TCS - Angles and Lines Lesson

Angles and Lines

In this lesson, we are going to focus on angles and lines, keeping in mind the transformations we studied in the previous lesson.

Let us first consider parallel lines cut by a transversal.  

The arrows on lines AB and CD indicate that these lines are, in fact, parallel. Remember, parallel just means that lines will never touch or intersect. They are an equal distance apart. In other words, they are translations of one another. The line that cuts through them forming the eight labeled angles is called a transversal. When referring to a picture like this, we say "parallel lines cut by a transversal."  

The eight angles that are labeled form what is known as pairs of corresponding angles resulting from parallel lines being cut by a transversal. These angles have some characteristics that will allow us to solve problems. There are 4 pairs of corresponding angles. Corresponding just means they represent the same place in relation to the transversal and "their" parallel line.

• 1 and 5 are corresponding.
• 4 and 8 are corresponding.

Can you find the other two pairs or corresponding angles? If you said 3 and 7 is a pair and 2 and 6 are a pair, you're exactly right! The cool thing about corresponding angles is that they are congruent. You can use that to help you solve for missing angles.

Now, there are also some other names for some different pairs of angles. Let's take a look at angles 3 and 5. These angles are referred to as alternate interior angles. The are on opposite sides of the transversal, which means alternate sides. They are also on the interior of the parallel lines, or on the "inside" of the parallel lines. Therefore, they are alternate interior angles. What is another pair of alternate interior angles?

• Solution: angles 2 and 8

There is also another pairing of angles called alternate exterior angles. Again, these angle pairs are on opposite, or alternate, sides of the transversal. However, they, as opposed to interior angles, are exterior, or on the "outside" of the parallel lines. Angles 4 and 6 represent one pair of alternate exterior angles. Can you name the other pair? 

• Solution: 1 and 7

 Same side interior angles are the pairs of angles that are on the same side of the transversal and on the interior, or inside, of the transversal. One pair of same side interior angles on the image is 2 and 5. Do you see the other pair? 

• Solution: 3 and 8

One last set of angles that we'll examine are vertical angles. Vertical angles are angles whose vertices are touching, like two Vs point to point. The vertical angles are the ones opposite each other and are congruent. Here's a picture of vertical angles.

Angles 1 and 3 are vertical and therefore congruent; and, angles 2 and 4 are vertical and therefore congruent.

You should recall from previous math courses what complementary and supplementary angles are. Complementary angles are two angles whose sum is 90 degrees. Supplementary angles are two angles whose sum is 180 degrees.

Another key thing to remember is that angles that make up a straight line must add together to be 180 degrees. It doesn't matter if there are two angles or five. If there are only two angles, this is called a linear pair.

Let's watch this video to solidify our understanding of parallel lines cut by a transversal.

Now, let's practice! Visit the link in the sidebar to practice measuring angles.

Earlier, we mentioned angles that form a straight line will add together to be 180. Let's do a little experiment. Go grab a piece of paper and a pair of scissors. Use a straight edge to draw a triangle of any size on your paper. Now take your scissors, cut the triangle out, and cut the angles off. Set those aside. Now, use your straight edge to draw a straight line on another sheet of paper. Take your angles that you cut off your triangle and arrange them on the line. What happened?   Your three angles should line up perfectly on your straight line. What does this tell you about the angles inside a triangle? Well, the angles inside a triangle will always sum to 180 degrees! That's right, any triangle will have this property. In fact, this is called the Triangle Sum Theorem!

Want to see the proof?

Another theorem we use when discussing a triangle's angle measures is the Exterior Angle Theorem. The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. In all other polygons, you must show that all angles are congruent for the figures to be similar. That isn't the case with triangles.

Let's take a look at one.

Which of these two triangles are similar?   Let's look at their angles.   We can look at these triangles and see that both Triangles A and C have two congruent angles; therefore, Triangles A and C are similar.   Now, let's not discount Triangle B just yet!   We can clearly see that it has a 32 degree angle as do Triangles A and C.   We've learned from the Triangle Sum Theorem that all the angles have to sum to 180 degrees.   What does that mean about the missing angle of triangle B?   That's right, it is 118 degrees.   What does this tell us?   All three of these triangles are similar.   Remember, this does not mean that the triangles are congruent, because they may or may not have the same side lengths.   However, we do know that they are similar.

Assignment - Homework Set 2

Now that you have spent some time learning about transformations, you are ready to complete your first homework set.   Click here to download the Transformation, Congruence & Similarity: Lesson 2 Homework, Homework Set 1. Links to an external site.

Once you have completed Homework Set 1, make sure you ask your teacher if you have any questions.   When you feel confident in your work, you'll need to take the Angles and Lines Homework Quiz  over Unit 1, Lesson 2: Angles and Lines.   Make sure you have your Homework Set 1 completed and with you when you log in to take it. This is a timed quiz and you'll only have time to enter the answers from your HW. You will not have time to work them out during the quiz.

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