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Learning Math: Geometry

Parallel Lines and Circles Part B: Parallel Lines (45 minutes)

Session 4, Part B

In this part:

  • Properties of Angles
  • Reasoning About Properties of Angles

Properties of Angles

Parallel lines are two lines in the same plane that never intersect. Another way to think about parallel lines is that they are “everywhere equidistant.” No matter where you measure, the perpendicular distance between two parallel lines is constant. With dynamic geometry software, you can draw two lines that look parallel, but you can’t be sure that they are parallel unless you construct them to be parallel. Note 2

 


Problem B1

Follow these steps to construct two parallel lines:

a. Using the Line Tool in Geometer’s Sketchpad, draw a line.
b. Pull down the Construct menu in Sketchpad. You’ll notice that the “Parallel Line” is gray — therefore, not an option to you. This is Sketchpad’s way of telling you that you don’t have the correct objects selected or you don’t have enough objects selected. What else do you need to construct a line parallel to your original line?
c. Continue your construction and record the steps you used to construct two parallel lines.

 


Problem B2

To measure an angle, select three points in the order of the angle. You may need to construct additional points on your lines before you can measure the angles.

Draw a transversal through your parallel lines. (A transversal is a line that passes through two parallel lines.)

a. Measure each of the angles formed.
b. Change the orientation of the transversal by dragging one of the defining points. Keep a record of what changes and what stays the same.

 

 


When a pair of parallel lines is cut by a transversal, several special pairs of angles are formed.

∠ABD and ∠EFB are corresponding angles.
∠ABF and ∠GFB are alternate interior angles.
∠ABD and ∠CBF are vertical angles.
∠ABD and ∠CBD form a linear pair.

 

 


Problem B3

Name another pair of corresponding angles, another pair of alternate interior angles, another pair of vertical angles, and another linear pair.

 


Problem B4

Using this new terminology, summarize the relationships you discovered in Problem B2.

Reasoning About Properties of Angles

In the previous subpart, you gathered evidence of some properties of angles. But evidence alone doesn’t explain why something is true, or even mean that it is true. Mathematics requires a reasoned argument that is general, not about a specific set of lines.

When two lines intersect, the vertical angles (angles opposite each other) have the same measure.

 

 


Problem B5

In this problem, you will look at an explanation for why vertical angles have the same measure.

a. m∠1 + m∠2 = 180°. Why?
b. Also m∠3 + m∠2 = 180°. Why?
c. So m∠1 must equal m∠3. Why?
d. What other pair of angles is equal in measure? Why?

 


When two parallel lines both intersect a third line, corresponding angles (angles in the same relative positions, like angles 1 and 7 or angles 3 and 5 in the picture below) have the same measure.

One way to understand this is to imagine sliding a copy of the picture above along line until line k sits on top of line l. (Note 3)

Now ∠1 sits exactly where ∠7 used to be, ∠3 sits exactly where ∠5 used to be, and so on.

To prove that corresponding angles are congruent, we could add another line segment, , parallel to line j. By doing so we have created a parallelogram, and thus we know that the adjacent angles of a parallelogram (in this case ∠2 and ∠7) equal 180°.

So, to prove that ∠1 and∠7 are congruent, we write the following:

∠1 +∠2 = 180° (because they form a straight line)
∠2 +∠7 = 180° (because they are adjacent angles of a parallelogram)

It follows that∠1 +∠2 = ∠2 +∠7 = 180°.

And thus, ∠1 =∠7

Alternate interior angles (angles on opposite sides of the transversal, and between the parallel lines, like∠7 and∠3 or∠2 and∠8), also have the same measure.


Problem B6

Create an argument to explain why the above statement about alternate interior angles would be true. You may want to use the facts about corresponding angles and vertical angles, or you may come up with another explanation.


Problem B7

Create any quadrilateral with Geometer’s Sketchpad. Construct the midpoints of all four sides of the quadrilateral, and then connect them in order.

a. Describe the quadrilateral in the center.
b. How could you test to see if you are right about the kind of quadrilateral you have in the center? What measurements could you make?
c. Move around your original quadrilateral, changing its shape. What changes on the inside quadrilateral? What stays the same?
d. How can you change your original quadrilateral to make the inside quadrilateral a square?

Problem B5 adapted from IMPACT Mathematics, Course 1, developed by Educational Development Center, Inc. p. 473. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

Problem B6 developed by Educational Development Center, Inc. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

Notes

Note 2

Discuss or reflect on why “in the same plane” is an important part of the definition. If you think in three dimensions, two lines can never intersect but also be not parallel. To demonstrate this, you can draw a line on one piece of paper, then draw a line on another piece of paper. Put one of the pieces of paper on a table or desk, and hold the second one above the first, parallel to it. You can rotate the second piece of paper and see that the two lines will never intersect as long as the planes (papers) stay parallel, but the lines are not always parallel to each other.

 


Note 3

If you are working in a group, you can also demonstrate this for the whole group with an overhead projector. Make two copies of the j, k, l setup on transparencies. Place both transparencies on the overhead projector, one on top of the other. Then slide the top transparency, keeping one line j on top of the other. Lines k and l very convincingly match up.

Solutions

Problem B1

a. Answers will vary. Remember that the Line Tool has two arrows; do not use the Segment Tool.
b. You need to select a line and a point.
c. Use the Line Tool to draw a line anywhere on the sketch. Use the Point Tool to create a point that is not on the line. Select both objects (point and line — remember to hold down the shift key!), and then choose “Parallel Line” from the Construct menu.

 


Problem B2

a. Answers will vary.
b. The points of intersection between the transversal and the pair of parallel lines will change, as will the measures of all of the angles. Also, the measures of linear pairs will add up to 180°. These are invariants for the diagram.

 


Problem B3

  • Corresponding:∠ABF and ∠EFH; ∠CBF and ∠GFH; ∠DBC and ∠BFG
  • Alternate interior: ∠EFB and ∠CBF
  • Vertical:∠EFH and ∠BFG; ∠ABF and ∠DBC; ∠BFE and ∠GFH
  • Linear pair: ∠ABF and ∠CBF; ∠EFB and ∠GFB; ∠EFH and ∠GFH; ∠DBA and ∠ABF; ∠BFE and ∠EFH; ∠DBC and ∠CBF; ∠BFG and ∠GFH

 


Problem B4

Corresponding angles are congruent (have the same measure).
Alternate interior angles are congruent.
Vertical angles are congruent.
Linear pairs are supplementary (angle measures add up to 180°).

 


Problem B5

a. When you add m1 and m2 together, you define the same angle as the angle defined by a straight line, namely an angle of 180°.
b. Same reasoning as in question (a).
c. Using questions (a) and (b), we can write
m1 + m2 = 180°
m2 + m3 = 180°
Subtracting the second equation from the first, we get
m1 – m3 = 0 or m1 = m3
d. We can repeat the arguments from questions (a)-(c) with 2, 3, and 4, and conclude that 2 and 4 have the same measure.

 


Problem B6

Let’s consider ∠3 and ∠7 from the picture. We need to show that this pair of alternate interior angles has the same measure. Notice that ∠1 and ∠7 are corresponding angles, and therefore they have the same measure. In addition, ∠1 and ∠3 have the same measure because they are vertical angles (see Problem B5). And so it follows that ∠3 and ∠7 have the same measure because they both have the same measure as ∠1.

 


Problem B7

 

a. The quadrilateral in the middle appears to be a parallelogram.
b. Ideally, you want to test if opposite sides are parallel. Any of the following tests would suffice: Opposite sides are congruent; opposite angles are congruent; adjacent angles are supplementary (add up to 180°); the diagonals bisect each other.
c. The two sides of the original quadrilateral opposite the vertex you choose and the angles between those sides remain the same as you move around the quadrilateral. Also, the side of the inscribed parallelogram opposite to the chosen vertex remains fixed. All other sides and angles change. The inscribed figure, however, remains a parallelogram.
d. In part (c), you noticed that moving one vertex leaves two sides fixed. The reason for this is that they depend on the length of the diagonal. If we make the diagonals of the original quadrilateral perpendicular and congruent, then the inside quadrilateral will be a square. The resulting figure is a square inscribed in a kite or even a square, but that’s not necessary.

Series Directory

Learning Math: Geometry

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Produced by WGBH Educational Foundation. 2003.
  • ISBN: 1-57680-597-2

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