https://file.aiquickdraw.com/mathbot/file/1729004342459miu1m2m7.jpg

Let's solve the problems one by one from the image:

### 3(a) - **ABCD is a rectangle**
Given angles:
- ∠A = 63°
- ∠B = 27°

For a rectangle, the opposite angles of a diagonal are equal.

To find \( x \) and \( y \):
1. **For \( y \)**: Use the fact that the sum of angles around point B is 180° (angle on a straight line):
   \[
   y = 180^\circ - 63^\circ - 27^\circ = 90^\circ
   \]
2. **For \( x \)**: In a rectangle, the diagonals bisect the angles. So, the angle near vertex D is 90° (half of the 180° angle on the diagonal). Therefore, \( x = 90^\circ \).

### 3(b) - **ABCD is a parallelogram**
Given angles:
- ∠D = \( x \)
- ∠B = 140°

In a parallelogram:
- Opposite angles are equal.
- Adjacent angles are supplementary (sum to 180°).

1. **For \( x \)**: The angle at D is supplementary to the angle at B:
   \[
   x = 180^\circ - 140^\circ = 40^\circ
   \]
2. **For \( y \)**: Since opposite angles are equal, \( y = 140^\circ \).

### 3(c) - **PQRS is a parallelogram**
Given angles:
- ∠P = 31°
- ∠Q = 51°

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary.

1. **For \( x \)**: Angle \( x \) is supplementary to ∠P (since PQRS is a parallelogram):
   \[
   x = 180^\circ - 31^\circ = 149^\circ
   \]
2. **For \( y \)**: Angle \( y \) is supplementary to ∠Q:
   \[
   y = 180^\circ - 51^\circ = 129^\circ
   \]

### 3(d) - **ABCD is a rhombus**
Given angle:
- ∠C = 62°

In a rhombus, opposite angles are equal, and adjacent angles are supplementary.

1. **For \( x \)**: In a rhombus, the diagonals bisect the angles. Thus, \( x \) is half of 62°:
   \[
   x = \frac{62^\circ}{2} = 31^\circ
   \]
2. **For \( y \)**: The diagonals of a rhombus are perpendicular (90°):
   \[
   y = 90^\circ
   \]

Would you like further details on any of these steps or a deeper explanation? Here are 5 questions to consider:

1. Why are diagonals of a rectangle important in solving angle problems?
2. What are the angle properties specific to parallelograms that help in these calculations?
3. How do the properties of diagonals in a rhombus assist in determining angles?
4. How would the solution change if the parallelogram in (b) was a rectangle?
5. What is the significance of bisecting angles in rhombuses and rectangles?

**Tip:** Always verify angle properties in quadrilaterals—opposite angles in parallelograms are equal, and diagonals often have specific properties that simplify calculations.

Find the value of x and y in the following diagrams: (a) ABCD is a rectangle, (b) ABCD is a parallelogram, (c) PQRS is a parallelogram, (d) ABCD is a rhombus.

Learn how to find the values of x and y in various quadrilaterals, including rectangles, parallelograms, and rhombuses. This guide explains key geometry concepts, including angle properties, diagonals, and supplementary angles. Suitable for students in Grades 7-10.

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