- Identify angles made by transversals: corresponding, alternate interior, alternate exterior and same-side/consecutive interior angles.
Angles and Transversals
Many geometry problems involve the intersection of three or more lines. Examine the diagram below:
In the diagram, lines g and h are crossed by line l. We have quite a bit of vocabulary to describe this situation:
- Line l is called a transversal because it intersects two other lines (g and h). The intersection of line l with g and h forms eight angles as shown.
- The area between lines g and h is called the interior of the two lines. The area not between lines g and h is called the exterior.
- Angles ∠1 and ∠2 are a linear pair of angles. We say they are adjacent because they are next to each other along a straight line: they share a side and do not overlap.
- There are many linear pairs of angles in this diagram. Some examples are ∠2 and ∠3,∠6 and ∠7, and ∠8 and ∠1.
A ________________________________ is a line that cuts across two or more lines.
The area in between lines g and h is called the __________________________ of the lines.
The area above line g and below line h is called the _______________________ of the lines.
The word _______________________________ means “next to.”
A linear _____________________ is a set of two angles that are adjacent to each other along a straight line.
The prefix “trans” means “across.”
A transversal cuts across two or more lines.
Other words that have the prefix “trans” are transport (to carry across) and transmit (to send across).
Can you think of some other words that begin with the prefix “trans”?
What do you remember about linear pairs from Unit 1?
- They are adjacent, which means they are right next to each other and they share a side.
- They are supplementary, which means they add up to 180∘.
Supplementary angles sum to ____________________.
∠1 and ∠3 in the diagram are vertical angles. They are nonadjacent angles (angles that are not next to each other) made by the intersection of two lines.
- Other pairs of vertical angles in the diagram on the previous page are ∠2 and ∠8,∠4 and ∠6, and ∠5 and ∠7.
Pairs of ________________________________ angles are across from each other at an intersection of two lines.
What do you remember about vertical angles from Unit 1?
Vertical angles get their name because they have the same vertex.
- You learned in Unit 1 that vertical angles are congruent. In other words, vertical angles have the same measure.
Corresponding angles are in the same position relative to both lines crossed by the transversal. ∠1 is on the upper left corner of the intersection of lines g and l. ∠7 is on the upper left corner of the intersection of lines h and l. So we say that ∠1 and ∠7 are corresponding angles.
Corresponding angles are in the ________________________ position at each intersection.
∠3 and ∠5 are _________________________________ angles because they are both in the bottom right corner of their intersections.
The word “corresponding” means “matching” or “similar.”
For example, Mexico City corresponds with Washington, D.C., because they are both the capitals of their countries.
Fill in the blanks with the angle that corresponds to each of the following angles in the diagram above. The first one has been done for you.
1. Angle 1 and angle ______.
2. Angle 8 and angle ______.
3. Angle 2 and angle ______.
4. Angle 5 and angle ______.
∠3 and ∠7 are called alternate interior angles. They are in the interior region of the lines g and h and are on opposite sides of the transversal.
Alternate __________________________ angles are inside lines g and h and on opposite sides of line l, the transversal.
∠4 and ∠8 are another example of __________________________ interior angles.
- Similarly, ∠2 and ∠6 are alternate exterior angles because they are on opposite sides of the transversal, and in the exterior of the region between g and h.
Alternate exterior angles are on opposite sides of the _____________________________.
∠1 and ∠5 are another example of ____________________ ___________________ angles.
If you alternate between two things, you switch between them.
Do you see how alternate angles switch sides?
One angle is to the right of the transversal and the other angle is to the left of the transversal.
To _______________________________ means “to switch.”
- Finally, ∠3 and ∠4 are consecutive interior angles. These are also known as same-side interior angles. They are on the interior of the region between lines g and h and are on the same side of the transversal. ∠8 and ∠7 are also consecutive interior angles.
Consecutive interior angles are the same thing as _______________ - ________________ interior angles.
Look at the picture of the lines being cut by a transversal below. Then, circle the type of angle pair represented by the angles given.
1. a and h corresponding alternate interior alternate exterior same-side interior
2. b and g corresponding alternate interior alternate exterior same-side interior
3. d and h corresponding alternate interior alternate exterior same-side interior
4. d and g corresponding alternate interior alternate exterior same-side interior