When two lines are cut by a transversal and are parallel corresponding angles are supplementary?

When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another word for opposite angles are vertical angles.

Vertical angles are always congruent, which means that they are equal.

Adjacent angles are angles that come out of the same vertex. Adjacent angles share a common ray and do not overlap.

The size of the angle xzy in the picture above is the sum of the angles A and B.

Two angles are said to be complementary when the sum of the two angles is 90°.

Two angles are said to be supplementary when the sum of the two angles is 180°.

If we have two parallel lines and have a third line that crosses them as in the ficture below - the crossing line is called a transversal

When a transversal intersects with two parallel lines eight angles are produced.

The eight angles will together form four pairs of corresponding angles. Angles 1 and 5 constitutes one of the pairs. Corresponding angles are congruent. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs e.g. 3 + 7, 4 + 8 and 2 + 6.

Angles that are in the area between the parallel lines like angle 2 and 8 above are called interior angles whereas the angles that are on the outside of the two parallel lines like 1 and 6 are called exterior angles.

Angles that are on the opposite sides of the transversal are called alternate angles e.g. 1 + 8.

All angles that are either exterior angles, interior angles, alternate angles or corresponding angles are all congruent.

Example

The picture above shows two parallel lines with a transversal. The angle 6 is 65°. Is there any other angle that also measures 65°?

6 and 8 are vertical angles and are thus congruent which means angle 8 is also 65°.

6 and 2 are corresponding angles and are thus congruent which means angle 2 is 65°.

6 and 4 are alternate exterior angles and thus congruent which means angle 4 is 65°.

Video lesson

Find the measure of all the angles in the figure

Whenever two parallel lines are cut by a transversal, an interesting relationship exists between the two interior angles on the same side of the transversal. These two interior angles are supplementary angles. A similar claim can be made for the pair of exterior angles on the same side of the transversal. There are two theorems to state and prove. I'll give formal statements for both theorems, and write out the formal proof for the first. The second theorem will provide yet another opportunity for you to polish your formal proof writing skills.

Theorem 10.4: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary angles.
Theorem 10.5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles.

Let the fun begin. As promised, I will show you how to prove Theorem 10.4.

Figure 10.6 illustrates the ideas involved in proving this theorem. You have two parallel lines, l and m, cut by a transversal t. You will be focusing on interior angles on the same side of the transversal: 2 and 3. You'll need to relate to one of these angles using one of the following: corresponding angles, vertical angles, or alternate interior angles. There are many different approaches to this problem. Because Theorem 10.2 is fresh in your mind, I will work with 1 and 3, which together form a pair ofalternate interior angles.

Figure 10.6l m cut by a transversal t.

Given: l m cut by a transversal t.
Prove: 2 and 3 are supplementary angles.
Proof: You will need to use the definition of supplementary angles, and you'll use Theorem 10.2: When two parallel lines are cut by a transversal, the alternate interior angles are congruent. That should be enough to complete the proof.

	Statements	Reasons
1.	l m cut by a transversal t	Given
2.	2 and 3 are same-side interior angles	Definition of same-side interior angles
3.	1 and 3 are alternate interior angles	Definition of alternate interior angles
4.	1 and 2 are supplementary angles, and m1 + m2 = 180º	Definition of supplementary angles
5.	1 ~= 3	Theorem 10.2
6.	m1 = m3	Definition of ~=
7.	m13 + m2 = 180º	Substitution (steps 4 and 6)
8.	2 and 3 are supplementary angles	Definition of supplementary angles

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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