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The diagram shows quadrilateral ABCD and two of its exterior angles, x°

(a) Find the value of x if angle BAD = 66°.

(b) Find the largest exterior angle of the quadrilateral.

We are given a quadrilateral \( ABCD \) with some of its interior and exterior angles labeled. Let's go through each part of the question.

### (a) Find the value of \( x \) if \( \angle BAD = 66^\circ \).

We know the following:
- \( \angle A = 66^\circ \)
- \( \angle B = x^\circ \) (exterior angle at \( B \))
- \( \angle C = 3x^\circ \) (exterior angle at \( C \))
- \( \angle D = 2x^\circ \) (exterior angle at \( D \))

Since these angles are exterior angles of the quadrilateral, the sum of all exterior angles of any polygon is always \( 360^\circ \). This gives us the equation:

\[
x^\circ + 3x^\circ + 2x^\circ + 66^\circ = 360^\circ
\]

Simplifying:

\[
6x^\circ + 66^\circ = 360^\circ
\]

Now subtract \( 66^\circ \) from both sides:

\[
6x^\circ = 294^\circ
\]

Dividing by 6:

\[
x = 49^\circ
\]

Thus, the value of \( x \) is \( 49^\circ \).

### (b) Find the largest exterior angle of the quadrilateral.

Now that we know \( x = 49^\circ \), we can calculate the exterior angles:

- Exterior angle at \( B = x = 49^\circ \)
- Exterior angle at \( C = 3x = 3 \times 49 = 147^\circ \)
- Exterior angle at \( D = 2x = 2 \times 49 = 98^\circ \)
- Exterior angle at \( A = 66^\circ \)

Clearly, the largest exterior angle is \( 147^\circ \) at vertex \( C \).

### Summary:
- (a) The value of \( x \) is \( 49^\circ \).
- (b) The largest exterior angle is \( 147^\circ \).

Would you like more details on any part of this? Here are some related questions you might find interesting:

1. What is the sum of the interior angles of quadrilateral \( ABCD \)?
2. How do we derive the exterior angle theorem for polygons?
3. If the quadrilateral were a parallelogram, would the exterior angles change?
4. What is the relationship between the interior and exterior angles of a polygon?
5. Can we generalize the process to any polygon, not just quadrilaterals?

**Tip:** Always remember that the sum of the exterior angles of any polygon, regardless of the number of sides, is always \( 360^\circ \).

The diagram shows quadrilateral ABCD and two of its exterior angles, x°. (a) Find the value of x if angle BAD = 66°. (b) Find the largest exterior angle of the quadrilateral.

This problem involves finding the value of x based on exterior angles in quadrilateral ABCD. It demonstrates the exterior angle sum rule, showing that the largest exterior angle is 147° after solving for x. Suitable for students in Grades 8-10.

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