Various polygon diagrams with internal angles and side markings.
A collection of geometric shapes including pentagons, hexagons, and irregular polygons with labeled angles and sides.
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Show Answer Key & Explanations
Step-by-step solution for: Interior/Exterior Angles of a Polygon Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Interior/Exterior Angles of a Polygon Worksheet
The task appears to involve analyzing and solving problems related to polygons, specifically focusing on their angles, sides, and properties. Below, I will explain the solution for each polygon shown in the image.
---
- Given: A quadrilateral with one angle labeled as \(60^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a quadrilateral is given by the formula:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- If only one angle is given (\(60^\circ\)), we cannot determine the other angles without additional information. However, we know the total sum of the angles is \(360^\circ\).
---
- Given: A pentagon with one angle labeled as \(108^\circ\) and another as \(90^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a pentagon is:
\[
(n-2) \times 180^\circ = (5-2) \times 180^\circ = 540^\circ
\]
- If two angles are given (\(108^\circ\) and \(90^\circ\)), the sum of the remaining three angles is:
\[
540^\circ - 108^\circ - 90^\circ = 342^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(342^\circ\).
---
- Given: A hexagon with one angle labeled as \(120^\circ\) and another as \(100^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a hexagon is:
\[
(n-2) \times 180^\circ = (6-2) \times 180^\circ = 720^\circ
\]
- If two angles are given (\(120^\circ\) and \(100^\circ\)), the sum of the remaining four angles is:
\[
720^\circ - 120^\circ - 100^\circ = 500^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(500^\circ\).
---
- Given: An irregular polygon with sides labeled as \(6\), \(8\), \(10\), \(12\), and \(14\).
- Task: Determine the number of sides or the sum of the interior angles.
- Solution:
- The polygon has 5 sides, so it is a pentagon.
- The sum of the interior angles of a pentagon is:
\[
(n-2) \times 180^\circ = (5-2) \times 180^\circ = 540^\circ
\]
---
- Given: A trapezoid with one angle labeled as \(120^\circ\) and another as \(60^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a trapezoid (which is a quadrilateral) is:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- If two angles are given (\(120^\circ\) and \(60^\circ\)), the sum of the remaining two angles is:
\[
360^\circ - 120^\circ - 60^\circ = 180^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(180^\circ\).
---
- Given: A right triangle with one angle labeled as \(45^\circ\).
- Task: Determine the measure of the third angle.
- Solution:
- In a triangle, the sum of the interior angles is always \(180^\circ\).
- Since it is a right triangle, one angle is \(90^\circ\). With another angle given as \(45^\circ\), the third angle is:
\[
180^\circ - 90^\circ - 45^\circ = 45^\circ
\]
- Therefore, the triangle is an isosceles right triangle.
---
- Given: A scalene triangle with one angle labeled as \(30^\circ\) and another as \(80^\circ\).
- Task: Determine the measure of the third angle.
- Solution:
- In a triangle, the sum of the interior angles is always \(180^\circ\).
- With two angles given as \(30^\circ\) and \(80^\circ\), the third angle is:
\[
180^\circ - 30^\circ - 80^\circ = 70^\circ
\]
---
- Given: A regular hexagon with one side labeled as \(10\).
- Task: Determine the measure of each interior angle.
- Solution:
- The sum of the interior angles of a hexagon is:
\[
(n-2) \times 180^\circ = (6-2) \times 180^\circ = 720^\circ
\]
- Since it is a regular hexagon, all interior angles are equal. Therefore, each interior angle is:
\[
\frac{720^\circ}{6} = 120^\circ
\]
---
\[
\boxed{360^\circ, 540^\circ, 720^\circ, 540^\circ, 180^\circ, 45^\circ, 70^\circ, 120^\circ}
\]
---
1. Quadrilateral (Top Left)
- Given: A quadrilateral with one angle labeled as \(60^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a quadrilateral is given by the formula:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- If only one angle is given (\(60^\circ\)), we cannot determine the other angles without additional information. However, we know the total sum of the angles is \(360^\circ\).
---
2. Pentagon (Top Right)
- Given: A pentagon with one angle labeled as \(108^\circ\) and another as \(90^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a pentagon is:
\[
(n-2) \times 180^\circ = (5-2) \times 180^\circ = 540^\circ
\]
- If two angles are given (\(108^\circ\) and \(90^\circ\)), the sum of the remaining three angles is:
\[
540^\circ - 108^\circ - 90^\circ = 342^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(342^\circ\).
---
3. Hexagon (Middle Left)
- Given: A hexagon with one angle labeled as \(120^\circ\) and another as \(100^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a hexagon is:
\[
(n-2) \times 180^\circ = (6-2) \times 180^\circ = 720^\circ
\]
- If two angles are given (\(120^\circ\) and \(100^\circ\)), the sum of the remaining four angles is:
\[
720^\circ - 120^\circ - 100^\circ = 500^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(500^\circ\).
---
4. Irregular Polygon (Middle Center)
- Given: An irregular polygon with sides labeled as \(6\), \(8\), \(10\), \(12\), and \(14\).
- Task: Determine the number of sides or the sum of the interior angles.
- Solution:
- The polygon has 5 sides, so it is a pentagon.
- The sum of the interior angles of a pentagon is:
\[
(n-2) \times 180^\circ = (5-2) \times 180^\circ = 540^\circ
\]
---
5. Trapezoid (Middle Right)
- Given: A trapezoid with one angle labeled as \(120^\circ\) and another as \(60^\circ\).
- Task: Determine the sum of the interior angles or any missing angles.
- Solution:
- The sum of the interior angles of a trapezoid (which is a quadrilateral) is:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- If two angles are given (\(120^\circ\) and \(60^\circ\)), the sum of the remaining two angles is:
\[
360^\circ - 120^\circ - 60^\circ = 180^\circ
\]
- Without further information, we cannot determine the individual values of the remaining angles, but their total is \(180^\circ\).
---
6. Right Triangle (Bottom Left)
- Given: A right triangle with one angle labeled as \(45^\circ\).
- Task: Determine the measure of the third angle.
- Solution:
- In a triangle, the sum of the interior angles is always \(180^\circ\).
- Since it is a right triangle, one angle is \(90^\circ\). With another angle given as \(45^\circ\), the third angle is:
\[
180^\circ - 90^\circ - 45^\circ = 45^\circ
\]
- Therefore, the triangle is an isosceles right triangle.
---
7. Scalene Triangle (Bottom Center)
- Given: A scalene triangle with one angle labeled as \(30^\circ\) and another as \(80^\circ\).
- Task: Determine the measure of the third angle.
- Solution:
- In a triangle, the sum of the interior angles is always \(180^\circ\).
- With two angles given as \(30^\circ\) and \(80^\circ\), the third angle is:
\[
180^\circ - 30^\circ - 80^\circ = 70^\circ
\]
---
8. Regular Hexagon (Bottom Right)
- Given: A regular hexagon with one side labeled as \(10\).
- Task: Determine the measure of each interior angle.
- Solution:
- The sum of the interior angles of a hexagon is:
\[
(n-2) \times 180^\circ = (6-2) \times 180^\circ = 720^\circ
\]
- Since it is a regular hexagon, all interior angles are equal. Therefore, each interior angle is:
\[
\frac{720^\circ}{6} = 120^\circ
\]
---
Final Answer
\[
\boxed{360^\circ, 540^\circ, 720^\circ, 540^\circ, 180^\circ, 45^\circ, 70^\circ, 120^\circ}
\]
Parent Tip: Review the logic above to help your child master the concept of interior exterior angles worksheet.
