3 Justification and Similarity

Corresponding Sides

Sides in two or more figures that are images of each other with respect to a sequence of transformations. If two figures are congruent, these sides are congruent to each other.

Flowchart

A diagram showing an argument for a conclusion from certain evidence. This diagram uses ovals connected by arrows to show the logical structure of the argument. When each oval has a reason stated next to it showing how the evidence leads to that conclusion, this represents a proof.

Enlarge

A figure that has been dilated and is larger than the original.

Logical Argument

A logical sequence of statements and reasons that lead to a conclusion. This can be written in a paragraph, represented with a flowchart, or documented in a two-column proof.

Tree Diagram

A model used to organize the possible outcomes, and the respective probabilities, of two or more events.

Similarity Statement

A statement that indicates that two figures are similar. The order of the letters in the names of the shapes determine which sides and angles correspond to each other. For example, ΔABC ~ ΔDEF is this. It indicated that ∠A must correspond to ∠D and side AB must correspond to side DE.

Dilation

A transformation which produces a figure similar to the original by proportionally shrinking or stretching the figure. In this, a shape is stretched (or compressed) proportionally from a point.

Proportional Equation

An equation stating that two ratios are equal. For example, the equation below is this. This is a useful type of equation to set up when solving problems involving proportional relationships. "68 votes for Mr.Mears" /"100 people surveyed" ="34 votes for Mr.Mears" /"50 people surveyed"

Similarity Transformations

Movements of figures that preserve their shape, but not necessarily their size. Examples of these are reflections, rotations, translations, and dilations.

Relationship

For this course, this is a way that two objects (such as two line segments or two triangles) are connected. When you know this holds between two objects, learning about one object can give you information about the other. These can be described in two ways: geometric (such as a pair of vertical angles or two line segments that are parallel) and this between the measures (such as two angles that are complementary or two sides of a triangle that have the same length). Common geometric these things are between two figures include being similar (when two figures have the shape, but not necessarily the same size) and being congruent (when two figures have the same shape and the same size).

AA Triangle Similarity (AA ~)

If two angles of one triangle are congruent to the two corresponding angles of another triangle, then the triangles are similar. For example, given ΔABC and ΔA′B′C′ with ∠A ≅ ∠A′ and ∠B ≅ ∠B′, then ΔABC ~ ΔA′B′C′. You can also show that two triangles are similar by showing that three pairs of corresponding angles are congruent (which would be called AAA ~), but two pairs of angles are sufficient to demonstrate similarity.

SSS Triangle Similarity (SSS ~)

If two triangles have all three pairs of corresponding sides that are proportional (this means that the ratios of corresponding sides are equal), then the triangles are similar.

SAS Triangle Similarity (SAS ~)

If two triangles have two pairs of corresponding sides that are proportional and have congruent included angles, then the triangles are similar.

Zoom Factor (Scale Factor)

The amount each side of a figure is multiplied by when the figure is proportionally enlarged or reduced in size. It is written as the ratio of a length in the new figure (image) to a length in the original figure (pre-image).

Point (Center) of Dilation

The point from which a figure is stretched proportionally when the figure is dilated. Notice that while this changes the size and location of the original figure, it does not rotate or reflect the original. While lengths can change, angles do not change under this.

Ratio of Similarity

The ratio of any pair of corresponding sides of two similar figures. This means that once it is determined that two figures are similar, all of their pairs of corresponding sides have the same ratio. This can also be called the linear scale factor. When the ratio is comparing a figure and its image after a dilation, this ratio can also be called the zoom factor.

Similar Figures

Two shapes are this if there is a sequence of rigid motions, followed by a dilation, that carries one onto the other. The corresponding angles of these polygons are congruent, and the corresponding sides are proportional. The symbol for this is ~.