Similarity and Quadrilaterals
Similarity
In geometry, two objects are similar (~) if the ratio of their corresponding sides are equal.
In the figures on the left:
Similar objects' corresponding angles are also equal. Figures could also be rotated, reflected, or translated, making similarities sometimes harder to identify.
Example 1. Find the values of the missing side lengths in the figure below.
We know that <ABC = <DEF and <ACB = <DFE, so <BAC = <EDF. Corresponding angles of both triangles are equal, so
Since corresponding side ratios are the same, we can set up equations and solve for the unknown lengths x and y.
Quadrilaterals
A quadrilateral is a 2D figure with four edges and vertices. The sum of the interior angles is 360 degrees.
Example 2. The perimeter of the rectangle is 12. Find the length of the longer side.
Since the sum of all four sides is 12, we can set up an algebraic equation and solve for x.
The longer side is 2x, so its length is 2*2 = 4.
Example 3. Quadrilaterals ABCD and WXYZ are similar. Find the measure of angle X.
Since ABCD and WXYZ are similar, <X = <B. We can find angle B by subtracting angles A, B, and C from 360 degrees.
Therefore,
It is important to remember that the remaining three angles in WXYZ are also equal to their corresponding angles in ABCD.