex]m \overset{\LARGE\frown}{GB} = 180^\circ - m \overset{\LARGE\frown}{GD}[/tex]
[tex]m \overset{\LARGE\frown}{GB} = 180^\circ - 104^\circ[/tex]
[tex]m \overset{\LARGE\frown}{GB} = 76^\circ[/tex]
4.
[tex]m \overset{\LARGE\frown}{GBD} = m \overset{\LARGE\frown}{GB} + 180^\circ[/tex]
[tex]m \overset{\LARGE\frown}{GBD} = 76^\circ +180^\circ[/tex]
[tex]m \overset{\LARGE\frown}{GBD} = 256^\circ[/tex]
5.
[tex]m \angle DBA= 90^\circ[/tex] (Tangent and diameter/ radius theorem)
If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another
6.
[tex]m \overset{\LARGE\frown}{GD} = 104^\circ[/tex] (As determined above)
7.
First, we will determine [tex]m\angle GFB[/tex]
[tex]m\angle GFB + m\angle EBD + 90^\circ = 180^\circ[/tex]
[tex]m\angle GFB = 180^\circ -(m\angle EBD + 90^\circ)[/tex]
[tex]m\angle GFB = 180^\circ - (34^\circ+90^\circ)[/tex]
[tex]m\angle GFB = 180^\circ - 124^\circ[/tex]
[tex]m\angle GFB = 56^\circ[/tex]
[tex]m\angle GFB = m\angle DFC[/tex] (Vertically opposite angles)
[tex]m\angle DFC= 56^\circ[/tex]
Hence, the different angle measures are
1. [tex]m\angle EBD = 34^\circ[/tex]
2. [tex]m\angle ACE = 52^{\circ}[/tex]
3. [tex]m \overset{\LARGE\frown}{GB} = 76^\circ[/tex]
4. [tex]m \overset{\LARGE\frown}{GBD} = 256^\circ[/tex]
5. [tex]m \angle DBA= 90^\circ[/tex]
6. [tex]m \overset{\LARGE\frown}{GD} = 104^\circ[/tex]
7. [tex]m\angle DFC= 56^\circ[/tex]
Learn more on Calculating the measures of angles here: https://brainly.com/question/21872415
SPJ1
