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Line WS is parallel to line KV. Line RT is perpendicular to line KV. Explain the relationship of line RT to line WS. Provide at least one reason to support your answer.

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If [tex]\displaystyle\mathsf{\overline{WS}}[/tex] ║ [tex]\displaystyle\mathsf{\overline{KV}}[/tex], and [tex]\displaystyle\mathsf{\overline{KV}}[/tex] ⊥ [tex]\displaystyle\mathsf{\overline{RT}}[/tex], then [tex]\displaystyle\mathsf{\overline{RT}}[/tex] ⊥ [tex]\displaystyle\mathsf{\overline{WS}}[/tex]

Step-by-step explanation:

We are given the diagram of parallel lines, [tex]\displaystyle\mathsf{\overline{WS}}[/tex] and [tex]\displaystyle\mathsf{\overline{KV}}[/tex], which are intersected by line [tex]\displaystyle\mathsf{\overline{KV}}[/tex]. The prompt requires us to determine the relationship between [tex]\displaystyle\mathsf{\overline{RT}}[/tex] and [tex]\displaystyle\mathsf{\overline{RS}}[/tex].


Parallel linesare non-intersecting lines that are coplanar and have the same slope.

Perpendicular lines: the intersection between two lines that form right angles. Perpendicular lines also have negative reciprocal slopes.

Transversal : a line that intersects at least two distinct coplanar lines in at least two or more definite points.


We will use the given diagram and the following theorems/postulates to prove the relationship between lines [tex]\displaystyle\mathsf{\overline{RT}}[/tex] and [tex]\displaystyle\mathsf{\overline{WS}}[/tex].

- The Parallel Postulate states that given line [tex]\displaystyle\mathsf{\overline{WS}}[/tex] and a point that is [tex]\displaystyle\sf\doubleunderline{\underline{\underline{not}}}[/tex] on line [tex]\displaystyle\mathsf{\overline{WS}}[/tex] , then there is exactly one line through that point that is parallel to line [tex]\displaystyle\mathsf{\overline{WS}}[/tex].
- The Perpendicular Postulate states that given line [tex]\displaystyle\mathsf{\overline{KV}}[/tex] and point R that is not on line [tex]\displaystyle\mathsf{\overline{KV}}[/tex], then there is a definite line through point R that is perpendicular to line [tex]\displaystyle\mathsf{\overline{KV}}[/tex].
- The Perpendicular Transversal Theorem states that given a transversal, [tex]\displaystyle\mathsf{\overline{RT}}[/tex], which is perpendicular to one of the parallel lines, [tex]\displaystyle\mathsf{\overline{WS}}[/tex], then it means that transversal [tex]\displaystyle\mathsf{\overline{RT}}[/tex] is also perpendicular to the other parallel line, [tex]\displaystyle\mathsf{\overline{KV}}[/tex].

Now that we have established the necessary postulates and theorem, we can state that:

- If [tex]\displaystyle\mathsf{\overline{WS}}[/tex] ║ [tex]\displaystyle\mathsf{\overline{KV}}[/tex], (Parallel Postulate) and [tex]\displaystyle\mathsf{\overline{KV}}[/tex] ⊥ [tex]\displaystyle\mathsf{\overline{RT}}[/tex], (Perpendicular Transversal Theorem ), then [tex]\displaystyle\mathsf{\overline{RT}}[/tex] ⊥ [tex]\displaystyle\mathsf{\overline{WS}}[/tex] (Perpendicular Postulate).



Parallel Lines

Perpendicular lines

Perpendicular Transversal Theorem


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