Find the missing angle x in each of the seven geometric figures.
A worksheet with seven geometric diagrams, each showing a polygon with angles marked, including one unknown angle labeled 'x' to be calculated. Diagrams are not to scale.
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Show Answer Key & Explanations
Step-by-step solution for: Exterior Angles of Convex Polygons - Opinions Nobody Asked For
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Show Answer Key & Explanations
Step-by-step solution for: Exterior Angles of Convex Polygons - Opinions Nobody Asked For
To solve for the angles marked \( x \) in each diagram, we will use geometric properties such as the sum of angles in a triangle, the sum of angles on a straight line, and the properties of parallel lines. Let's go through each problem step by step.
---
[Diagram 1]
- Given angles: \( 80^\circ \), \( 60^\circ \)
- The shape is a quadrilateral with one right angle (\( 90^\circ \)).
#### Solution:
1. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
2. Let the unknown angle be \( x \).
3. The equation for the sum of the angles is:
\[
80^\circ + 60^\circ + 90^\circ + x = 360^\circ
\]
4. Simplify:
\[
230^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 230^\circ = 130^\circ
\]
Answer for Problem 1:
\[
\boxed{130^\circ}
\]
---
[Diagram 2]
- Given angles: \( 130^\circ \), \( 60^\circ \), \( 110^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
130^\circ + 60^\circ + 110^\circ + x = 360^\circ
\]
5. Simplify:
\[
300^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 300^\circ = 60^\circ
\]
Answer for Problem 2:
\[
\boxed{60^\circ}
\]
---
[Diagram 3]
- Given angles: \( 110^\circ \), \( 120^\circ \), \( 50^\circ \), \( 70^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
110^\circ + 120^\circ + 50^\circ + x = 360^\circ
\]
5. Simplify:
\[
280^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 280^\circ = 80^\circ
\]
Answer for Problem 3:
\[
\boxed{80^\circ}
\]
---
[Diagram 4]
- Given angles: \( 120^\circ \), \( 30^\circ \), \( 70^\circ \)
#### Solution:
1. The shape involves a triangle and an exterior angle.
2. The exterior angle \( 120^\circ \) is equal to the sum of the two non-adjacent interior angles of the triangle.
3. Let the unknown angle be \( x \).
4. The equation for the exterior angle is:
\[
120^\circ = x + 30^\circ
\]
5. Solve for \( x \):
\[
x = 120^\circ - 30^\circ = 90^\circ
\]
Answer for Problem 4:
\[
\boxed{90^\circ}
\]
---
[Diagram 5]
- Given angles: \( 75^\circ \), \( 120^\circ \), \( 75^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
75^\circ + 120^\circ + 75^\circ + x = 360^\circ
\]
5. Simplify:
\[
270^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 270^\circ = 90^\circ
\]
Answer for Problem 5:
\[
\boxed{90^\circ}
\]
---
[Diagram 6]
- Given angles: \( 25^\circ \), \( 45^\circ \), \( 95^\circ \)
#### Solution:
1. The shape involves a triangle.
2. The sum of the interior angles of a triangle is \( 180^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
25^\circ + 45^\circ + x = 180^\circ
\]
5. Simplify:
\[
70^\circ + x = 180^\circ
\]
6. Solve for \( x \):
\[
x = 180^\circ - 70^\circ = 110^\circ
\]
Answer for Problem 6:
\[
\boxed{110^\circ}
\]
---
[Diagram 7]
- Given angles: \( 120^\circ \), \( 120^\circ \), \( 60^\circ \)
#### Solution:
1. The shape involves a triangle.
2. The sum of the interior angles of a triangle is \( 180^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
120^\circ + 60^\circ + x = 180^\circ
\]
5. Simplify:
\[
180^\circ + x = 180^\circ
\]
6. Solve for \( x \):
\[
x = 180^\circ - 180^\circ = 0^\circ
\]
However, this result seems incorrect because it implies a degenerate triangle. Re-evaluating the diagram, we notice that the \( 120^\circ \) angles are exterior angles to the triangle. The correct approach is to use the fact that the sum of the interior angles of the triangle is \( 180^\circ \).
Revised Solution:
1. The exterior angle \( 120^\circ \) corresponds to an interior angle of \( 180^\circ - 120^\circ = 60^\circ \).
2. The triangle has angles \( 60^\circ \), \( 60^\circ \), and \( x \).
3. The equation for the sum of the angles is:
\[
60^\circ + 60^\circ + x = 180^\circ
\]
4. Simplify:
\[
120^\circ + x = 180^\circ
\]
5. Solve for \( x \):
\[
x = 180^\circ - 120^\circ = 60^\circ
\]
Answer for Problem 7:
\[
\boxed{60^\circ}
\]
---
\[
\boxed{130^\circ, 60^\circ, 80^\circ, 90^\circ, 90^\circ, 110^\circ, 60^\circ}
\]
---
Problem 1:
[Diagram 1]
- Given angles: \( 80^\circ \), \( 60^\circ \)
- The shape is a quadrilateral with one right angle (\( 90^\circ \)).
#### Solution:
1. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
2. Let the unknown angle be \( x \).
3. The equation for the sum of the angles is:
\[
80^\circ + 60^\circ + 90^\circ + x = 360^\circ
\]
4. Simplify:
\[
230^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 230^\circ = 130^\circ
\]
Answer for Problem 1:
\[
\boxed{130^\circ}
\]
---
Problem 2:
[Diagram 2]
- Given angles: \( 130^\circ \), \( 60^\circ \), \( 110^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
130^\circ + 60^\circ + 110^\circ + x = 360^\circ
\]
5. Simplify:
\[
300^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 300^\circ = 60^\circ
\]
Answer for Problem 2:
\[
\boxed{60^\circ}
\]
---
Problem 3:
[Diagram 3]
- Given angles: \( 110^\circ \), \( 120^\circ \), \( 50^\circ \), \( 70^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
110^\circ + 120^\circ + 50^\circ + x = 360^\circ
\]
5. Simplify:
\[
280^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 280^\circ = 80^\circ
\]
Answer for Problem 3:
\[
\boxed{80^\circ}
\]
---
Problem 4:
[Diagram 4]
- Given angles: \( 120^\circ \), \( 30^\circ \), \( 70^\circ \)
#### Solution:
1. The shape involves a triangle and an exterior angle.
2. The exterior angle \( 120^\circ \) is equal to the sum of the two non-adjacent interior angles of the triangle.
3. Let the unknown angle be \( x \).
4. The equation for the exterior angle is:
\[
120^\circ = x + 30^\circ
\]
5. Solve for \( x \):
\[
x = 120^\circ - 30^\circ = 90^\circ
\]
Answer for Problem 4:
\[
\boxed{90^\circ}
\]
---
Problem 5:
[Diagram 5]
- Given angles: \( 75^\circ \), \( 120^\circ \), \( 75^\circ \)
#### Solution:
1. The shape is a quadrilateral.
2. The sum of the interior angles of a quadrilateral is \( 360^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
75^\circ + 120^\circ + 75^\circ + x = 360^\circ
\]
5. Simplify:
\[
270^\circ + x = 360^\circ
\]
6. Solve for \( x \):
\[
x = 360^\circ - 270^\circ = 90^\circ
\]
Answer for Problem 5:
\[
\boxed{90^\circ}
\]
---
Problem 6:
[Diagram 6]
- Given angles: \( 25^\circ \), \( 45^\circ \), \( 95^\circ \)
#### Solution:
1. The shape involves a triangle.
2. The sum of the interior angles of a triangle is \( 180^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
25^\circ + 45^\circ + x = 180^\circ
\]
5. Simplify:
\[
70^\circ + x = 180^\circ
\]
6. Solve for \( x \):
\[
x = 180^\circ - 70^\circ = 110^\circ
\]
Answer for Problem 6:
\[
\boxed{110^\circ}
\]
---
Problem 7:
[Diagram 7]
- Given angles: \( 120^\circ \), \( 120^\circ \), \( 60^\circ \)
#### Solution:
1. The shape involves a triangle.
2. The sum of the interior angles of a triangle is \( 180^\circ \).
3. Let the unknown angle be \( x \).
4. The equation for the sum of the angles is:
\[
120^\circ + 60^\circ + x = 180^\circ
\]
5. Simplify:
\[
180^\circ + x = 180^\circ
\]
6. Solve for \( x \):
\[
x = 180^\circ - 180^\circ = 0^\circ
\]
However, this result seems incorrect because it implies a degenerate triangle. Re-evaluating the diagram, we notice that the \( 120^\circ \) angles are exterior angles to the triangle. The correct approach is to use the fact that the sum of the interior angles of the triangle is \( 180^\circ \).
Revised Solution:
1. The exterior angle \( 120^\circ \) corresponds to an interior angle of \( 180^\circ - 120^\circ = 60^\circ \).
2. The triangle has angles \( 60^\circ \), \( 60^\circ \), and \( x \).
3. The equation for the sum of the angles is:
\[
60^\circ + 60^\circ + x = 180^\circ
\]
4. Simplify:
\[
120^\circ + x = 180^\circ
\]
5. Solve for \( x \):
\[
x = 180^\circ - 120^\circ = 60^\circ
\]
Answer for Problem 7:
\[
\boxed{60^\circ}
\]
---
Final Answers:
\[
\boxed{130^\circ, 60^\circ, 80^\circ, 90^\circ, 90^\circ, 110^\circ, 60^\circ}
\]
Parent Tip: Review the logic above to help your child master the concept of interior angles of a polygon worksheet.
