LINES AND ANGLES (LECTURE 4)

PARALLEL LINES AND A TRANSVERSAL.

Good morning boys........

Guidelines for the class:

1. Note down  the work in your register on a regular basis.

2. Please feel free to ask, if you have any doubt by dropping a message in the comment box.

3. Make a column on the right hand side, if you need to do any rough work

4. Write  today's date.

5. Write the chapter number and name.

6. Text in red has to be noted down as its your class work.

7. Text in green has to be read thoroughly and understood.

 Let us recall that a line which intersects two or more lines at distinct points is called a transversal.

Today's learning outcomes are:

• I will be able to comprehend the relation between the angles formed by two or more parallel lines are intersected by a transversal at distinct points.
• Corresponding angles axiom and its converse.
•   Lines parallel to the same line.

Now you see two parallel lines and a transversal..................   

Observe that four angles are formed at each of the intersecting point. C, D,X and W are called the interior angles, while A,B,Y and Z are called exterior angles.

 TRY THESE

ANSWER?

Problem 2

Answer?

Note: Interior angles on the same side of the transversal are also referred to as consecutive interior angles or allied angles or co-interior angles.

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What is corresponding angles axiom?

If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

This is to read and understand 👇

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, that if two lines are parallel, the corresponding angles created by a transversal line are congruent.

Now let’s prove an important converse theorem: that if 2 corresponding angles are congruent, then the lines are parallel.

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Prove:

If 2 corresponding angles formed by a transversal line intersecting two other lines are congruent, then the two lines are parallel.

To prove this, we will introduce the technique of “proof by contradiction,” which will be very useful down the road.

When we use a “proof by contradiction” method, we start by assuming that what we are trying to prove is NOT true, and we proceed from there until we reach an impossible conclusion – a contradiction.

This means that our original assumption is false (as it leads to this impossible situation), and thus what we were trying to prove must be true.

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Proof: Converse of the Corresponding Angles Axiom

So, let’s say we have two lines L1, and L2 intersected by a transversal line, L3, creating 2 corresponding angles, 1 & 2 which are congruent (∠1 ≅ ∠2, m∠1=∠2).

We want to prove the L1 and L2 are parallel, and we will do so by contradiction. Assume L1 is not parallel to L2. Then, according to the parallel line axiom we started with (“for every straight line and every point not on that line, there is one straight line that passes through that point which is parallel to the first line”), there is a different line than L2 that passes through the intersection point of L2 and L3 (point A in the drawing above), which IS parallel to L1.

Let’s draw that line, and call it P. Let’s also call the angle formed by the traversal line and this new line angle 3, and we see that if we add some other angle, call it angle 4, to it, it will be the same as angle 2.

Now, P||L1 so ∠1 ≅ ∠3, as corresponding angles formed by a transversal of parallel lines, and so m∠1=m∠3, from the definition of congruent angles. But we also know that m∠2=m∠3+m∠4, so now perform the substitution m∠2=m∠1+m∠4, and so m∠2 is larger, and not equal to m∠1!

This contradicts what was given,  that angles 1 and 2 are congruent! This contradiction means our assumption (“L1 is not parallel to L2”) is false, and so L1 must be parallel to L2.

LINES PARALLEL TO THE SAME LINE 

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  I know that it's bit heavy for you .... so I am not asking you to do any sum as home work, However, you are requested to sit and practice this class work.