The Consecutive Interior Angles Theorem states that the consecutive interior angles on the same side of a transversal line intersecting two parallel lines are supplementary (That is, their sum adds up … [Read more...] about Consecutive Interior Angles Converse Theorem
What Are Parallel Lines in Geometry?
As we wrote earlier, an axiom of Euclidean geometry is that for every straight line and every point not on that line, there is one straight line that passes through that point, and never intersects the first line. Such lines are called Parallel lines.
Parallel lines are always the same distance from each other. We use this notation - ||- to describe two parallel line segments, for example: AB||CD
When a third line intersects two parallel lines, as above, it creates 4 angles with each one of the lines. The angles in the same position, for example, above the line and to the right, like 1 and 5, above, are called corresponding angles. The two pairs of angles that are on the “inside” of the parallel lines (4 & 5, and 3 & 6) are called “interior angles”.
The two pairs of angles that are on the “outside ” the parallel lines (1 & 8, and 2 & 7) are called “exterior angles”.
The line crossing and intersecting the parallel lines is called the transversal line. Another way of stating the parallel line axiom (”for every line and every point not on that line, there is one straight line that passes through that point, and never intersects the first line“) is that a transversal line intersects parallel lines creating corresponding angles that are congruent. This has no proof- it is another way of stating the axiom.
We will now prove several theorems about the angles formed by the intersection of the transversal line and the two parallel lines. You can click on each one of the proofs below. Once you review them all you will find that we have shown that when two parallel lines are intersected by a transversal line, the following angles are all the same:
And also that the following angles are all the same:
Now that we've explained the basic concept of parallel lines in geometry, let's scroll down to work on specific geometry problems relating to this topic.
The three parallel lines theorem is another theorem that provides a ratio between line segments created by a transversal of parallel lines, similar to the Intercept Theorem.It states that if three … [Read more...] about Three Parallel Lines Theorem
The Intercept theorem provides the ratios between the line segments created when two parallel lines are intercepted by two intersecting lines. It is sometimes called "Thales' Theorem" (not to be … [Read more...] about Intercept Theorem
One way to show that two lines are parallel to each other is to find two corresponding angles created by a transversal line that are congruent, as we have proven here.We can use this theorem to … [Read more...] about Two Lines that are Parallel to a Third Line are Parallel to Each Other
We have shown that when two parallel lines are intersected by a transversal line, the interior alternating angles and exterior alternating angles are congruent (that is, they have the same measure of … [Read more...] about Converse Alternate Interior Angles Theorem
We had earlier said axiomatically, with no proof, that if two lines are parallel, the corresponding angles created by a transversal line are congruent. Now let’s prove an important converse theorem: … [Read more...] about Converse of the Corresponding Angles Theorem
Prove:Interior alternating angles and exterior alternating angles are congruent (that is, they have the same measure of the angle.)How to approach this problem?So how do we go about … [Read more...] about Alternate Interior Angles Theorem
The TheoremThe two interior angles formed by a transversal line intersecting two parallel lines are supplementary (i.e: sum up to 180°).The problemAB||CD, prove m∠5 + m∠4 = 180°So … [Read more...] about Consecutive Interior Angles Theorem