The image shows a trapezium $ABCD$ where:

• $AD \parallel BC$
• $\angle A = 135^\circ$
• $\angle C = 42^\circ$
• $\angle D_2 = 118^\circ$

The task is to calculate the size of $\angle B_1$.

Approach:

1. In a trapezium, consecutive interior angles between the parallel sides are supplementary, meaning their sum is $180^\circ$.

2. For $\angle A$ and $\angle B_1$, we have: $\angle A + \angle B_1 = 180^\circ$ Substituting $\angle A = 135^\circ$: $135^\circ + \angle B_1 = 180^\circ$ Solving for $\angle B_1$: $\angle B_1 = 180^\circ - 135^\circ = 45^\circ$

Thus, the size of $\angle B_1$ is $45^\circ$.

Let me know if you'd like more details or have any other questions!

Here are 5 related questions to extend your understanding:

1. How would you calculate the sum of the interior angles of a quadrilateral?
2. What are the properties of a trapezium that make it distinct from other quadrilaterals?
3. How can we calculate the size of $\angle D_1$?
4. What is the relationship between the angles $\angle D_1$ and $\angle C$ in this trapezium?
5. How do the properties of parallel lines affect the angles in a trapezium?

Tip: When working with trapeziums, always remember that the angles between the parallel sides add up to $180^\circ$.