In the figure , If AB||CE, then find the values of p , q and r.

Answer

To find the values of \( p \), \( q \), and \( r \) given that \( AB \parallel CE \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We know that \( AB \parallel CE \).
- \( AC \) acts as a transversal line intersecting the parallel lines.

2. **Use Alternate Angles**:
- Since \( AB \parallel CE \) and \( AC \) is a transversal, we can use the property of alternate interior angles.
- Therefore, \( q = 32^\circ \) (as given in the problem).

**Hint**: Recall that alternate interior angles are equal when two lines are parallel and cut by a transversal.

3. **Apply the Triangle Angle Sum Property**:
- In triangle \( ABC \), the sum of the angles is \( 180^\circ \).
- We know one angle is \( 90^\circ \) (let's assume \( \angle ABC = 90^\circ \)) and \( q = 32^\circ \).
- Therefore, we can write the equation:
\[ p + 90^\circ + q = 180^\circ \]
Substituting \( q \):
\[ p + 90^\circ + 32^\circ = 180^\circ \]

**Hint**: Remember that the sum of angles in a triangle is always \( 180^\circ \).

4. **Solve for \( p \)**:
- Rearranging the equation gives:
\[ p + 122^\circ = 180^\circ \]
\[ p = 180^\circ - 122^\circ = 58^\circ \]

**Hint**: Isolate \( p \) by subtracting the known angle sum from \( 180^\circ \).

5. **Determine \( r \)**:
- The angle \( r \) is an exterior angle to triangle \( ACE \).
- The exterior angle theorem states that the exterior angle is equal to the sum of the two opposite interior angles.
- Therefore:
\[ r = \theta + q \]
Since \( \theta = p = 58^\circ \) and \( q = 32^\circ \):
\[ r = 58^\circ + 32^\circ = 90^\circ \]

**Hint**: Recall that the exterior angle is equal to the sum of the two non-adjacent interior angles.

### Final Values:
- \( p = 58^\circ \)
- \( q = 32^\circ \)
- \( r = 90^\circ \)