REASONING ABOUT SIMILAR TRIANGLES

LESSON OBJECTIVE:Identify similar polygons and determine the scale

factor of similar polygons.

VOCABULARY: Scalene Triangle: No two sides

of a triangle are congruent. Isosceles Triangle: At least

two sides of a triangle are congruent.

Equilateral Triangle: All the sides of a triangle are congruent.

Equiangular Triangle: An acute triangle with all angles congruent.

Congruent Triangle: Triangles that are the same size and shape.

Congruence Transformation: If you slide, flip or turn a triangle, the shape does not change.

Similar Polygons: Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

Scale Factor: The ratio of the lengths of two corresponding sides of two similar polygons.

Proportion: A statement of equality between ratios.

Above, A’B’C’D’ ABCD. The symbol ~ means “is similar to.”

14

410

6

15

25

10

35

A’

D

B

C

A

C’D’

B’

𝑚 𝑚 and

== or equivalent AB== or equivalent = or equivalent == or equivalent

The constant is called the scale factor from quadrilateral ABCD TO quadrilateral A’B’C’D’. It scales (multiplies) the length of each side of quadrilateral ABCD to produce the length of the corresponding side of quadrilateral A’B’C’D’.a) What is the scale factor from

quadrilateral A’B’C’D’ to quadrilateral ABCD?

b) If two pentagons are similar, describe how to find the scale factor from the smaller pentagon to the larger pentagon. Then describe how to find the scale factor from the larger pentagon to the smaller pentagon.

c) Suppose and the scale factor from to is Write as many mathematical statements as you can about pairs of corresponding angles and about pairs of corresponding sides. Compare your statements with other students.

a) 2/5b) length of side larger

pentagon length of corresponding side of smaller pentagon

c) ; ∠Q= ∠Y; 𝑚 𝑚 ∠𝑚 R= ∠Z𝑚

Knowing that two triangles are similar allows you to conclude that the three pairs of corresponding angles are congruent and that the three pairs of corresponding sides are related by the same scale factor.

Conversely, if you know that the three pairs of corresponding angles are congruent and the three pairs of corresponding sides are related by the same scale factor, you can conclude that the triangles are similar.

Solving proportions:1) = 2) = 3) = 4) = 5) =

Several students at Black River High School made conjectures about families of polygons. Each student tried to outdo the previous student. For each claim, explain as precisely as you can why it is true or give a counterexample.

1. Monisha conjectured that all isosceles right triangles are similar.

2. Ahmed conjectured that all equilateral triangles are similar.

3. Loreen claimed that all squares are similar.

4. Jeff conjectured that all rhombi are similar.

5. Amy claimed that all regular hexagons are similar.

1. True. All isosceles right triangles are similar. Since the base angles of isosceles triangles are congruent, the two base angles for any isosceles right triangle measure 180 - 90/2 = 45º.

2. True. All equilateral triangles are similar. Since the angles of any equilateral triangle each measure 60º, corresponding angles are congruent.

3. True. All squares are similar. Since all angles are right angles, corresponding angles have the same measure.

4. False. All rhombi are not similar. A rhombus is a square, but a square is not a rhombus. A square has four right angles but a rhombus does not.

5. True. All regular hexagons are similar because a regular hexagon is a 6-sided polygon in which all sides and angles are congruent.

HOMEWORK!!

Complete the two homework sheets. You are allowed to work in groups. Have a great day!!