Angles in Polygons Worksheets - Math Monks - Free Printable

To solve the problems in the "Finding Missing Angles in a Polygon Worksheet," we need to use the properties of polygons, specifically the sum of the interior angles. The formula for the sum of the interior angles of a polygon with \( n \) sides is:

\[
\text{Sum of interior angles} = (n-2) \times 180^\circ
\]

We will apply this formula and the given angles to find the missing angles \( x \).

Problem 1:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 135^\circ, 60^\circ, 50^\circ \)

\[
135^\circ + 60^\circ + 50^\circ + x = 360^\circ
\]

\[
245^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 245^\circ = 115^\circ
\]

\[
\boxed{115}
\]

Problem 2:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 90^\circ, 90^\circ, 50^\circ \)

\[
90^\circ + 90^\circ + 50^\circ + x = 360^\circ
\]

\[
230^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 230^\circ = 130^\circ
\]

\[
\boxed{130}
\]

Problem 3:

The polygon is a triangle (3 sides).

\[
\text{Sum of interior angles} = (3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\]

Given angles: \( 60^\circ, 60^\circ \)

\[
60^\circ + 60^\circ + x = 180^\circ
\]

\[
120^\circ + x = 180^\circ
\]

\[
x = 180^\circ - 120^\circ = 60^\circ
\]

\[
\boxed{60}
\]

Problem 4:

The polygon is a pentagon (5 sides).

\[
\text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

Given angles: \( 130^\circ, 130^\circ, 130^\circ, 122^\circ, 138^\circ \)

\[
130^\circ + 130^\circ + 130^\circ + 122^\circ + 138^\circ + x = 540^\circ
\]

\[
650^\circ + x = 540^\circ
\]

\[
x = 540^\circ - 650^\circ = -110^\circ
\]

This result seems incorrect. Let's recheck the problem. The correct approach should be:

\[
130^\circ + 130^\circ + 130^\circ + 122^\circ + 138^\circ = 650^\circ
\]

Since the sum of the given angles exceeds 540°, there might be a mistake in the problem setup or interpretation. Let's assume the problem is correctly stated and re-evaluate:

\[
x = 540^\circ - (130^\circ + 130^\circ + 130^\circ + 122^\circ + 138^\circ) = 540^\circ - 650^\circ = -110^\circ
\]

This indicates a potential error in the problem statement. Assuming the problem is correct as stated, the answer is:

\[
\boxed{-110}
\]

Problem 5:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 120^\circ, 107^\circ, 90^\circ \)

\[
120^\circ + 107^\circ + 90^\circ + x = 360^\circ
\]

\[
317^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 317^\circ = 43^\circ
\]

\[
\boxed{43}
\]

Problem 6:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 140^\circ, 70^\circ, 50^\circ \)

\[
140^\circ + 70^\circ + 50^\circ + x = 360^\circ
\]

\[
260^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 260^\circ = 100^\circ
\]

\[
\boxed{100}
\]

Problem 7:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 60^\circ, 60^\circ, 120^\circ \)

\[
60^\circ + 60^\circ + 120^\circ + x = 360^\circ
\]

\[
240^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 240^\circ = 120^\circ
\]

\[
\boxed{120}
\]

Problem 8:

The polygon is a quadrilateral (4 sides).

\[
\text{Sum of interior angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\]

Given angles: \( 140^\circ, 40^\circ, 140^\circ \)

\[
140^\circ + 40^\circ + 140^\circ + x = 360^\circ
\]

\[
320^\circ + x = 360^\circ
\]

\[
x = 360^\circ - 320^\circ = 40^\circ
\]

\[
\boxed{40}
\]

Final Answers:

\[
\boxed{115, 130, 60, -110, 43, 100, 120, 40}
\]