Xayma
Lacking creativity
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But just to prove it:
Consider two parallel lines
y=m<sub>1</sub>x+b
y=m<sub>2</sub>x+c
Because they are parallel m<sub>1</sub>=m<sub>2</sub>
If a perpendicular line is drawn from each of the parallel lines so that they intersect, the slope of both lines would be m<sub>3</sub> where m<sub>1</sub>=1/m<sub>3</sub> since m<sub>1</sub>m<sub>3</sub>=-1 for perpendicular lines.
ie the equation of the line from the first parallel line would be
y=m<sub>3</sub>x+d
y=m<sub>3</sub>x+e
As they intersect the angle between the lines can be found by using the formula tan @=|m<sub>a</sub>-m<sub>b</sub>/(1+m<sub>a</sub>m<sub>b</sub>)|
Since the gradients are equal tan @=0 therefore @=180 degrees. Therefore they form a straight line ie they are the same line.
OR
A simpler way.
Since a line is perpendicular the angle between it and the line it is projected from is 90 degrees.
Corresponding angles between a transversal and two parallel lines = 180 degrees.
Therefore the other angle is 90 degrees therefore it is perpendicular to the other line, similarly the other line is perpendicular to both hence they are the same line and hence form a straight angle on interception.
Consider two parallel lines
y=m<sub>1</sub>x+b
y=m<sub>2</sub>x+c
Because they are parallel m<sub>1</sub>=m<sub>2</sub>
If a perpendicular line is drawn from each of the parallel lines so that they intersect, the slope of both lines would be m<sub>3</sub> where m<sub>1</sub>=1/m<sub>3</sub> since m<sub>1</sub>m<sub>3</sub>=-1 for perpendicular lines.
ie the equation of the line from the first parallel line would be
y=m<sub>3</sub>x+d
y=m<sub>3</sub>x+e
As they intersect the angle between the lines can be found by using the formula tan @=|m<sub>a</sub>-m<sub>b</sub>/(1+m<sub>a</sub>m<sub>b</sub>)|
Since the gradients are equal tan @=0 therefore @=180 degrees. Therefore they form a straight line ie they are the same line.
OR
A simpler way.
Since a line is perpendicular the angle between it and the line it is projected from is 90 degrees.
Corresponding angles between a transversal and two parallel lines = 180 degrees.
Therefore the other angle is 90 degrees therefore it is perpendicular to the other line, similarly the other line is perpendicular to both hence they are the same line and hence form a straight angle on interception.
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