Proportions and Similarity
Using Proportions
An equation that equates two ratios is a proportion. For instance, if the ratio a/b is equal to the ratio c/d , then the following proportion can be written:
 The numbers a and d are the extremes of the proportion. The numbers b and c are the means of the proportion.
PROPERTIES OF PROPORTIONS
1. Cross Product Property The Product of the extremes equals the product of the means.
If  , then ad = bc .
2. Reciprocal Property If two ratios are equal, then their reciprocals are also equal.
If  , and 
Example: Solving Proportions
Solve the proportions.
Solution:  Write original proportion.
 Reciprocal property
x =  Multiply each side by 4.
 Simplify.
The geometric mean of two positive numbers a and b is the positive number x such that  . If you solve this proportion for x , you find that  , which is a positive number.
For example, the geometric mean of 8 and 18 is 12, because  , and also because 
Similar Polygons
IDENTIFYING SIMILAR POLYGONS
When there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons .
In the diagram, ABCD is similar to EFGH . The symbol ~ is used to indicate similarity. So, ABCD ~ EFGH .
Example: Writing Similarity Statements
Pentagons JKLMN and STUVW are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.
Solution: Because JKLMN ~ STUVW , you can write Ð J @ Ð S , Ð K @ Ð T , Ð L @ Ð U , Ð M @ Ð V , and Ð N @ Ð W .
You can write the statement of proportionality as follows:
If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. In Example 2 on the previous page, the common ratio of  is the scale factor of WXYZ to PQRS .
THEOREM
If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.
If KLMN ~ PQRS , then
POSTULATE - Angle - Angle(AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
If Ð JKL @ Ð XYZ and Ð KJL @ Ð YXZ ,
then D JKL ~ D XYZ .
THEOREMS
Theorem 1 Side - Side - Side ( SSS ) Similarity Theorem
If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
If  ,
then D ABC ~ D PQR,
Theorem 2 Side - Angle - Side (SAS) Similarity Theorem
If an angle of one triangle 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 Ð X @ Ð M and 
then D XYZ ~ D MNP.
Theorem 3 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
If  , then 
Theorem 4 Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
If  , then 
Theorem 5
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
If r || s and s || t, and t and m intersect r, s, and t, then  .
Theorem 6
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
If  bisects Ð ACB, then  .
Congruence And Triangles
Identifying Congruent Figures
Two geometric figures are congruent if they have exactly the same size and shape.
When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. For the triangles below, you can write D ABC @ D PQR, which is read “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence.
Corresponding angle Corresponding sides
Ð A @ Ð P 
Ð B @ Ð Q 
Ð C @ Ð R 
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There is more than way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write D BCA @ D QRP.
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Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If Ð A @ Ð D and Ð B @ Ð E
Then Ð C @ Ð F.
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