Question

the diagram shows a convex polygon. angles: (4c - 50^circ), (15c + 1^circ), (18c + 44^circ), (16c + 23^circ), (22c). what is the value of (c)? (c = square^circ)

Explanation:

Step1: Recall the sum of exterior angles of a convex polygon.

The sum of the exterior angles of any convex polygon is \( 360^\circ \). So we add up all the given exterior angles and set the sum equal to \( 360^\circ \).
The angles are \( 4c - 50^\circ \), \( 15c + 1^\circ \), \( 18c + 44^\circ \), \( 22c \), and \( 16c + 23^\circ \).
So the equation is:
\( (4c - 50) + (15c + 1) + (18c + 44) + 22c + (16c + 23) = 360 \)

Step2: Combine like terms.

First, combine the \( c \)-terms:
\( 4c + 15c + 18c + 22c + 16c = (4 + 15 + 18 + 22 + 16)c = 75c \)
Then, combine the constant terms:
\( -50 + 1 + 44 + 23 = (-50 + 1) + (44 + 23) = -49 + 67 = 18 \)
So the equation simplifies to:
\( 75c + 18 = 360 \)

Step3: Solve for \( c \).

Subtract 18 from both sides:
\( 75c = 360 - 18 = 342 \)
Then divide both sides by 75:
\( c = \frac{342}{75} = \frac{114}{25} = 4.56 \)? Wait, that can't be right. Wait, maybe I made a mistake in combining terms. Let's check again.

Wait, let's re - add the constant terms:
\( -50 + 1 = -49 \); \( -49 + 44 = -5 \); \( -5 + 23 = 18 \). That's correct.
Wait, the sum of exterior angles is \( 360^\circ \). Wait, maybe the angles are: \( 4c - 50 \), \( 15c + 1 \), \( 18c + 44 \), \( 22c \), \( 16c + 23 \). Let's add them again:

\( 4c - 50+15c + 1+18c + 44+22c+16c + 23 \)

Group \( c \) terms: \( 4c+15c + 18c+22c+16c=(4 + 15+18+22+16)c=75c \)

Group constants: \( - 50+1 + 44+23=(-50)+(1 + 44+23)=-50 + 68 = 18 \). So equation is \( 75c+18 = 360 \)

Subtract 18: \( 75c=360 - 18 = 342 \)

\( c=\frac{342}{75}=\frac{114}{25}=4.56 \)? But that seems odd. Wait, maybe I misread the angles. Let's check the diagram again. Wait, maybe the angle is \( 4c - 50 \), \( 15c + 1 \), \( 18c + 44 \), \( 22c \), \( 16c + 23 \). Wait, maybe the problem has a typo or I misread. Wait, let's check the arithmetic again.

Wait, \( 360-18 = 342 \), \( 342\div75 = 4.56 \). But maybe I made a mistake in the sum of exterior angles? No, the sum of exterior angles of any convex polygon is \( 360^\circ \), regardless of the number of sides. Wait, how many sides does this polygon have? The number of exterior angles is equal to the number of sides. Let's count the exterior angles: 5, so it's a pentagon. The sum of exterior angles of a pentagon is \( 360^\circ \), so that part is correct.

Wait, maybe the original problem has different angle expressions. Wait, the user wrote "18c + 44°", "16c + 23°", "22c", "4c - 50°", "15c + 1°". Let's add them again:

\( 4c-50 + 15c + 1+18c + 44+22c+16c + 23 \)

\(=(4c + 15c+18c+22c+16c)+(-50 + 1+44+23) \)

\(=75c+( - 50+68) \)

\(=75c + 18 \)

Set equal to 360:

\( 75c=360 - 18=342 \)

\( c = 342\div75 = 4.56 \). But that is a decimal. Maybe there is a mistake in the problem or my reading. Wait, maybe the angle is \( 4c - 50 \), \( 15c + 1 \), \( 18c + 44 \), \( 22c \), \( 16c + 23 \). Wait, maybe I misread \( 4c - 50 \) as \( 4c - 50 \), but maybe it's \( 4c + 50 \)? Let's check. If it's \( 4c + 50 \), then the constant term would be \( 50 + 1+44+23=118 \), \( 75c+118 = 360 \), \( 75c=242 \), \( c\approx3.23 \), still not nice. Wait, maybe \( 4c - 5 \) instead of \( 4c - 50 \)? Then constant term: \( - 5+1+44+23=63 \), \( 75c+63 = 360 \), \( 75c=297 \), \( c = 3.96 \). No. Wait, maybe the angle is \( 4c - 5 \), \( 15c + 1 \), \( 18c + 4 \), \( 22c \), \( 16c + 23 \). No, the user's problem says \( 4c - 50^\circ \), \( 15c + 1^\circ \), \( 18c + 44^\circ \), \( 22c \), \( 16c + 23^\circ \).

Wait, maybe I made a mistake in the number of exterior angles. Let's count the diagram: the convex polygon has 5 exterior angles, so pentagon, sum of exterior angles \( 360^\circ \). So the equation is correct. Then \( c=\frac{342}{75}=4.56 \). But maybe the problem has a typo. Alternatively, maybe I misread the angles. Let's check again:

First angle: \( 4c - 50^\circ \)

Second: \( 15c + 1^\circ \)

Third: \( 18c + 44^\circ \)

Fourth: \( 22c \)

Fifth: \( 16c + 23^\circ \)

Sum: \( 4c-50 + 15c + 1+18c + 44+22c+16c + 23 \)

\(=(4 + 15+18+22+16)c+(-50 + 1+44+23) \)

\(=75c+( - 50+68) \)

\(=75c + 18 \)

\( 75c+18 = 360 \)

\( 75c=342 \)

\( c = 342\div75 = 4.56 \). So the value of \( c \) is \( 4.56 \). But maybe the problem expects an integer, so perhaps there is a mistake in the problem statement or my reading.

Wait, maybe the angle is \( 4c + 50 \) instead of \( 4c - 50 \). Let's try:

Sum: \( 4c + 50+15c + 1+18c + 44+22c+16c + 23 \)

\(=75c+(50 + 1+44+23)=75c + 118 \)

\( 75c+118 = 360 \)

\( 75c=242 \)

\( c\approx3.23 \), not integer.

Wait, maybe \( 4c - 15 \) instead of \( 4c - 50 \):

Sum: \( 4c-15+15c + 1+18c + 44+22c+16c + 23 \)

\(=75c+( - 15+1+44+23)=75c + 53 \)

\( 75c+53 = 360 \)

\( 75c=307 \)

\( c\approx4.09 \)

No. Alternatively, maybe the third angle is \( 18c - 44 \):

Sum: \( 4c-50+15c + 1+18c - 44+22c+16c + 23 \)

\(=75c+( - 50+1-44+23)=75c - 70 \)

\( 75c-70 = 360 \)

\( 75c=430 \)

\( c\approx5.73 \)

No. Maybe the fifth angle is \( 16c - 23 \):

Sum: \( 4c-50+15c + 1+18c + 44+22c+16c - 23 \)

\(=75c+( - 50+1+44-23)=75c - 28 \)

\( 75c-28 = 360 \)

\( 75c=388 \)

\( c\approx5.17 \)

No. Alternatively, maybe the second angle is \( 15c - 1 \):

Sum: \( 4c-50+15c - 1+18c + 44+22c+16c + 23 \)

\(=75c+( - 50-1+44+23)=75c + 16 \)

\( 75c+16 = 360 \)

\( 75c=344 \)

\( c\approx4.59 \)

Still not integer. Maybe the problem has a decimal answer. So according to the given angles, \( c = \frac{342}{75}=4.56 \). But let's check the division again: \( 342\div75 \), divide numerator and denominator by 3: \( 114\div25 = 4.56 \).

Answer:

\( c = 4.56 \)