Math worksheet for finding missing angles in triangles and a regular pentagon.

Problem Overview:

The task involves solving for missing angles in various triangles and a regular pentagon. The key concepts used are:

1. Sum of Angles in a Triangle: The sum of the interior angles of a triangle is always \(180^\circ\).
2. Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
3. Regular Polygon Properties: For a regular polygon with \(n\) sides, each interior angle is given by \(\frac{(n-2) \times 180^\circ}{n}\).

Solution:

#### Section A: Missing Angles in Triangles

1. Triangle 1:
- Given angles: \(100^\circ\) and \(30^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(a\):
\[
a = 180^\circ - 100^\circ - 30^\circ = 50^\circ
\]
- Answer: \(a = 50^\circ\).

2. Triangle 2:
- Given angles: \(50^\circ\) and \(45^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(b\):
\[
b = 180^\circ - 50^\circ - 45^\circ = 85^\circ
\]
- Answer: \(b = 85^\circ\).

3. Triangle 3:
- Given angles: \(60^\circ\) and one right angle (\(90^\circ\)).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(c\):
\[
c = 180^\circ - 90^\circ - 60^\circ = 30^\circ
\]
- Answer: \(c = 30^\circ\).

4. Triangle 4:
- Given angles: \(34^\circ\) and one right angle (\(90^\circ\)).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(d\):
\[
d = 180^\circ - 90^\circ - 34^\circ = 56^\circ
\]
- Answer: \(d = 56^\circ\).

5. Triangle 5:
- Given angles: \(26^\circ\) and \(6^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(e\):
\[
e = 180^\circ - 26^\circ - 6^\circ = 148^\circ
\]
- Answer: \(e = 148^\circ\).

6. Triangle 6:
- Given angles: One right angle (\(90^\circ\)) and another angle (\(30^\circ\)).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(f\):
\[
f = 180^\circ - 90^\circ - 30^\circ = 60^\circ
\]
- Answer: \(f = 60^\circ\).

#### Section B: Missing Angles in Triangles (with Exterior Angles)

1. Triangle 1:
- Given exterior angle: \(116^\circ\).
- Interior angle adjacent to the exterior angle: \(180^\circ - 116^\circ = 64^\circ\).
- Other given angle: \(41^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(a\):
\[
a = 180^\circ - 64^\circ - 41^\circ = 75^\circ
\]
- Answer: \(a = 75^\circ\).

2. Triangle 2:
- Given exterior angle: \(139^\circ\).
- Interior angle adjacent to the exterior angle: \(180^\circ - 139^\circ = 41^\circ\).
- Other given angle: \(49^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(b\):
\[
b = 180^\circ - 41^\circ - 49^\circ = 90^\circ
\]
- Answer: \(b = 90^\circ\).

3. Triangle 3:
- Given angles: \(93^\circ\) and \(40^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(c\):
\[
c = 180^\circ - 93^\circ - 40^\circ = 47^\circ
\]
- Answer: \(c = 47^\circ\).

4. Triangle 4:
- Given angles: \(62^\circ\) and \(38^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(d\):
\[
d = 180^\circ - 62^\circ - 38^\circ = 80^\circ
\]
- Answer: \(d = 80^\circ\).

5. Triangle 5:
- Given angles: \(56^\circ\) and \(10^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(e\):
\[
e = 180^\circ - 56^\circ - 10^\circ = 114^\circ
\]
- Answer: \(e = 114^\circ\).

6. Triangle 6:
- Given angles: \(53^\circ\) and \(147^\circ\) (exterior angle).
- Interior angle adjacent to the exterior angle: \(180^\circ - 147^\circ = 33^\circ\).
- Sum of angles in a triangle: \(180^\circ\).
- Missing angle \(f\):
\[
f = 180^\circ - 53^\circ - 33^\circ = 94^\circ
\]
- Answer: \(f = 94^\circ\).

#### Extension: Regular Pentagon

- A regular pentagon has 5 sides, so each interior angle is:
\[
\text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
\]
- The given diagram shows a regular pentagon with one interior angle split into two parts: \(72^\circ\) and \(x\).
- Since the total interior angle is \(108^\circ\):
\[
72^\circ + x = 108^\circ
\]
\[
x = 108^\circ - 72^\circ = 36^\circ
\]
- Answer: \(x = 36^\circ\).

Final Answers:

\[
\boxed{50^\circ, 85^\circ, 30^\circ, 56^\circ, 148^\circ, 60^\circ, 75^\circ, 90^\circ, 47^\circ, 80^\circ, 114^\circ, 94^\circ, 36^\circ}
\]