Let's work through the questions in the image:

1.2 Complete the following statements.

1.2.1 If two lines are perpendicular to the same line, they are parallel.

• Explanation: If two lines are both perpendicular to a third line, they must be parallel to each other, as they share the same orientation relative to the third line.

1.2.2 If two lines are parallel to the same line, they are parallel to each other.

• Explanation: This follows from the transitive property of parallel lines, meaning that if two lines are parallel to the same third line, they will also be parallel to each other.

1.3 Name four pairs of angles which you could use to prove that $AB \parallel EF$.

Looking at the diagram provided:

• Lines $AB$ and $EF$ are suspected to be parallel.
• To prove they are parallel, we can use various angle pair relationships formed by the transversal lines intersecting $AB$ and $EF$.

Here are four pairs of angles to prove parallelism based on the diagram:

1. Corresponding angles: $\angle 1$ (near $A$) and $\angle 1$ (near $E$)
2. Alternate interior angles: $\angle 2$ (near $D$) and $\angle 2$ (near $F$)
3. Co-interior (Consecutive interior) angles: $\angle 1$ (near $D$) and $\angle 3$ (near $F$)
4. Alternate exterior angles: $\angle 3$ (near $A$) and $\angle 3$ (near $E$)

These angle relationships can be used to establish that $AB \parallel EF$ using properties of parallel lines and transversals.

Would you like further details or explanations on any of these concepts?

Here are five related questions to expand your understanding:

1. What is the converse of the corresponding angle postulate?
2. How do you prove that two lines are perpendicular?
3. Can two skew lines ever be parallel? Why or why not?
4. How can the angles formed by a transversal be used to determine if two lines are perpendicular?
5. What is the definition of co-interior angles and how do they help prove parallelism?

Tip: Always remember that parallel lines never meet, no matter how far they are extended! This can help visualize angle properties involving transversals.