Question

Complete the proof below.
Given: h || k; j ⊥ h
Prove: j ⊥ k

Answer 1

The proof is complete. Since
`\( h \)` is parallel to
`\( k \)` and
`\( j \)` is perpendicular to
`\( h \),` it follows that
`\( j \)` is perpendicular to
`\( k \)` by the transitive property of parallel lines.

Step-by-step explanation:

The proof begins by establishing that
`\( h \)` is parallel to
`\( k \)` and
`\( j \)` is perpendicular to
`\( h \)`. The given information implies that
`\( j \)` forms a right angle with
`\( h \)`. Now, by the transitive property of parallel lines, if
`\( h \)` is parallel to
`\( k \)`, and
`\( j \)` forms a right angle with
`\( h \)`, then
`\( j \)` must also be perpendicular to
`\( k \)`. This logical progression completes the proof.

Understanding the properties of parallel and perpendicular lines is crucial in geometry proofs. In this case, the parallelism of
`\( h \)` and
`\( k \)` sets the stage for the transitive property, allowing us to deduce the perpendicular relationship between
`\( j \)` and
`\( k \)`. It's important to recognize and apply these geometric principles to construct a sound proof.

This proof demonstrates the use of logical reasoning and the transitive property, showcasing the interconnected nature of parallel and perpendicular relationships in geometry. It serves as a model for applying these principles in similar proofs involving parallel and perpendicular lines.