Answer:
16. 19 degrees
17. 152 degrees
18. 103 degrees
19-20. 108 degrees
21-23. 140 degrees
24-25. 30 degrees
Step-by-step explanation:
B.
16.
[tex]\sf 88^{\circ} + 73 ^{\circ} + a = 180^{\circ}[/tex]
[tex]\implies \sf 161^{\circ}+a = 180^{\circ}[/tex]
[tex]\implies \sf a = 180^{\circ}-161^{\circ}[/tex]
[tex]\implies \boxed{\boxed{\sf a = 19^{\circ}}}[/tex]
17.
[tex]\sf 87^{\circ} + 104^{\circ} +107^{\circ}+90^{\circ}+b=540^{\circ}[/tex]
[tex]\implies \sf 388^{\circ}+b=540^{\circ}[/tex]
[tex]\implies \sf b = 540^{\circ} - 388^{\circ}[/tex]
[tex]\implies \boxed{\boxed{\sf b = 152^{\circ}}}[/tex]
18.
[tex]\sf 113^{\circ}+2(72^{\circ})+c=360^{\circ}[/tex]
[tex]\implies \sf 113^{\circ} + 144^{\circ} +c = 360^{\circ}[/tex]
[tex]\implies \sf 257^{\circ}+c=360^{\circ}[/tex]
[tex]\implies \sf c = 360^{\circ} - 257^{\circ}[/tex]
[tex]\implies \boxed{\boxed{\sf c = 103^{\circ}}}[/tex]
19-20.
Solving for d:
Let a, b, c, and f be the other 4 interior angles of the pentagon.
[tex]\sf a + b + c + d + f = 540^{\circ}[/tex]
From the given image, the angles of the pentagon are equal.
Hence,
[tex]\sf a = b = c = d = f[/tex]
Substituting,
[tex]\sf d + d + d +d + d = 540^{\circ}[/tex]
[tex]\implies \sf 5d = 540^{\circ}[/tex]
[tex]\implies \sf d = \frac{540^{\circ}}{5}[/tex]
[tex]\implies \boxed{\boxed{\sf d = 108^{\circ}}}[/tex]
Solving for e:
Let e' be such that e' and e are supplementary angles.
[tex]\sf e' +e = 180^{\circ}[/tex]
Observe that [tex]\sf e' = d[/tex]
We know that [tex]\sf d = 108^{\circ}[/tex]
Thus
[tex]\sf 108^{\circ}+e = 180^{\circ}\\\implies \sf e = 180^{\circ}-108^{\circ}\\\implies \boxed{\boxed{\sf e = 72^{\circ}}}[/tex]
C.
21-23.
The formula in finding the sum of the interior angles of a polygon is
[tex]\sf s = (n-2)180^{\circ}[/tex]
The formula in finding the sum of the interior angles of a polygon can also be written as
[tex]\sf \angle 1 + \angle 2 + ... +\angle n = (n-2)180^{\circ}[/tex]
We know that a nonagon has 9 sides.
[tex]\sf \angle 1 + \angle 2 +... + \angle 9 = (9-2)180^{\circ}[/tex]
Since the nonagon is a regular nonagon, its interior angles are equal.
This implies,
[tex]\sf \angle 1 = \angle 2 = ... = \angle 9[/tex]
Hence,
[tex]\sf \angle 1 + \angle 1 + \angle 1 + \angle 1 + \angle 1 + \angle 1 + \angle 1 + \angle 1 + \angle 1=(9-2)180^{\circ}\\\implies \sf 9\angle 1 = (7)180^{\circ}\\\implies \sf 9\angle 1 = 1260^{\circ}\\\implies \sf \angle 1= \frac{1260^{\circ}}{9}\\\implies \sf \angle 1 = 140^{\circ}[/tex]
Hence, the measure of each interior angle of a regular nonagon is 140°.
24-25.
The other form of the formula sum of the interior angles of a polygon is
[tex]\sf \angle 1 + \angle 2 + ... +\angle n = (n-2)180^{\circ}[/tex], (as we've shown to 21-23)
We know that a regular dodecagon has 12 sides.
[tex]\sf \angle 1 + \angle 2 + ... + \angle 12 = (12-2)180^{\circ}[/tex]
Also, a regular dodecagon has equal interior angles.
Hence,
[tex]\sf \angle 1 = \angle 2 = ... = \angle 12[/tex]
Substituting and adding yields,
[tex]\sf 12\angle 1 = (12-2)180^{\circ}\\\implies \sf 12 \angle 1 = (10)180^{\circ}\\\implies \sf 12\angle 1 = 1800^{\circ}\\\implies \sf \angle 1 = \frac{1800^{\circ}}{12}\\\implies \sf \angle 1 = 150^{\circ}[/tex]
Therefore the measure of each interior angles of a regular dodecagon is 150°
Now, we know that interior angles and exterior angles of a polygon are supplementary angles
We also know that the measure of each interior angle of a regular dodecagon is 150°.
Hence,
[tex]\sf 150^{\circ} + Exterior \ Angle = 180^{\circ}[/tex]
[tex]\implies \sf Exterior \ Angle = 180^{\circ} - 150^{\circ}[/tex]
[tex]\implies \boxed{\boxed{\sf Exterior \ Angle = 30^{\circ}}}[/tex]
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