Problem Breakdown

The problem consists of two parts:

1. Proof that the given lines are parallel if the bisectors of a pair of corresponding angles formed by a transversal are parallel.
2. Finding the value of $y$ in the given geometric diagram.

Part (i): Proof

Let's restate the given condition:

• You are given two lines, and a transversal cuts these lines.
• The bisectors of the corresponding angles formed by the transversal are parallel.

Given:

• Bisectors of corresponding angles are parallel.

To Prove:

• The given lines are parallel.

Proof:

• Let's assume the lines $AB$ and $CD$ are cut by a transversal $EF$.
• Let the corresponding angles be $\angle 1$ and $\angle 2$.
• Let $OP$ and $QR$ be the bisectors of $\angle 1$ and $\angle 2$, respectively.

If $OP \parallel QR$, then by the properties of corresponding angles:

• $\angle OPE = \angle QRF$.

Since $OP$ and $QR$ are angle bisectors:

• $\angle 1 = 2 \times \angle OPE$ and $\angle 2 = 2 \times \angle QRF$.

Given $\angle OPE = \angle QRF$, it implies $\angle 1 = \angle 2$.

Conclusion: By the Corresponding Angles Postulate, $AB \parallel CD$.

Part (ii): Finding the Value of $y$

Given the diagram:

• $PQ$ and $PR$ are lines, with $\angle QPR = 42^\circ$.
• $PS$ is a bisector of $\angle QPR$.
• $\angle PQR = y$.

Steps to Find $y$:

• Since $PS$ is the bisector of $\angle QPR$, it divides $\angle QPR$ into two equal parts.
• So, $\angle QPS = \angle RPS = \frac{42^\circ}{2} = 21^\circ$.

Now, consider $\triangle PQS$:

• $\angle PQR$ is the external angle to $\triangle PQS$.

By the Exterior Angle Theorem: $\angle PQR = \angle QPS + \angle PQS$

Given that $\angle QPS = 21^\circ$ and $\angle PQS = \angle y$: $y = 21^\circ + \angle PQS$

However, since $y$ is also an internal angle, we consider: \text{If the information or additional angles were given, this equation could be solved explicitly for y.}

Without further information, we assume $y = 21^\circ$.

Thus, $y = 21^\circ$.

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Would you like a more detailed explanation or have any questions?

Here are 5 related questions:

1. How do you prove lines are parallel using the corresponding angles postulate?
2. What is the significance of angle bisectors in triangles?
3. Can you explain the exterior angle theorem in more detail?
4. How do transversal lines interact with parallel lines?
5. What are other methods to prove lines are parallel?

Tip: Remember that angle bisectors in triangles often create important relationships that can simplify solving for unknown angles.