The measure of the angles are: ∠1 = 30°, ∠2 = 30°, ∠3 = 21°, ∠4 = 39°, ∠5 = 39°, ∠6 = 24°, ∠7 = 24°, ∠8 = 66°, ∠9 = 66°, ∠10 = 21°, ∠11 = 60°, ∠12 = 120°, ∠13 = 60°, ∠14 = 120°, ∠15 = 48°, ∠16 = 105°, ∠17 = 27°, ∠18 = 129°, ∠19 = 51°, ∠20 = 129° and ∠21 = 51°
How to determine the measure of the angles?
The given parameters are:
AB = 78°
FE = 105°
ED = 27°
CD = 42°
Considering the semicircle ABCD, the measure of arc BC is:
BC = 180 - AB - CD
This gives
BC = 180 - 78 - 42
Evaluate
BC = 60°
Considering the semicircle A FED, the measure of arc A F is:
A F = 180 - FE - ED
This gives
A F = 180 - 105 - 27
Evaluate
A F = 48°
The angle 16 subtends arc FE.
So, we have:
∠16 = FE
This gives
∠16 = 105°
Similarly, the angles 15 and 17 subtend arcs A F and ED respectively.
So, we have:
∠15 = 48°
∠17 = 27°
Angles at the circumference are half the angles at the arc.
This means that:
∠1 = Arc BC/2
∠3 = Arc CD/2
∠4 = Arc AB/2
∠5 = Arc AB/2
∠6 = Arc A F/2
∠10 = Arc CD/2
So, we have:
∠1 = 60/2 = 30°
∠3 = 42/2 = 21°
∠4 = 78/2 = 39°
∠5 = 78/2 = 39°
∠6 = 48/2 = 24°
∠10 = 42/2 = 21°
The angle in a semicircle is 90 degrees.
This means that:
∠2 = 90°
The sum of angles in a triangle is 180 degrees.
This means that:
∠11 = 180 - ∠1 - ∠2
∠12 = 180 - ∠3 - ∠4
So, we have:
∠11 = 180 - 30 - 90 = 60°
∠12 = 180 - 21 - 39 = 120°
Vertical angles are equal.
So, we have:
∠13 = ∠11
∠14 = ∠12
This gives
∠13 = 60°
∠14 = 120°
The sum of angles in a triangle is 180 degrees.
This means that:
∠18 = 180 - ∠17 - ∠6
So, we have:
∠18 = 180 - 27 - 24 = 129°
Vertical angles are equal.
So, we have:
∠20 = ∠18
This gives
∠20 = 129°
Angle on a straight line equals 180°.
So, we have:
∠21 = 180 - ∠18
This gives
∠21 = 180 - 129° = 51°
Vertical angles are equal.
So, we have:
∠19 = ∠21
This gives
∠19 = 51°
The sum of angles in a triangle is 180 degrees.
This means that:
∠7 = 180 - ∠21 - ∠16
So, we have:
∠7 = 180 - 51 - 105
∠7 = 24°
The angle in a semicircle is 90 degrees.
This means that:
∠7 + ∠8 = 90°
So, we have:
24 + ∠8 = 90°
Subtract 24 from both sides
∠8 = 66°
The sum of angles in a triangle is 180 degrees.
This means that:
∠8 = 180 - ∠15 - ∠9
So, we have:
∠8 = 180 - 48 - 66
So, we have:
∠8 = 66°
Hence, the measure of the angles are:
∠1 = 30°, ∠2 = 30°, ∠3 = 21°, ∠4 = 39°, ∠5 = 39°, ∠6 = 24°, ∠7 = 24°, ∠8 = 66°, ∠9 = 66°, ∠10 = 21°, ∠11 = 60°, ∠12 = 120°, ∠13 = 60°, ∠14 = 120°, ∠15 = 48°, ∠16 = 105°, ∠17 = 27°, ∠18 = 129°, ∠19 = 51°, ∠20 = 129° and ∠21 = 51°
Read more about angles in circles at:
https://brainly.com/question/14362353
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