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.