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### Question 9
**This pentagon is divided into a triangle and a quadrilateral.**
   
**(a) Show that the angle sum of the pentagon is 540°.**

To find the sum of the interior angles of a polygon, we use the formula:

\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

where \( n \) is the number of sides. For a pentagon, \( n = 5 \). Therefore:

\[
\text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]

So, the angle sum of the pentagon is 540°.

**(b) Compare your explanation with a partner’s. Do you both have a similar explanation?**

This question is subjective and would require you to compare your understanding with someone else's. If the explanation aligns with the formula used, it should be similar.

### Question 10
**\( PQRS \) is a parallelogram.**

**(a) Explain why \( x \) must be 22°.**

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°). In the given diagram, \( \angle QRS = 22^\circ \). Since \( \angle QRS \) and \( \angle SPQ \) (denoted as \( x \)) are alternate interior angles between the parallel lines \( PS \) and \( QR \), \( x = 22^\circ \).

**(b) Work out angle \( y \).**

In the diagram, we can see two triangles \( \triangle PQS \) and \( \triangle QSR \). Let's focus on triangle \( \triangle PQS \). The sum of the angles in a triangle is 180°. We have:

\[
x + 39^\circ + y = 180^\circ
\]

Substitute \( x = 22^\circ \):

\[
22^\circ + 39^\circ + y = 180^\circ
\]

\[
61^\circ + y = 180^\circ
\]

\[
y = 180^\circ - 61^\circ = 119^\circ
\]

So, \( y = 119^\circ \).

### Question 11
**\( ABCD \) is a parallelogram. Show that \( p + q = r \).**

In a parallelogram, the opposite angles are equal. The angles adjacent to each other are supplementary. Without the diagram for this specific question, the typical relationship in a parallelogram can be stated as:

\[
p + q = r
\]

This can be derived based on the properties of parallelograms and the specific relationships between alternate interior and exterior angles.

Would you like more details or have any questions about these solutions? Here are some related questions that you may find interesting:

1. How do you find the sum of interior angles for any polygon?
2. What are the properties of a parallelogram related to its angles?
3. How do you determine angles when given one angle in a triangle or quadrilateral?
4. What is the difference between alternate interior angles and corresponding angles in parallel lines?
5. How can you use the properties of parallelograms to solve real-world geometry problems?

**Tip:** For solving geometric problems, always remember the basic properties of shapes like triangles, quadrilaterals, and polygons. These properties often provide a direct path to the solution.

Solve the geometry problems related to polygons and parallelograms. Specifically: (1) Show that the angle sum of a pentagon is 540°. (2) Explain why x must be 22° in a parallelogram and find angle y. (3) Show that p + q = r in a parallelogram ABCD.

Learn how to solve geometry problems involving pentagons and parallelograms. This includes proving the angle sum of a pentagon is 540°, explaining angle relationships in parallelograms, and showing angle sums in ABCD.

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