1. Module 5

1.24. Page 4

Mathematics 10-3 Module 5 Lesson 5

Module 5: Angles

 

Bringing Ideas Together
This illustration shows two parallel lines labelled line 1 and line 2, and a third line crossing the first two lines is labelled transversal. The angles created are numbered from 1 to 8. The upper right angle is labelled 1, with the other three angles numbered 2, 3, and 4 going counterclockwise from angle 1. The bottom right angle is labelled 8, and the other three angles are numbered 7, 6, and 5 and go counterclockwise from angle 8.

 

In the Explore section you compared the angles formed when two parallel lines are cut by a transversal. You should have discovered that vertically opposite angles are congruent.

 

In the diagram the vertically opposite angles are ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, and ∠6 and ∠8.

 

In TT 6 to TT 8 you compared the angles formed when two parallel lines are cut by a transversal. You discovered that angles in the same corresponding positions at the two points of intersection are congruent.

 

In the diagram, the corresponding angles are ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8.

 

corresponding angles: angles in the same relative positions when two lines are intersected by a transversal



The applet “Corresponding Angles” highlights these angle relationships. Move the lines around by dragging the black dots on the line. Notice how the corresponding angles remain congruent.

 

Investigate the following questions as you explore “Corresponding Angles.”

  • How many different angle measures are there?

  • How does this number compare with the colouring diagram you did in the Share activity?

Now rotate the lines by dragging anywhere on the line except on the black dot. Watch for the answer to this question: Do the corresponding angles remain constant?

 

Example 1

 

This is a photograph of a construction worker climbing a ladder that is propped against a flat roof.

© Lisa F. Young/shutterstock

A ladder on level ground is propped against the side of a building with a flat roof (the roof is parallel to the ground). The angle between the ladder and the roof is 84°. What is the measure, x, of the angle between the foot of the ladder and the ground?

 

View the animated “Lesson 5: Example 1 Solution.”

 

This illustration shows a horizontal line labelled ground, and a vertical line rising from the ground that meets a horizontal line segment labelled roof. A red line is shown starting at the ground and touching the corner of the vertical line and the roof, and the line extends above the roof. The angle between the line and the ground on the side nearest the vertical line is labelled x. The angle above the roof and between the roof and the line is 84 degrees.

 

Example 2

 

This is an image of a horse and rider pulling a travois. The travois is also displayed vertically in the image.

Jefferys, C. W. The Picture Gallery of Canadian History. Toronto: Ryerson Press, 1942. p 34. (left) Sparrow Illustration

 

Travois were used by people—like the Nehiyawak and the Siksika—on the plains and prairies to transport goods. Before the Spanish introduced the horse to North America, dogs were used to pull travois. A travois consisted of two poles lashed together and stabilized by one or more poles crossing the first two.

 

The following diagram matches the design of a travois.

 

This illustration shows two long line segments. One segment goes from upper left to lower right, and a second segment goes from lower left to upper right. They cross near their right-hand ends. A third line crosses both segments near their left-hand ends. The upper right angle formed between the crossing line and one of the segments is labelled 2. The upper right angle formed between the crossing line and the other segment is labelled 1.

 

Classify ∠1 and ∠2. Are they congruent? Why or why not?

 

Solution

 

∠1 and ∠2 are corresponding angles because they lie in the same relative position at the cross pole.

 

They are not congruent because the other two poles (lines) are not parallel, as ∠1 is acute and ∠2 is obtuse.

 

Other Angles at the Transversal

 

This illustration shows Line 1 and Line 2 as parallel horizontal lines. The Transversal crosses both lines as it rises from the lower left of the illustration. The angles formed between the transversal and line 1 are numbered 1, 2, 3, and 4 starting in the upper right and moving counterclockwise. Labels 3 and 4 are in a pink box. Labels 1 and 2 are in a blue box. The angles formed between the transversal and line 2 are numbered 5, 6, 7, and 8. Labels 5 and 6 are in a pink box. Labels 7 and 8 are in a blue box.

 

In the Explore section you discovered that corresponding angles were not the only congruent pairs.

 

interior angles: angles lying between two lines cut by a transversal

 

exterior angles: angles outside two lines cut by a transversal

Look at the four angles between the parallel lines. ∠3, ∠4, ∠5, and ∠6 are interior angles.

 

The other four angles, ∠1, ∠2, ∠7 ,and ∠8, are exterior angles.

 

Self-Check

 

SC 3. In the preceding diagram, which interior angles are congruent?

 

Compare your answer.

 

alternate interior angles: interior angles lying on opposite sides of the transversal—one is on the left side and the other is on the right side

∠3 and ∠5 are alternate interior angles. ∠4 and ∠6 are also alternate interior angles, because they lie on alternate sides of the transversal.

 

Use the applet “Alternate Angles (of a Transversal)” to describe the relationship between alternate angles.

 

From the applet you can draw the following conclusions:

  • When two parallel lines are cut by a transversal, the alternate interior angles are congruent (equal in measure).

  • When two parallel lines are cut by a transversal, the alternate exterior angles are congruent (equal in measure).

This illustration shows line 1 and line 2 as parallel horizontal lines. The transversal crosses both lines as it rises from the lower left of the illustration. The angles formed between the transversal and line 1 are numbered 1, 2, 3, and 4 starting in the upper right and moving counterclockwise. The angles formed between the transversal and line 2 are numbered 5, 6, 7, and 8. Labels for interior angles 3, 4, 5, and 6 are in pink boxes. Labels for exterior angles 1, 2, 7, and 8 are in blue boxes.

 

alternate exterior angles: exterior angles lying on opposite sides of the transversal—one is on the left and the other is on the right

So, in the diagram, if the lines are parallel, the alternate exterior angles are congruent. The angles coloured pink are alternate exterior angles, so . Also, the angles coloured blue are alternate exterior angles, thus .

 

Study these examples.

 

Example 3

 

This is a photograph of a crossbraced wooden gate next to a ploughed field.

© Paul.J.West/shutterstock

A wooden gate is crossbraced as shown. If the horizontal boards are parallel, determine the measure of the angle between one of the braces and the top horizontal board.

 

This illustration shows a zed shape, representing a tracing of the top and bottom boards of the wooden gate and one of the crossbraces. The top and bottom boards are parallel, and the angle between the crossbrace and the bottom board is 42 degrees.

 

 

 

 

 

 

 

 

 

Solution

 

The two angles in the outline of the boards are alternate interior angles. The diagonal crossbrace is a transversal cutting across the parallel lines of the horizontal boards.

 

So, x = 42°.

 

Example 4

 

This illustration shows a four-legged electrical tower, which tapers inward as it rises. A zed shape is drawn over the photo. The shape runs along one leg of the four-legged tower, down along one cross-brace, and up another leg of the four tower legs.

© Gulei Ivan/shutterstock

Classify the two angles in the diagram formed by the sides of the transmission tower and one of the diagonal braces. Do you have enough information to determine the value of x? Why or why not?

 

This illustration shows a zed shape representing the tracing on the electrical tower. The one angle between the brace and a tower leg is labelled x, the other is labelled 45 degrees.

 

Solution

 

The angles in the outline are alternate interior angles. However, the sides of the tower are not parallel, so there is not enough information to find the value of x.

 

Another Angle Relationship

 

There is one more angle relationship that can be determined from the intersection of parallel lines by a transversal.

 

This illustration shows line 1 and line 2 as parallel horizontal lines. The transversal crosses both lines as it rises from the lower left of the illustration. The angles formed between the transversal and line 1 are numbered 1, 2, 3, and 4 starting in the upper right and moving counterclockwise. The angles formed between the transversal and line 2 are numbered 5, 6, 7, and 8 and moving counterclockwise. A single red arc marks angle 4, and a double red arc marks angle 5.

 

Consider the angle pair ∠4 and ∠5. What is the relationship between these two angles? The angles are not congruent, but a relationship does exist. Can you see it?

 

Use the applet “Transversal Angles” to help determine the relationship.

 

co-interior angles: interior angles that lie on the same side of the transversal

From the applet, you should have discovered that ∠4 and ∠5 are co-interior angles and are supplementary (add up to 180°). Consider this explanation as to why this is true.

 

The adjacent angles ∠1 and ∠4 form a straight angle—they are supplementary. So, ∠1 + ∠4 = 180°.

 

But ∠1 = ∠5, since they are corresponding angles. Therefore, you can substitute ∠5 for ∠1. So, ∠5 + ∠4 = 180°.

 

These two co-interior angles are supplementary. Similarly, the co-interior angles ∠3 and ∠6 are supplementary. So, ∠3 + ∠6 = 180°.

 

co-exterior angles: exterior angles that lie on the same side of the transversal

You may have noticed in the applet that co-exterior angles are supplementary if the lines are parallel.

 

This illustration shows line 1 and line 2 as parallel horizontal lines. The transversal crosses both lines as it rises from the lower left of the illustration. The angles formed between the transversal and line 1 are numbered 1, 2, 3, and 4 starting in the upper right and moving counterclockwise. The angles formed between the transversal and line 2 are numbered 5, 6, 7, and 8. A single red arc marks angle 1, and a red arc and a blue arc mark angle 8.

 

 

Note: All of the angle relationships are reversible. If the corresponding angles are congruent, the lines are parallel. If the alternate interior or alternate exterior angles are equal, the lines are parallel. If the co-interior or co-exterior angles are supplementary, the lines are parallel.

 

Work through the following examples before you practise your skills.

 

Example 5

 

A cardboard box is partially flattened into a parallelogram, as shown in the diagram.

 

This illustration shows a parallelogram with angles labelled a, b, c, and 55 degrees. The angles are labelled starting in the lower right corner and moving counterclockwise around the parallelogram.

 

The angle at the lower left is 55°. Find the measures of the other three angles.

 

Solution

 

This illustration shows a parallelogram with angles labelled a, b, c, and 55 degrees. The angles are labelled starting in the lower right corner and are moving counterclockwise around the parallelogram. The lower side is highlighted in blue, and red arrowheads pointing upward are drawn on the upward-sloping sides.

 

 

 


 

This illustration shows a parallelogram with angles labelled a, b, c, and 55 degrees. The angles are labelled starting in the lower right corner and are moving counterclockwise around the parallelogram. The right side is highlighted in blue, and red arrowheads pointing to the right are drawn on the top and bottom sides.

 

 

 



 

This illustration shows a parallelogram with angles labelled a, b, c, and 55 degrees. The angles are labelled starting in the lower right corner and moving counterclockwise around the parallelogram. The left side is highlighted in blue, and red arrowheads pointing to the right are drawn on the top and bottom sides.


Example 6

 

Are line 1 and line 2 parallel? Why or why not?

 

This illustration shows two lines sloping slightly upward from left to right. The top line is labelled line 1. The bottom line is labelled line 2. A third line crosses both of these lines. The lower left angle of the third line meets with line 2 and measures 61 degrees. The upper left angle of the third line meets with line 1 and measures 118 degrees.

 

Solution

 

The given angles are co-exterior angles. If the lines were parallel, the co-exterior angles would be supplementary (add up to 180°).

 

Since 118° + 61° = 179°, line 1 and line 2 are not parallel.

 

Your turn!

 

Self-Check

 

Complete the following questions.

 

SC 4. Play “Grocery Store Game (Exploring Parallel Lines).”

 

Try to collect at least 14 tokens. Select [Hints] if you need to review the meaning of vertically opposite, corresponding, and alternate angles. Consider a second game if you have trouble remembering the difference between vertically opposite, corresponding, and alternate angles.

 

SC 5. The following figure was found as a pattern in a painting.

 

This illustration shows a triangle with its vertical side labelled AB. The angle between this vertical side and the hypotenuse is 50 degrees. A line segment, CD, is drawn parallel to side AB. The larger angle formed between line CD and the hypotenuse is labelled x.

 

Based on the figure, answer the following questions:

  1. Is Justify your answer.

  2. Calculate the value of x. State the angle property you used to justify your answer.

SC 6. This illustration shows a pair of vertical bridge supports and a crossbeam running from the top of one support to the base of the other. The angle between the second support and the beam is labelled x.

 

This is a photograph of a railway bridge against a blue sky.

© basel101658/shutterstock

This illustration shows a pair of vertical bridge supports, and a crossbeam running from the top of one support to the base of the other. The angle between the first support and the beam is 32 degrees. The angle between the second support and the beam is labelled x.



Calculate the value of x. Justify your answer. Assume the vertical bridge supports are parallel.

 

SC 7. Line 1 is parallel to Line 2.

 

This illustration shows two parallel horizontal lines labelled line 1 and line 2. Line 1 is above line 2. A third line crosses both of these lines. The upper right angle the third line makes with line 1 measures 143 degrees. The lower right angle the third line makes with line 2 is labelled x.

 

Find the value of x.

 

Compare your answer.

 

Mastering Concepts

 

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

 

The angles within a triangle are called interior angles. If a side of a triangle is extended, an exterior angle is formed. In DABC, ∠A, ∠B, and ∠ACB are interior angles. ∠ACD is an example of an exterior angle.

 

This illustration shows triangle ABC. Side BC is extended to D. Ray CE is drawn parallel to side AB.

 

Use parallel line relationships.

 

Compare your answer.