Chapter 3 math terms

Proportional Equation

An equation stating that two ratios are equal. For example, the equation below is a proportion. A proportion is a useful type of equation to set up when solving problems involving proportional relationships.

Flowchart

A diagram showing an argument for a conclusion from certain evidence. A flowchart 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, the flowchart represents a proof. See the example below.

Ratio

A ratio compares two quantities by division. A ratio can be written using a colon, but is more often written as a fraction. For example, in the two similar triangles below, a ratio can be used to compare the length of in ΔABC with the length of in ΔDEF. This ratio can be written as 5:11 or as the fraction .

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, is a similarity statement. It indicated that ∠A must correspond to ∠D and must correspond to .

Translate

A transformation that preserves the size, shape, and orientation of a figure while sliding (moving) it to a new location. The result is called the image of the original figure (preimage). Note that a translation is sometimes referred to as a "slide."

Dilation

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

Vertex

For a two-dimensional geometric shape, a vertex is a point where two or more line segments or rays meet to form a "corner," such as in a polygon or angle. (b) For a three-dimensional polyhedron, a vertex is a point where the edges of the solid meet. See also apex. (c) On a graph, a vertex can be used to describe the highest or lowest point on the graph of a parabola or absolute value function (depending on the graph's orientation).

Relationship

For this course, a relationship is a way that two objects (such as two line segments or two triangles) are connected. When you know that the relationship holds between two objects, learning about one object can give you information about the other. Relationships can be described in two ways: a geometric relationship (such as a pair of vertical angles or two line segments that are parallel) and a relationship between the measures (such as two angles that are complementary or two sides of a triangle that have the same length). Common geometric relationships 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).

Angle Triangle Similarity

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

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

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

Angle

In general, an angle is formed by two rays joined at a common endpoint. Angles in geometric figures are usually formed by two segments, with a common endpoint (such as the angle shaded in the figure below).

Similarity Transformation

Movements of figures that preserve their shape, but not necessarily their size. Examples of similarity transformations are reflections, rotations, translations, and dilations. See also rigid transformations.

Zoom 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 (preimage). For the triangles at right, the zoom factor is or . See also ratio of similarity.

Perimeter

The distance around the exterior of a figure on a flat surface. For a polygon, the perimeter is the sum of the lengths of its sides. The perimeter of a circle is also called a circumference.

Hypotenuse

The longest side of a right triangle (the side opposite the right angle). Note that legs of a right triangle are always shorter than its hypotenuse.

Similar

Two shapes are similar if they have exactly the same shape but are not necessarily the same size. Similar polygons have congruent angles, but not congruent sides - the corresponding sides are proportional. The symbol for similar is ~ .

Corresponding Sides

When two lines are intersected by a third line (called a transversal), angles on the same side of the two lines and on the same side of the transversal are called corresponding angles. For example, the shaded angles in the diagram at right are corresponding angles. Note that if the two lines cut by the transversal are parallel, the corresponding angles are congruent. Conversely, if the corresponding angles are congruent, then the two lines intersected by the transversal are parallel; (b) Angles in two figures may also correspond, as shown in the figure in corresponding parts. See also alternate interior angles and same-side interior angles. corresponding parts

Congruent

two shapes are congruent if they have exactly the same shape and size. Congruent shapes are similar and have a scale factor of 1. The symbol for congruence is