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Properties of Parallel Lines Cut by a Transversal

Subject:

Math  

Grades:

9, 10  

By – M. Wilson
Primary Subject – Math
Grade Level – 9-10

PA Academic Standards:

  • 2.4.11.A — Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.
  • 2.4.11.B — Construct valid arguments from stated facts.
  • 2.9.8.B — Draw, label, measure and list the properties of complementary, supplementary and vertical angles.
  • 2.9.8.E — Construct parallel lines, draw a transversal, and measure and compare angles formed (e.g., alternate interior and exterior angles).

Goal of this lesson:

  • To understand the angle theorems related to parallel lines
  • To apply the theorems to find angle measurements when given the measurement of some angle
  • To practice the geometric proof technique
  • To discover learning by comparing angle measurements and types of angles
  • To encourage working cooperatively in groups

Materials

  • Overhead projector
  • Lesson transparencies
  • Blank transparencies for writing on
  • Markers to use on transparency
  • Work sheets — 4 for activity
  • Tracing Paper
  • Scissors
  • Rulers
  • Paper clips
  • Protractors (if time permits activity)

Developmental Activities: (Instructional components that provide opportunities for students to make progress toward intended instructional objectives)

(5 minutes) Tell students that we will start with a review of the terminology we learned yesterday. Use overhead to conduct the review of the following terms:

  • Parallel lines – Coplanar lines that do not intersect
  • Transversal – A line that intersect 2 coplanar lines
  • Alternate Interior Angles — 2 nonadjacent interior angles on opposite sides of a transversal (form “Z” shape)
  • Alternate Exterior Angles — 2 nonadjacent exterior angles on opposite sides of a transversal
  • Same-side Interior Angles — 2 interior angles on same side of a transversal (form “[” shape)
  • Corresponding Angles — 2 nonadjacent angles on the same side of a transversal such that one is an exterior angle and the other is an interior angle (form “F” shape)
  • Vertical Angles — 2 nonadjacent angles formed by a pair of intersecting lines

Transition: At end of review, tell students that today we will discover some special properties of these angles formed by parallel lines.

(5 minutes) Exploration:

      Introduce the explore activity (adapted from idea in Addison-Wesley’s

Geometry

    , p.111): Tell students they are going to discover these properties by measuring and comparing angles. Show, and tell them that I have a set of directions to guide them through the activity. Explain that the tracing paper and diagram will be used to look at the angles. The packet also includes worksheets which they will complete in groups. The two white worksheets are to be completed first. Each group can decide how to split up this work. Then the green worksheet will be done collaboratively as a group by comparing the 2 worksheets. The green sheet will be handed in at the conclusion of the group activity.

(10 minutes)

    Class works on the activity while I walk around to give students guidance, answer their questions, make sure they are on task, and verify that their answers on worksheets are correct. Also pass out another green sheet, 1 per group, to hand in.

Transition: When groups seem to be finishing, refocus them as a whole for the explanation. Collect their green group sheets for assessment. Praise their work and tell them that we will now formalize what they discovered in their groups.

(10 minutes) Explanation:

      Ask students, based on their exploration, what can we say about corresponding angles? Wait for their answer: They are congruent.
      Show transparency with 1

st

      postulate of lesson showing: If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.

Next question: What did we discover about alternate interior angles? Wait for answer: They are congruent.

Move down transparency to show theorem: If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. Explain further by telling class that we can prove this using what we already presented as a postulate. Show proof and explain, marking angles as we refer to them:

 
a || b Given
L2 = L3 Corresponding L s are congruent
L 3 = L 1 Vertical L s are congruent
L 2 = L 1 Congruence is transitive

Next question: What can we say about alternate exterior angles? Wait for answer: They are congruent. Move transparency down to show next theorem: If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent. Show proof and explain:

 
a || b Given
 
L1 = L 3 Vertical Ls are congruent
 
L 3 = L 2 Corresponding Ls are congruent
 
L 1 = L 2 Congruence is transitive

Next question: What did you discover about same side interior angles? Wait for answer: They are supplementary. Move transparency down to show theorem: If 2 parallel lines are cut by a transversal, then same-side interior angles are supplementary.

Tell class that we can also prove this theorem based on the theorems we have already proved. We are building more knowledge based on facts we already know. This proof will be assigned as homework, or done in class if time permits. Remind them that the 2-column proof is one we use over and over in geometry, so it is important to understand them and use them frequently.

Do another example: If one alternate exterior angle = 2x + 10 and another = 15x – 40, solve for x. Solve this on transparency: 2x + 10 = 3x – 40. Solution x = 50.

If time permits:

      1) Have students try on their own to prove the same-side interior angles are supplementary theorem. Walk around to give hints. After a few minutes, go over it together. It is on the transparency.
    2) The group activity could have been done with protractors instead of tracing paper. Have students measure all the angles on the group diagram sheet with a protractor and compare to reinforce the day’s lesson.

Conclusion

Summarize lesson by asking students to remind the class of the four properties of parallel lines. Tell them that we must know the lines are parallel to use these properties; “looking parallel” is not enough. In today’s lesson, we started knowing the lines are parallel. Tomorrow we will look at what we can discover about lines, knowing certain information about the angles formed by them.



Discover Properties of Parallel Lines

 

Pair 1 members:

    1. Using tracing paper, trace L1, mark “1″ at the vertex and cut it out.
    2. Use the cut-out to find congruent angles. Record them here:

L1 is congruent to _____ _____ _____ _____ _____

    1. Choose another angle not listed in 2). Trace it, mark the vertex with the angle number and cut it out.
    2. Use the cut-out to find congruent angles. Record them here:

L___ is congruent to _____ _____ _____ _____ _____

    1. Repeat steps 1) through 4) with the second example.

L1 is congruent to _____ _____ _____ _____ _____

L___ is congruent to _____ _____ _____ _____ _____

    1. Line up the vertices of the two angle cut-outs so that they form adjacent angles (the angles share a common side). What do they form?

We can say that these two angles are _____________________

 


Discover Properties of Parallel Lines

 

Group members:

    1. Name the pairs of corresponding angles:

_____ _____ _____ _____

_____ _____ _____ _____

    1. Name the pairs of alternate interior angles:

_____ _____

_____ _____

    1. Name the pairs of alternate exterior angles:

_____ _____

_____ _____

    1. Name the pairs of same side interior angles:

_____ _____

_____ _____

 


(green handout)

Discover Properties of Parallel Lines

 

Group members:

As a group

Using the data gathered by comparing the angles, what conclusions can you draw about these special pairs of angles?

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