What you should learn

To find the unknown measures of the sides of two similar triangles

NCTM Curriculm Standards 2 - 4, 6 - 9

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Corresponding Angles

Corresponding Sides

Introduction: "Rock n' roll is here to stay," is what you might hear after visiting the Rock and Roll Hall of Fame and Museum in Cleveland, Ohio. Memorabilia from musicians such as the Beatles, U2, and Carlos Santana cover the walls inside.

The building itself, designed by architect I.M. Pei, is made up of several geometric forms - a triangle, a cylinder, a rectangle, and a trapezoid. The triangular section of the building is similar to the one Pei designed for the Louvre in Paris, France. Both are made up of similar triangles.

Two figures are similar if they have the same shape, but not necessarily the same size.

If corresponding angles of two triangles have equal measures, the triangles are similar. The two triangles above are similar. We write this as ABC ~ DEF. The order of the letters indicates the angles that correspond.

A and D

B and E

C and F

The side opposite corresponding angles are called corresponding sides. Compare the measures of the corresponding sides. Note that AC means the measure of AC, DF means the measure of DF and so on.

AB/DE = 6/18 = 1/3

BC/EF = 5/15 = 1/3

AC/DF = 3/9 = 1/3

When the measures of the corresponding sides form equal ratios, the measures are said to be proportional.

Similar Triangles: If two triangles are similar, the measures of their corresponding sides are proportional, and the measures of their corresponding angles are equal.

 

Proportions can be used to find the measures of the sides of similar triangles when some measurements are known.

Exercise 1: Surveyors normally use instruments to measure objects that are too large or too far away to measure by hand. They also can use the shadows that objects cast to find the height of the objects without measuring them directly. How can a surveyor use a telephone pole that is 25 feet tall and casts a shadow 20 feet long to find the height of a building that casts a shadow 52 feet long?

NOP is similar to MRQ

RQ/OP = QM/PN

25/x = 20/52

20x = 25(52)

20x = 1300

x = 65

The building is 65 feet high.

Similar triangles may be positioned so that the corresponding parts are not obvious.

Exercise 2: Find the missing measures of the sides if each pair of trianges below are similar.

Remember that the sum of the measures of the angles in a triangle is 180 degrees.

a.

The measure of Q is 180 - (135 + 27) or 36

The measure of T is 180 - (135 + 36) or 27

Since the corresponding angles have equal measures, QRS ~ VUT. This means that the lengths of the corresponding sides are proportional...

b. NRT is similar to MST

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 13 - 33 odd, 35 - 41

Alternative Homework: Enriched: 12 - 28 even, 30 - 41

Extra Practice: Students book page 764 Lesson 4-2

Extra Practice Worksheet: Click Here.

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