Module 5

1. Module 5

1.29. Page 4

Mathematics 10-3 Module 5 Lesson 6

Module 5: Angles

Bringing Ideas Together

Try This

In this multimedia piece you will explore the relationship between the sum of co-interior angles and the sum of co-exterior angles for parallel and non-parallel lines. As you explore this relationship between the sums, think about the following question.

Does the relationship between pairs of co-interior angles or pairs of co-exterior angles depend on the way the parallel lines are positioned on the plane?

TT 1. Go to “Exploring Parallel Lines—Explore It.” Click on “Explore It 2,” (in the middle of your screen), and follow the directions to complete the table that follows in this lesson.

Step 1: To begin, choose any transversal with parallel lines.

Step 2: Add the co-interior angles together, and record the sum in the table.

Step 3: Add the co-exterior angles together, and record the sum in the table.

Step 4: Repeat Steps 1 to 3 using parallel and non-parallel lines to fill in the table.

	Sum of Co-Interior Angles	Sum of Co-Exterior Angles
Parallel Lines
Example 1
Example 2
Non-Parallel Lines
Example 1
Example 2

You used “Exploring Parallel Lines—Explore It” to explore co-interior and co-exterior angles. You looked at both parallel and non-parallel lines. Share your results from TT 1 with a partner or a group, and then discuss the following question.

How does the way parallel or non-parallel lines are positioned on a plane affect the relationship between pairs of co-interior or co-exterior angles?

Write a one-sentence response to the discussion question. Save a copy of your response in your course folder.

In the Explore section of this lesson, you formed parallel lines by positioning supplementary angles along one side of a transversal. Then, in the Try This section, you saw that it does not matter which way the parallel lines are pointed on a plane. As long as the lines are parallel, the co-interior or co-exterior pairs add up to 180°.

These important relationships are summarized in the following information, and the relationships will be important as you solve problems later in this lesson.

One of the co-interior pairs is ∠4 and ∠5. If those two angles add up to 180°, line 1 will be parallel to line 2.

Or, vice versa, if lines 1 and 2 are parallel, that makes ∠4 and ∠5 supplementary (the angles add up to 180°).

This is also true for the other pair of co-interior angles, ∠3 and ∠6.

One of the co-exterior pairs is ∠1 and ∠8. If those two angles add up to 180°, line 1 will be parallel to line 2.

Or, vice versa, if lines 1 and 2 are parallel, then ∠1 and ∠8 will be supplementary (the angles add up to 180°).

This is also true for the other pair of co-exterior angles, ∠2 and ∠7.

Two Lines and a Transversal

Examples 1 to 3 involve angle relationships for the following types of angles formed when two lines are cut by a transversal:

vertically opposite
supplementary
corresponding
alternate interior
alternate exterior
co-interior
co-exterior

This diagram shows the linear path of a football across the 35-yard and 40-yard lines. The path is equivalent to a transversal to the two lines. The path makes a 70-degree angle with the 35-yard line. The vertically opposite angle to the 70-degree angle is marked with the letter a. The interior angle adjacent to the 70-degree angle is marked with the letter b. The corresponding angle to the 70-degree angle is marked with the letter c.

Example 1

A forward pass in football crosses the 35- and 40-yard lines as shown.

Determine the measure of angles a, b, and c. State the relationship you used to find each measure.

Solution

View the animated “Lesson 6: Example 1 Solution.”

Example 2

The outline of a kite is shown. Are the two edges drawn in red parallel? Justify your answer.

This illustration shows the outline of a diamond-shaped kite. The upper-right and lower-left sides are drawn in red. A pair of perpendicular segments are drawn on the kite. The vertical segment forms angles of 70 degrees with the upper-right side of the kite and 69 degrees with the lower-left side of the kite.

Solution

This illustration shows two lines and a transversal. One interior angle measures 70 degrees. The corresponding interior angle measures 69 degrees.

The 69° and 70° angles are alternate interior angles formed by the vertical transversal. As these angles are not congruent, the two red edges are not parallel.

Example 3

This illustration shows a covered deck attached to a house. The side of the covered deck is parallel to the side of the house. The angle between the side of the deck and the roof is 123 degrees. The angle between the roof and the wall of the house is labelled a.

A covered deck is attached to the side of a house. Find the angle with measure a.

Solution

This illustration shows two parallel lines with a transversal. The interior angles formed are 123 degrees and “a” degrees.

It is now your turn!

Self-Check

Complete the following questions.

SC 7.

This illustration shows an A-frame chalet with an overlay on some of the window frames. The overlay has three vertical line segments joined at their tops by a fourth upward sloping line.

This illustration shows three vertical line segments joined at their tops by a fourth upward sloping line segment. The angle between the left vertical line and the top line is labelled a. The angle between the middle line and the top line on the right of the middle line is labelled b. The angle between the right-most vertical line and the sloping top line is labelled 40 degrees.

This illustration shows the measures of the angles in the overlay to the A-frame chalet’s windows. Find the missing measures for the windows of the chalet.

SC 8.

This illustration shows two vertical parallel lines with a transversal crossing them from lower left to upper right. The left line is labelled line 1, and the right line is labelled line 2. The lower-left angle, between the transversal and line 1, is 75 degrees. The upper-right angle, between the transversal and line 2, is labelled a.

If line 1 is parallel to line 2, find a. What angle relationship did you use?

SC 9. Find the measure of ∠B.

This illustration shows triangle ABC with angle A measuring 60 degrees. A line parallel to AC joins a point on AB with a point on BC. The angle formed between this line and BC, nearest B, is 40 degrees.

SC 10.

This illustration shows parallel lines, line 1 and line 2. They are joined by a segment that is perpendicular to line 1 at point A. The segment meets line 2 at point B.

Compare your answers.

Mastering Concepts

Try this problem. When you are finished, check your answer.

A line through point A is drawn parallel to the base of Without using a protractor, justify that the three angles of add up to 180°.

This illustration shows triangle ABC with a line through point A that is parallel to side BC. Angle B is labelled 1, and angle C is labelled 3. Angle BAC is labelled 2. The angle between the line through A and side AB is labelled a. The angle between the line through A and side AB is labelled b.

Compare your answer.