Given: <2 and <5 are supplementary angles Prove: l || m
To prove that l || m, we need to show that the alternate interior angles are congruent.
Let's assume that l is not parallel to m and hence intersects with m at point P.
Since l is not parallel to m, there must be a transversal. Consider the transversal that intersects l at point Q and intersects m at point R.
Since <2 and <5 are supplementary angles, we have:
<2 + <5 = 180°
Now, let's consider the alternate interior angles formed by l and m:
<2 and <3: Since <2 and <5 are supplementary, we can write:
<2 = 180° - <5
Since <2 and <3 are corresponding angles (formed by l and the transversal), they must be congruent:
<3 = 180° - <5
<3 and <4: Since <3 and <5 are corresponding angles (formed by l and the transversal), they must be congruent:
<3 = <5
<5 and <4: Since <5 and <4 are vertical angles (formed by l and m at point P), they must be congruent:
<5 = <4
Therefore, we have:
<3 = <5 = <4
However, this implies that all three angles surrounding point P (formed by l and m) are congruent. This is a contradiction because the sum of interior angles of a triangle is 180°. Hence, our assumption that l is not parallel to m must be false.
Therefore, l || m.