Answer:
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].
Definitions:
Parallel lines are 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.
Explanation:
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).
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Keywords:
Parallel Lines
Perpendicular lines
Perpendicular Transversal Theorem
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