1. Module 5

1.19. Page 4

Mathematics 10-3 Module 5 Lesson 4

Module 5: Angles

 

Bringing Ideas Together

 

This illustration shows two lines intersecting. The lines are labelled line 1 and line 2. The angles are labelled counterclockwise from the top as 1, 2, 3, and 4.

You worked with the angles in a triangle. You demonstrated that the sum of the angles is 180° by showing that these angles can be arranged to form a straight angle. You have also demonstrated that any pair of adjacent angles formed at the intersection of lines adds up to 180°. There is another angle relationship that is based on 180° and involves intersecting lines.

 

vertically opposite angles: angles lying across from each other at the point where two lines intersect

 

Vertically opposite angles are also referred to as opposite angles.

Line 1 and line 2 intersect to form two pairs of angles called vertically opposite angles or, simply, opposite angles. In the diagram, ∠1 and ∠3 are vertically opposite angles. Also, ∠2 and ∠4 are vertically opposite angles.

 

Try This

 

In this activity you will explore how vertically opposite angles are related.

 

On the applet “Opposite Angles,” change the position of lines 1 and 2 to change the way the lines intersect.

 

Based on your investigation of vertically opposite angles in “Opposite Angles,” what conclusion could you make about the angles’ measures?

 

Answer the following questions to show, mathematically, how the sizes of the vertically opposite angles compare.

 

Self-Check

 

This illustration shows two lines intersecting. The lines are labelled line 1 and line 2. The angles are labelled counterclockwise from the top as 1, 2, 3, and 4.

First, you will show that ∠1 is congruent to ∠3.

 

SC 9. Complete the following statement. Then justify your answer.

 

 

 

∠1 + ∠2 = ____°.

 

SC 10. Complete the following statement. Then justify your answer.

 

 

 

∠3 + ∠2 = ____°.

 

SC 11. Using the results from SC 9 and SC 10, show (in a mathematical way) why ∠1 + ∠2 = ∠3 + ∠2.

 

SC 12. Use a similar approach to show that ∠2 is congruent to ∠4.

 

Compare your answers.

 

Try This

 

TT 1. Next, you will demonstrate that opposite angles are congruent by folding paper.

 

Step 1: Draw a pair of intersecting lines on a blank sheet of paper. Label the opposite angles as shown.

 

 

This illustration shows two intersecting lines. The angles are labelled counterclockwise from the top as 1, 2, 3, and 4.

 

Step 2: Use your protractor to measure the opposite angles. What do you notice?

 

Step 3: Explain how to fold your sheet of paper to demonstrate that the opposite angles are congruent.

 

Using Angle Relationships

 

In the following examples and practice questions, you will apply the angle relationships you examined. These angle relationships will include the following angles in a triangle:

  • vertically opposite
  • adjacent
  • complementary
  • supplementary

Example 2

 

A student studying geometry in architecture used the following picture to illustrate the angles made by the roof of a structure.

 

This is a photograph of a Kwakwaka’wakw “big house” located in Thunderbird Park in Victoria, BC. An overlay shows how the roof lines make up the sides of an isosceles triangle.

© David Strand

 

The angle at the top of the roof was measured to be 150°. The two angles at the base are equal. What is the measure of each angle?

This illustration shows an equilateral triangle with two angles of size x and one angle of size 150 degrees.

Solution

 

The sum of the measures of the three angles of the triangle is 180°. Therefore,

 

 

 

Check

 

 

 

Each angle at the base is 15°.

 

Sum of the Angles of a Triangle

 

Remember that when you rip off the corners of any triangle, you will see they form a line and add up to 180°.

 

This graphic shows a triangle and, to its right, its ripped-off angles are repositioned to form a straight angle.

 

Example 3

 

The W-type roof truss is the most common type of truss in simple wood-frame construction.

 

This illustration shows a W-type roof truss. Overlaid on the truss is a triangle. One side of the triangle joins the left end of the horizontal beam to the top point of the truss. The second triangle side joins the top to the bottom of the W. The third side joins the left end of the horizontal beam to the bottom left of the W and is extended farther to the right.

 

The angles will depend on the roof’s pitch. If the pitch of the roof is 32°, calculate the values of x and y.

 

This illustration shows the same shape as the overlay in the previous diagram. Added are the measures of the top and left angle, which are each 32 degrees. At the lower right, the angles are labelled x (inside the triangle) and y (outside the triangle).

 

Solution

 

First, find the value of x.

 

The sum of the angles in the triangle is 180°.

 

 

 

The angles with measures x and y are supplementary.

 

 

 

Example 4

 

Two straight paths cross at 30° as shown. Find the measures of the other three angles, which are represented by a, b, and c.

 

This illustration shows two intersecting line segments. The angles are labelled. The top angle is labelled b, while clockwise from the top the angles are 30 degrees, c, and a.

 

Solution

 

The angle with measure a is opposite the 30° angle. Therefore, a = 30°.

 

The angles with measures a and b are supplementary. So,

 

 

 

The angle with measure c is opposite b.

 

Therefore, c = b = 150°.

 

Example 5

 

Angles A and B are angles of a right triangle.

 

This illustration shows a right triangle labelled ABC with BC longer than AB.

 

Show that ∠A and ∠C are complementary.

 

Solution

 

Watch “Lesson 4: Example 5 Solution.”

 

Self-Check

 

Respond to the following questions.

 

SC 13.

 

This illustration shows right triangle ABC with the added line BD. Point D is the midpoint of side AC.

 

  1. Name two pairs of complementary angles.
  2. Name one pair of supplementary angles.

SC 14. Calculate the angle measures of a and b.

This illustration shows a vertical and a horizontal line segment that intersect. A third segment shares the meeting point and is drawn at an angle of 50 degrees below the left side of the horizontal line.

 

SC 15. Find the angle measures of a, b, c, and d for the rectangular envelope shown in the photograph. In the figure, ∠b = ∠c.

 

This illustration shows the flap side of a white envelope. Lines are drawn from the upper left corner along the top, down the left side, and along the left side of the flap. A segment is drawn from the lower left corner to the line along the flap, forming a triangle with both angles on the left side of the triangle equal.

© mayamaya/shutterstock

This illustration shows the overlay from the previous illustration. The angle between the top of the envelope and the flap is labelled d. The angle between the flap and the left side of the envelope is labelled b. The lower left angle in the triangle is labelled c. The third angle in the triangle is labelled a. The angle outside the triangle, between the flap and the line from the bottom of the envelope, is shown as measuring 110 degrees.

 


 

SC 16. The layout for a sheet-metal front-cap flashing to be positioned at the front of a chimney is shown. Find the missing measures a and b.

 

This illustration shows how a flat sheet of metal will become flashing. The flashing fits on a chimney and under the shingles. Fold marks are shown on the flat sheet of metal. There is an overlay on the right side of the flashing. The overlay consists of lines from the top right corner along the top and down the angled right side. A third line runs along the bottom horizontal fold and extends to the right.

 

This illustration shows the overlay from the flashing illustration. There is an additional line perpendicular to the top and bottom lines. It meets the angled line where they connect with the bottom horizontal line. The top angle in the triangle is labelled b. The bottom angle is labelled a. The exterior angle is shown measuring 50 degrees.

 

SC 17. Find the indicated measures.

 

This illustration shows two line segments intersecting. The angles from the top going clockwise are labelled b, a, c, and 20 degrees. On the right side of the illustration, a line is drawn between the ends of the two segments. The upper angle in the triangle is 100 degrees and the bottom angle is labelled d.

 

Compare your answer.

 

Mastering Concepts

 

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

 

This illustration shows triangle ABC with side BC extending from C and ending at D.

The angles within a triangle are called interior angles. If a side of a triangle is extended, an exterior angle is formed.

 

In ΔABC, ∠A, ∠B, and ∠ACB are interior angles. ∠ACD is an example of an exterior angle.

 

Show that the following is true: ∠A + ∠B = ∠ACD.

 

Compare your answer.

From the demonstration applet, it appears that vertically opposite angles are congruent (equal in measure).
The opposite angles are equal in measure.