Lines Parallel to the Same Line

Multiple parallel lines cut by a tranverse

If two lines are parallel to the same line, are the two lines parallel to each other? In the above figure:

line m || line l and

line n || line l.

If we draw a transversal line t for the parallel lines, we already know that:

∠ 1 = ∠ and ∠ 2 = ∠ (Corresponding angles axiom)

So, ∠ 1 = ∠

Since, ∠ 1 and ∠ 3 are the corresponding angles for the parallel lines m and n, we can therefore, say that:

Line m || Line n (Converse of corresponding angles axiom)

Thus, we can state this in the form of the following theorem:

Given the same reasoning, this property can be extended to more than twofourthree lines. Now, let's use this property to solve problems.

Example 4: In the figure, if PQ || RS, ∠ MXQ = 135° and ∠ MYR = 40°, ∠ XMY will be equal to:

First construct a line AB parallel to line PQ and passing through point M.

Now, AB || PQRS and PQ || RSAB and as a result

AB ||

Instructions

Example 5: If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

Let's take the following: PQ and RS are two lines with a transversal AD as shown. The corresponding angles ∠ ABQ and ∠ BCS have angle bisectors BE and CG, respectively which are parallel to each other.

Instructions

Example 6: In the figure, AB || CD, CD || EF and EA ⊥ AB. If ∠ BEF = 55°, find the values of x, y and z.

Instructions