Unit 4: Similarity

Question 1

If a and b are two numbers or quantities and b≠0, then the ratio of a to b is…

There are 2 ways!

Answer 1

a/b (as a fraction)
OR
a:b

Question 2

Here’s an example:

Answer 2

The ratio of a side length in △ABC to a side length in △DEF can be written as 2/1 or 2:1

Question 3

Two ratios that have the same simplififed form are called…

Answer 3

Equivalent Ratios

Ratios are usually expressed in simplest form.

Question 4

Here’s an example:

Answer 4

The ratios 7:14 and 1:2 in the example below are equivalent:

width of RSTU : length of RSTU = 7ft:14ft = 1:2

Question 5

An equation that states 2 ratios are equal is called a…

Answer 5

Proportion

Question 6

Cross Products Property is…

Answer 6

…the product of the extremes equals the products of the means:

extreme -> a c <- mean
— = —
mean –> b d <– extreme

If a/b = c/d where b≠0 and d≠0, then ad=bc:

2/3 = 4/6 –> 3(4) = 2(6) –> 12 = 12

Question 7

What is the geometric mean?

Answer 7

The geometric mean of two positive a and b is the positive number x that satisfies:

a/x = x/b –> x^2 = ab –> x = √ab

Question 8

What is Reciprocal Property?

Answer 8

If two rations are equal, then their reciprocals are also equal.

If you interchange the means of a proportion, then you for another true proportion.

In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another true proportion.

Question 9

A Scale Drawing is…

Answer 9

…a drawing that is the same shape as the object it represents. The SCALE is a ratio that describes how the dimensions in a drawing are related to the actual dimensions of the object.

Question 10

Two polygons are Similar Polygons if…

Answer 10

…corresponding angles are congruent AND corresponding side lengths are proportional.

“△ABC is similar to △XYZ” is written as “△ABC~△XYZ”

Order of the letters MATTER!! (Just like congruent polygons)

Question 11

Perimeters of Similar Polygons:

Answer 11

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

If △KLM~△PQR, then…

KL+LM+KM/PQ+QR+PR –>

KL/PQ = LM/QR = KM/PR

Question 12

Corresponding Lengths in Similar Polygons:

Answer 12

If two polygons are similar, then the ratio of any two corresponding lengths is equal to the scale factor of the similar polygon.

Question 13

Here’s an example:

Answer 13

In △ABC, AB = 5, BC = 7, AC = 6
In △DEF, DE = 5, EF = 7, DF = 6

In △ABC and △DEF, the scale factor is 5/5 = 1.

You can write △ABC~△DEF and △ABC≅△DEF.

NOTICE that any two congruent figures are also similar. Their scale factor is 1:1.

Question 14

Angle-Angle (AA) Similarity Postulate:

Answer 14

If two angles of one triangle are congurent to two angles of another triangle, then the two triangles are similar.

Question 15

Side-Side-Side (SSS) Similarity Theorem:

Answer 15

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

If AB/RS = BC/ST = CA/ RT, then △ABC~△RST

Question 16

Side-Angle-Side (SAS) Similarity Theorem:

Answer 16

If an angle of one triangel is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

If <X≅<M, and ZX/PM = XY/MN, then △XYZ~△MNP

Question 17

Triangle Similarity Theorem:

Answer 17

If two triangles are similar, then pairs of corresponding sides have the same ratio and pairs of corresponding angles are congruent.

Question 18

Equilateral Triangle Similarity Theorem:

Answer 18

All equilateral triangles are similar to each other.

Question 19

If we know that two triangles are congruent, we can conclude that any pair of corresponding parts (sides or angles), often abbreviated as CPCTC. With similar triangles we need to make a distinction. We can say that:

Corresponding —— of similar triangles are congruent.

Corresponding —– of similar triangles are proportional.

Answer 19

Corresponding ANGLES of similar triangles are congruent. (CASTC)

Corresponding SIDES of similar triangles are proportional. (CSSTP)

TIp: Remember abbrevations by remembering words!

Question 20

Triangle Proportionality Theorem:

Answer 20

If a line parallel to one side of a trangle intersects the other two sides, then it divides the two sides proportionally.

Question 21

Converse of the Tirangle Proportionality Theorem:

Answer 21

If a lie divides two sides of a triangle proportionally, then it is parallel to the third side.

Question 22

Three Parallel Lines Theorem:

Answer 22

If three parallel lines intersect at two transversals, then they divide the transversals proportionally.

Question 23

Triangle Angle Bisector Theorem:

Answer 23

If a ray bisects an angle of a tringle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.