"Equidistant" means the same distance (from the prefix "equi-", which means equal, and "distance"). Parallel lines are equidistant from each other. This means that every point on one line is always … [Read more...] about Parallel Lines are Equidistant
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.
In today's lesson, we will show a simple method for proving the Consecutive Interior Angles Converse Theorem. The Consecutive Interior Angles Theorem states that the consecutive interior angles on … [Read more...] about Consecutive Interior Angles Converse Theorem
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
In today's lesson, we will show that two lines parallel to a third line are parallel to each other. In geometry, parallel lines are lines that do not meet. They do not intersect or touch each other … [Read more...] about Two Lines Parallel to a Third are Parallel to Each Other
In today's geometry lesson, we'll prove the converse of the Alternate Interior Angles Theorem. We have shown that when two parallel lines are intersected by a transversal line, the interior … [Read more...] about Converse Alternate Interior Angles Theorem
In today's lesson, we will prove the Converse of the Corresponding Angles Theorem. We will use the very useful technique of proof by contradiction. We had earlier said axiomatically, with no proof, … [Read more...] about Converse of the Corresponding Angles Theorem
In today's lesson, we will prove the alternate interior theorem, stating that interior alternating angles and exterior alternating angles between parallel lines are congruent. Prove: Interior … [Read more...] about Alternate Interior Angles Theorem
The Theorem The Consecutive Interior Angles Theorem states that the two interior angles formed by a transversal line intersecting two parallel lines are supplementary (i.e: they sum up to … [Read more...] about Consecutive Interior Angles Theorem