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Let's break down and solve the problems from the image you uploaded:

### Part a) (i)
This is a triangle, and we are asked to calculate angles \(a\) and \(b\).
- We know that the sum of the angles in a triangle is always \(180^\circ\).
- Given that one of the angles is \(42^\circ\), and the triangle is isosceles (since two sides are marked as equal), the two unknown angles, \(a\) and \(b\), must be equal.

Therefore, we can calculate:
\[
a + b + 42^\circ = 180^\circ
\]
Since \(a = b\), we can substitute \(a\) for \(b\):
\[
2a + 42^\circ = 180^\circ
\]
\[
2a = 180^\circ - 42^\circ
\]
\[
2a = 138^\circ
\]
\[
a = b = \frac{138^\circ}{2} = 69^\circ
\]

So, \(a = 69^\circ\) and \(b = 69^\circ\).

### Part a) (ii)
This is a triangle with two angles \(c\) and \(d\) unknown, and we are given a \(130^\circ\) exterior angle and a \(45^\circ\) angle inside the triangle.
- We know that the exterior angle is equal to the sum of the two opposite interior angles:
\[
c + 45^\circ = 130^\circ
\]
Thus:
\[
c = 130^\circ - 45^\circ = 85^\circ
\]
- Now, to find \(d\), we use the fact that the sum of the angles in a triangle is \(180^\circ\):
\[
c + d + 45^\circ = 180^\circ
\]
Substituting the value of \(c\):
\[
85^\circ + d + 45^\circ = 180^\circ
\]
\[
d + 130^\circ = 180^\circ
\]
\[
d = 50^\circ
\]

So, \(c = 85^\circ\) and \(d = 50^\circ\).

### Part b)
In this irregular quadrilateral, we are asked to calculate the angles \(p\) and \(q\).

- We know the sum of the angles in any quadrilateral is \(360^\circ\).
- Given that the known angles are \(112^\circ\), \(80^\circ\), and \(95^\circ\), we can calculate \(p\) using:
\[
p + 112^\circ + 80^\circ + 95^\circ = 360^\circ
\]
\[
p + 287^\circ = 360^\circ
\]
\[
p = 360^\circ - 287^\circ = 73^\circ
\]

- Next, \(q\) is the supplementary angle to \(p\), as they are on a straight line:
\[
p + q = 180^\circ
\]
\[
73^\circ + q = 180^\circ
\]
\[
q = 180^\circ - 73^\circ = 107^\circ
\]

So, \(p = 73^\circ\) and \(q = 107^\circ\).

---

Would you like further details or have any questions? 

Here are some related questions for practice:
1. If one angle in a triangle is \(60^\circ\), and the triangle is equilateral, what are the other angles?
2. How do you calculate an exterior angle of a triangle?
3. What is the sum of the interior angles of a pentagon?
4. If two angles in a quadrilateral are \(90^\circ\) and \(85^\circ\), what is the sum of the remaining two angles?
5. In an isosceles triangle, if one of the equal angles is \(70^\circ\), what is the third angle?

**Tip**: Always remember that the sum of interior angles depends on the number of sides of the polygon—\( (n-2) \times 180^\circ \), where \(n\) is the number of sides.

Calculate the unknown angles in the following triangles and diagram.

Learn how to calculate unknown angles in triangles and irregular quadrilaterals using key geometric theorems. This problem involves the Triangle Angle Sum Theorem, Exterior Angle Theorem, and Supplementary Angles Theorem. Suitable for students in Grades 6-8.

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