1 ▶︎▶︎▶︎ MEANS-EXTREMES PROPERTY, or CROSS-PRODUCTS PROPERTY (also known as Property 1):

➜ If a/b = c/d, then a • d = b • c.

➜ Conversely, if a • d = b • c, and neither a • d nor b • c equals zero, then a/b = c/d and b/a = d/c

For example, for the proportion 8/10 = 4/5,

the Means-Extremes Property (Property 1) specifies 8 • 5 = 10 • 4, or 40 = 40.

2 ▶︎▶︎▶︎ MEANS OR EXTREMES SWITCHING PROPERTY (also known as Property 2):

➜ If a/b = c/d and is a proportion, then both d/b = c/a and a/c = b/d are proportions.

For example, for the proportion 8/10 = 4/5,

the Means or Extremes Switching Property (Property 2) specifies that if you were to switch the 8 and 5 or switch the 4 and 10, then the new statement is still an accurate proportion.

If 8/10 = 4/5, then 5/10 = 4/8, OR if 8/10 = 4/5, then 8/4 = 10/5.

3 ▶︎▶︎▶︎ UPSIDE-DOWN PROPERTY (also known as Property 3):

➜ If a/b = c/d, then b/a = d/c.

For example, if 9 • a = 5 • b, and the product ≠ 0, then find the ratio for a/b.

First, apply the converse of the Cross Products Property and obtain 9/5 = b/a.

Next, proceed in one of the following two ways:

☛ Apply Property 3 to 9/5 = b/a:

Turn each side upside-down.

5/9 = a/b, or a/b = 5/9

☛ Apply Property 2 to 9/b = 5/a:

Switch the 9 and the a, so that a/b = 5/9

4 ▶︎▶︎▶︎ DENOMINATOR ADDITION/SUBTRACTION PROPERTY (also known as Property 4):

➜ If a/b = c/d, then (a + b)/ b = ( c + d)/ d or (a − b)/ b = (c − d)/ d.

Copy and paste the following link into your browser to learn more about using the four properties of proportion in geometry:

https://youtu.be/fvtlFmRA5sY Two polygons with the same shape are called SIMILAR POLYGONS. The symbol for "is similar to" is ∼. Notice that it is a portion of the "is congruent to" symbol, ≅. When two polygons are similar, these two facts BOTH must be true:

➜ Corresponding angles are equal.

➜ The ratios of pairs of corresponding sides must all be equal.

Two geometrical objects are called SIMILAR if they both have the same shape, or one has the same shape as the mirror image of the other.

More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.

This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object.

If two objects are similar, each is CONGRUENT to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other.

On the other hand, ellipses are NOT all similar to each other, rectangles are NOT all similar to each other, and isosceles triangles are NOT all similar to each other.

Copy and paste the following link into your browser to learn more about using similar polygons in geometry:

https://youtu.be/AFEDUm4bPHk