1. Module 6

1.14. Page 4

Mathematics 10-3 Module 6 Lesson 3

Module 6: Triangles and Other Polygons

 

Bringing Ideas Together

 

In the Explore section, you investigated three methods for drawing similar triangles. These methods are based on the three conditions listed in the following table.

 

TRIANGLE CONDITIONS FOR SIMILARITY

Graphic Representation

Description in Words

Symbolic Description

Nickname

Condition 1

 

This illustration shows triangles ABC and A'B'C'. Angles A and A' are marked with a five-pointed star. Angles B and B' are marked with a single arc. Angles C and C' are marked with a four-pointed star.

If the corresponding angles of two triangles are equal in measure, then the triangles are similar.

e.g., AAA Similarity

Condition 2

 

This illustration shows triangles ABC and DEF. Sides AB and DE are marked with five-pointed stars. Sides BC and EF are marked with four-pointed stars. Sides AC and DE are marked with a lightning bolt.

If the corresponding sides of two triangles are proportional, the triangles are similar.

 

Condition 3

 

This illustration shows triangles ABC and DEF. Sides AB, AC, DE, and DF are drawn in red. Angles A and D are marked with a five-pointed star.

If two pairs of sides of two triangles are proportional, and the angles between those pairs of sides are congruent, the triangles are similar.

 

 

Nicknames for the conditions may help you keep the three conditions straight. A possible nickname is provided in the table for Condition 1.

 

On a copy of “Triangle Conditions for Similarity,” fill in your own nicknames for the conditions.

 

Place your completed table in your course folder.

 

Note: When two figures are similar, you can use the symbol ~ to replace the words “is similar to.”

 

The following examples show how these conditions can be used to solve problems involving similar triangles.

 

Example 2: Using Condition 1

 

A 3-m pole and a 4-m pole are leaning against a vertical wall. Each pole makes an angle of 80° with the ground.

  1. Are the triangles formed by the poles similar? Why?
  2. If the 4-m pole reaches 3.94 m up the wall, how far up the wall does the 3-m pole reach?

Solution

  1. This illustration shows two right triangles of different sizes. In both triangles the base is horizontal and is the shortest side. The base angles are 90 degrees and 80 degrees in both triangles. In the smaller triangle, the non-base sides are labelled x and 3 metres. The corresponding sides in the larger triangle are labelled 3.94 metres and 4 metres.

    Assume the vertical wall is perpendicular to the ground. All of the corresponding angles are equal.

    The two 80° angles are equal in measure. The two right angles are equal in measure. So, the third angle in each triangle must be 10°, since 10° + 80° + 90° = 180°.

    Note: This example confirms the fact that if two pairs of angles are equal in measure (congruent), so too is the third pair.

    As the three pairs of corresponding angles are congruent, these triangles are similar.

  2. As the triangles are similar, the sides are proportional.

    Let x be the height up the wall the 3-m pole reaches.

     


    The 3-m pole reaches approximately 2.96 m up the wall.

Example 3: Using Condition 2

 

Two triangular sails from a model ship have the following dimensions.

 

This illustration shows two triangular sails, ABC and DEF. ABC has sides measuring 6 centimetres, 8 centimetres, and 10 centimetres. DEF has sides that measure 4.8 centimetres, 6.4 centimetres, and 8 centimetres.

 

Are the corresponding angles congruent?

 

Solution

 

Use the indicated lengths of the sides to see if corresponding sides are proportional.

 

 

 

 

Because the ratios are equal, the triangles are similar.

 

Since the corresponding angles are congruent.

 

Example 4: Using Condition 3

 

In the following figure there are two triangles—one triangle is inside the other.

 

This illustration shows triangles ABC and ADE. DE is a segment that joins point D on side AB with point E on side AC. AD has a measure of 2 units. DB has a measure of 3 units. AE has a measure of 4 units. EF has a measure of 6 units.

 

Is

 

View the animated “Example 4 Solution: Finding Lengths in Similar Triangles.”

 

Similar Right Triangles

 

From a couple of the examples involving right angles, you may have noticed this condition—if just one pair of corresponding acute angles are equal in measure, the triangles are similar.

 

More generally, for any two triangles, if two pairs of corresponding angles are equal in measure, the triangles are similar.

 

This general statement can be represented this way.

 

This illustration shows triangles PQR and ABC. Angles Q and B are marked with a single arc. Angles R and C are marked with a double arc.

 

The general statement follows from Condition 1. Why?

 

 

 

 

The reasoning is this: If triangles PQR and ABC have two pairs of angles that are equal, the other pair of angles must also be equal because the angles in each triangle must add up to 180°.

 

Self-Check

 

SC 4. Jill said to her partner, “For any two triangles, if two pairs of corresponding angles are equal in measure, the triangles are similar.”

 

Harlon said that, from Jill’s statement, he could then conclude the following about right triangles:

 

 

 

For right triangles, if just one pair of corresponding acute angles are equal in measure, the right triangles are similar.

 

How could Harlon make this conclusion about right triangles?

 

SC 5. Look at in the following diagram.

 

This illustration shows Triangle ABC with angle A marked as a right angle. Segment DE joins point D on side BC with point E on side AC. Angle EDC is a right angle.

 

Are similar? Why or why not?

 

SC 6. Name the equal ratios in SC 5. If there are none, explain why there are none.

 

SC 7. In the following diagram,

 

This illustration shows segments AB and DC meeting at point X. Segments AC and BD are marked as parallel.

 

Are similar? Why or why not?

 

SC 8. Look at the following triangles.

 

This illustration shows two right triangles, ABC and DEF. Angles B and D are right angles. AB measures 6 cm, BC measures 12.6 cm, DE measures 3 cm, and EF measures 6.4 cm.

 

Are similar? If they are, explain why they are. If they are not, what change could you make in one or more of the measures given so that the triangles would be similar?

 

SC 9. Identify in the diagram.

 

This illustration shows triangle ABC with segment DE drawn parallel to side BC. D is a point on AB, and E is a point on AC.

 

Suppose Are similar triangles? Why or why not?

 

SC 10. Expand your copy of the table “Triangle Conditions for Similarity,” presented near the beginning of this lesson, to include the two new conditions explored in this Self-Check activity. You should have a copy of this table in your course folder; if not, go to “Triangle Conditions for Similarity” for another copy.

 

Graphic Representation

Description in Words

Symbolic Description

Nickname

Condition 4

 

This illustration shows right triangles PQR and ABC. Angles Q and B are right angles, and angles R and C are marked with a single arc.

 

 

 

Condition 5

 

This illustration shows triangles ABC and A'B'C'. Angles A and A' are marked with five-pointed stars. Angles C and C' are marked with four-pointed stars.

 

 

 

 

SC 11. Which diagram makes it easier to see the corresponding parts of the two triangles? Explain your answer.

 

  1. This illustration shows triangles ABC and A'B'C'. Angles A and A' are marked with five-pointed stars. Angles C and C' are marked with four-pointed stars. Triangle ABC is shown above triangle A'B'C', and is placed with side BC horizontal and point A above BC. Triangle A’B’C’ is placed so side B'C' is vertical and point A is to the right of B'C'.

  2. This illustration shows triangles ABC and A'B'C'. Angles A and A' are marked with five-pointed stars. Angles C and C' are marked with four-pointed stars. Triangle ABC is shown above triangle A'B'C', and is placed with side BC horizontal and point A above BC. Triangle A'B'C' is placed so side B'C' is vertical, but A' is to the left of B'C'.

  3. This illustration shows triangles ABC and A'B'C'. Angles A and A' are marked with five-pointed stars. Angles C and C' are marked with four-pointed stars. Triangle ABC is shown above triangle A'B'C', and is placed with side BC horizontal and point A above BC. Triangle A'B'C' is placed so side B'C' is horizontal with point A' above B'C'.

Compare your answers.

 

Mastering Concepts

 

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

 

The diagram shows . joins point D on side AB with point E on side AC.

 

This illustration shows triangle ABC with segment DE joining point D on side AB with point E on side AC. D bisects side AB. E bisects side AC. AD and DB have a length of 3 units. AE and EC have a length of 4 units. Side BC has a length of 10 units.

  1. In the diagram, is Why or why not?
  2. What is the length of

Compare your answers.