Given: <2 and <5 are supplementary Prove: l || m
To prove that lines l and m are parallel, we need to show that the alternate interior angles formed by the lines are congruent.
Given that <2 and <5 are supplementary:
<2 + <5 = 180°
Suppose that line l intersects line m at point A. Then, the angles formed at point A are the alternate interior angles.
We can define the alternate interior angles as <2 and <6 (opposite angles) and <5 and <7 (opposite angles).
Since <2 and <5 are supplementary, we know that <2 + <5 = 180°.
Also, since <6 and <7 are opposite angles formed by parallel lines, we know that <6 = <7.
Now, we can substitute the values of <6 and <7 into the equation:
<2 + <5 = <6 + <7
Since <6 = <7:
<2 + <5 = <6 + <6
Combining like terms:
<2 + <5 = 2<6
Since <2 + <5 = 180°:
180° = 2<6
Dividing both sides by 2:
90° = <6
This shows that angle <6 is congruent to 90°.
Therefore, we have shown that the alternate interior angles formed by lines l and m, namely <2 and <6, are congruent. This is sufficient to prove that lines l and m are parallel.
Hence, it has been proven that l || m.