Author: GB

Mathematics teacher, technology enthusiast, blogger.

Parallel Lines and Transversals

High School Geometry, Questions and Quandaries

Given a triangle and a line segment, let us place the triangle such that one of its side and the line segment coincide. Next, we replicate the triangle and slide to the right hand side  such that two of their vertices meet as shown in Figure 1.

Figure 1 – Translated triangles have parallel corresponing sides.

The angles with the same color are basically the same angle, so we will call them corresponding angles. The sides which are parallel are also the same side, so we will call them corresponding sides.

What can you say about their corresponding angles? Their corresponding sides?

Figure 2 – The sum of the angles in point T is 180 degrees.

Next, we replicate one of the triangles, rotate it and place it between the two triangles as shown in Figure 2. Click here to rotate the triangles using GeoGebra. In the figure, we can observe and deduce the following:

  1. The measure of the three adjacent angles in T is half the circle and is therefore 180 degrees. This implies that the sum of the interior angles of triangle PQT is also 180 degrees, since they are the sum of the three adjacent angles in T.
  2. PQRT and QRST are parallelograms. (Explain why the opposite sides are parallel).
  3. The opposite angles of parallelogram are congruent, and their opposite sides are also congruent. (Explain why).
  4. We also observe that the sum of the consecutive angles of a parallelogram is 180 degrees. For example, the sum of angle PQR and angle QRT is 180 degrees.
  5. The interior angles of a parallelogram consists of 2 green angles, 2 yellow angles and 2 red angles, which means that the angle sum of the interior angles of a parallelogram is twice 180 or 360 degrees.
  6. PQRS is a trapezoid and the sum of its interior angles is also 360 degrees. (Explain why QR and PS are parallel).

If we extend QR and QT, then we will two parallel line segments and a transversal. In Figure 3, OR and PS are parallel lines a NU is a transversal. From Figure 1 and Figure 2, the yellow angles are corresponding angles, so RQT and PTQ are congruent.  We can also see that  the blue angle and yellow angle are supplementary (the sum of their measure is 180 degrees).

Figure 3 – NU is a transversal to the parallel lines OR and PS.

With the knowledge that the blue angle and the yellow angle are supplementary, we can deduce the following:

  1. Interior angles on the same side of a transversal of parallel lines (RQT and STQ) are supplementary.
  2. Angles OQN and OQT are supplementary so NQT must have the same measure as that of the yellow angle.
  3. Angles RQT and RQN are supplementary so RQN must have have the same measure as that of the blue angle.
  4. From 2 and 3, it follows that PTU is a blue angle and STU is a yellow angle.

With all the observations and deductions above, we came up with the following diagram. Note that angles symbols with the same color are congruent angles.

Figure 4 – Angle signs with the same color are congruent.

Pairs of these angles have special names  for distinction purposes. Most of the names are descriptive, so it is easy to determine other pairs. Some pairs of angles and their names are listed below.

1.) TQR and QTS are same side interior angles.

2.) NQR and STU are same side exterior angles.

3.) RQT and PTQ are alternate interior angles.

4.) STU and NQO are alternate exterior angles.

5.) NQO and QTP are corresponding angles.

Parallel lines and its relationship with its interval was mentioned in Euclid’s book The Elements. It is called the Fifth postulate or the Parallel Postulate. The parallel postulate states that

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

All the relationships we have deduced above can be derived from this single statement.

I have also written why the angle sum of the interior angles of a triangle is 180 degrees. C

Leave a comment , , , , ,

Maxima Tutorial 2: Algebraic Manipulation and Graphing

Maxima, Software Tutorials

In Maxima Tutorial 1, we have discussed how Maxima can be used to perform numerical computations.  In this tutorial, we are going to enumerate some of the capabilities of Maxima to perform symbol manipulation.  Note than in each equation below, do not forget to press the ENTER key after the semicolon.

Maxima can perform algebraic operations and here are a few of its capabilities:

  • Assigning values to variables and using them in computations.
  1. p:5; q:12; (Press the Enter key on your keyboard after the second semicolon)
  2. r: sqrt(p^2 + q^2);

In a, we used the colon sign to assign values to p and q and assign them as the value of r in b.

  • Solve equations in one variable
  1. solve(2*x  = 3, x);
  2. solve(a^3 + 3*a^2 + 5*a = 15,a);
  3. ratsimp(2*a + 3*b + c – a –b +2*c);
  4. ratsimp(x^2 – 3*x*y + y^2 – 2*x^2 – 5*y^2);

In c, we solve for the 2x = 3 for the value of x. In d, we use * for multiplication, and ^ for exponentiation. The ratsimp command simplifies expressions.

  • Expand and factor algebraic expressions
  1. expand((x + y)^6);
  2. factor (x^6 – 1);

Figure 1 – The Maxima command line.

  • Solve systems of linear equations
  1. linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
  2. eq_1: x^2 + 3*x*y + y^2 = 0;
  3. eq_2: 3*x + y = 1;
  4. solve (eq_1,eq_2);

In a, we solved for the values of x and y in the systems of linear equation 3x + 4y = 7 and 2x + ay = 3. In b and we defined two equations, eq_1 and eq_2, then used the two variables in d.

  • Plot graphs in 2 and 3 dimensions
  1. plot2d (sin(x)/x, [x, -20, 20]); (See Figure 1)
  2. plot3d(sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2), [x,-12, 12], [y,-12,12] (See Figure 2)

Figure 2 – Graph of sin(x)/x.

Figure 3 – Graph of (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2).

2 comments , , , , ,

Geometer’s Sketchpad Tutorial 1: Constructing an Equilateral Triangle

Geometer's Sketchpad, High School Geometry, Software Tutorials

In this tutorial, we will use the Geometer’s Sketchpad to mimic the compass and straightedge construction tool we use in elementary geometry.  Geometer’s Sketchpad is a proprietary interactive geometry software developed by Key Curriculum Press.

The idea in constructing an equilateral triangle is to use the centers of two circles and one of their intersections as vertices of the triangle (see Figure 6). First, we construct a circle with center A passing through B, then construct a circle with center B and passing through point A, and then determine their intersections. Next, we hide all the unnecessary objects and connect the three points with segments. The final output of the tutorial is shown in Figure 1.

 

Figure 1 – The appearance of the expected of this tutorial.

 

In constructing our triangle, we will learn the following:

  • select, deselect and move objects
  • construct objects such as circles and  segments
  • display the label of objects
  • hide objects
  • display the measure of segments
  • construct points of intersection of two objects

Construction Steps

1.)  Click the circle tool, determine the center by clicking a location on the drawing pad, move the mouse to determine the radius, and click another location to draw the circle. If you made a mistake, just click the Edit menu and choose Undo from the list.

 

Figure 2 – The toolbar contains the different tools of Sketchpad used for drawing.

 

2.) Click the arrow selection tool and try to drag the two points. What do you observe?

3.) We now show the label of the two points. Click the two points, then click the Display menu from the menu bar, then click Show labels from the drop-down menu. Notice that Geometer’s Sketchpad names the objects in alphabetical order, the center of the circle is named A and the point on the circle is named B.

 

Figure 3 – The Display drop-down menu.

 

4.) To construct a circle with center point B and passing through A, click the compass tool, click point B and click point A. Your drawing should look like the one shown in Figure 4.

 

Figure 4 – Circe with center A and passing through B and circle with center B passing through A.

 

5.) Next, we construct the intersections of two circles. Click the selection arrow tool, click then select the two circles. (Be sure that only the two circles are selected. To deselect an object, click anywhere on the drawing pad).

 

Figure 5 – The two circles are selected.

 

6.) To construct the intersection, click Construct from the menu bar, then Intersections from the drop-down menu bar.

7.) Display the name of the two intersections by clicking both of them and clicking Show labels from the Display menu.

8.) Next, we hide the two circles and point D leaving only points A, B and C on the drawing area. To do this click the selection arrow tool, click the two circles and point D.

 

Figure 6 – The two circles and point D are selected.

 

9.) To hide the selected objects, click the Display menu and choose Hide Objects. Notice that the only left objects on the drawing area are points A, B and C.

10.) Select segment tool (be sure that you choose the segment tool, not the ray tool or line tool) from the straightedge tool, then connect the three points to form triangle ABC. Using the selection arrow tool, move the points on the vertices of the triangle.

  • What do you observe?
  • What kind of triangle is triangle ABC? Why do you say so?

11.) Let us try to see if your conjecture about triangle ABC is true. We will display the length of the segments. To do this, select one of the sides of the triangle, click Measure from the menu bar and choose Length from the drop-down menu.

  • Measure the other two sides. What do you observe?
  • Move the vertices of the triangle. Is your observation still true?
  • Explain why triangle ABC is always an equilateral triangle.

12.) If you want to save your file, click the File menu from the menu bar and click Save from the drop-down menu.

13.) Type the file name of your first sketch, choose a location to save, then click Save.

I also created a similar tutorial on creating an equilateral triangle but using GeoGebra. If you are interested, click here.

Congratulations, you just finished your first construction. Click here to go to Geometer’s Sketchpad Tutorial 2.

 

3 comments , , , , ,
1 … 395 396 397 398 399 … 405