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We are tasked with finding the values of angles \(x\) and \(y\) in the trapezium shown, where one of the angles is labeled as \(155^\circ\).

### Key properties we can use:
1. **Sum of interior angles of a quadrilateral**: The sum of all interior angles in any quadrilateral is always \(360^\circ\).
2. **Adjacent angles in a trapezium**: Since two sides of a trapezium are parallel, adjacent angles formed on the same side of a transversal are supplementary (i.e., they add up to \(180^\circ\)).

### Solving for angle \(x\):
- The angle adjacent to \(x\) (on the same straight line) is given as \(155^\circ\).
- Since these two angles are on a straight line, they must sum to \(180^\circ\) (they are supplementary).
\[
x + 155^\circ = 180^\circ
\]
Solving for \(x\):
\[
x = 180^\circ - 155^\circ = 25^\circ
\]

### Solving for angle \(y\):
- Now we know \(x = 25^\circ\), and the sum of all interior angles in the trapezium must be \(360^\circ\).
- We can sum all known angles, including the two angles at the bottom left and right (which are right angles because the shape is a trapezium):
\[
155^\circ + 90^\circ + 90^\circ + y = 360^\circ
\]
Simplifying:
\[
335^\circ + y = 360^\circ
\]
Solving for \(y\):
\[
y = 360^\circ - 335^\circ = 25^\circ
\]

Thus, \(x = 25^\circ\) and \(y = 25^\circ\).

Would you like any further clarifications or detailed steps?

Here are 5 related questions to expand your understanding:
1. How do we know that the sum of angles in a quadrilateral is always \(360^\circ\)?
2. Can a trapezium have non-right angles as its base angles?
3. What are the properties of parallel lines and how do they affect angle calculations in quadrilaterals?
4. How would the angle calculations change if the trapezium was not isosceles?
5. How can we use the exterior angle theorem in a trapezium?

**Tip:** When dealing with quadrilaterals, always remember to use the property that their interior angles sum to \(360^\circ\), which often simplifies solving for unknown angles.

A trapezium is shown. Work out the sizes of angles x and y. Give your answers in degrees (°).

Learn how to calculate angles x and y in a trapezium by using the properties of supplementary angles and the sum of interior angles in quadrilaterals. This step-by-step solution helps in understanding key geometry concepts, suitable for students in Grades 8-10.

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