2 Angles and Measurements

Dimension

(a) Flat figures have two of these (which can be labeled "base" and "height"); (b) Solids have three of these (such as "width," "height," and "depth").

Base

(a) Triangle: Any side of a triangle to which a height is drawn. There are three of these possible in each triangle; (b) Trapezoid: the two parallel sides; (c) Parallelogram (and rectangle, rhombus, and square): Any side to which a height is drawn. There are four of these possible.

Height

(a) Triangle: the length of a segment that connects a vertex of the triangle to a line containing the opposite base (side) and is perpendicular to that line; (b) Trapezoid: the length of any segment that connects a point on one base of the trapezoid to the line containing the opposite base and is perpendicular to that line; (c) Parallelogram (includes rectangle, rhombus, and square): the length of any segment that connects a point on one base of the parallelogram to the line containing the opposite base and is perpendicular to that line; (d) Pyramid and cone: the length of the segment that connects the apex to a point in the plane containing the figure's base and is perpendicular to that plane; (e) Prism or cylinder: the length of a segment that connects one base of the figure to the plane containing the other base and is perpendicular to that plane.

Corresponding Angles

(a) 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 this. Note that if the two lines cut by the transversal are parallel, these angles are congruent. Conversely, if these angles are congruent, then the two lines intersected by the transversal are parallel; (b) Angles in two figures may also be this.

Theorem

A conjecture that has been proven to be true.

Slope-Intercept Form

A form of a linear equation: y = mx + b. In this form, m is the slope and the point (0, b) is the y-intercept.

Transversal

A line that intersects two or more other lines on a flat surface (plane).

Substitution Method

A method for solving a system of equations by replacing one variable with an expression involving the remaining variable(s). (See Math Note 2.1.3 for example of it being used.)

Square Root

A number a is a square root of b if a squared = b. For example, the number 9 has two of these, 3 and −3. A negative number has no real number of this; a positive number has two; and zero has just one, namely, itself. In a geometric context, the principal square root of a number x (written √x ) represents the length of a side of a square with area x.

Arrow Diagram

A pictorial representation of a conditional statement. The arrow points toward the conclusion of the conditional (the "then" part of the "If... then" statement).

Point(s) of Intersection

A point that the graphs of two or more equations have in common. Graphs may intersect in one, two, several, many or no points. The set of coordinates of this are a solution to the equation for each graph.

Proof by Contradiction

A proof that begins by assuming that an assertion is true and then shows that this assumption leads to a contradiction of a known fact. This demonstrates that the assertion is false. Also known as an indirect proof.

Trapezoid

A quadrilateral with at least one pair of parallel sides.

Rhombus

A quadrilateral with four congruent sides.

Square

A quadrilateral with four right angles and four congruent sides.

Rectangle

A quadrilateral with four right angles.

Parallelogram

A quadrilateral with two pairs of parallel sides.

Unit of Measure

A standard quantity (such as a centimeter, second, square foot, or gallon) that is used to measure and describe an object. A single object can be measured using different ones of these, which will usually yield different results. For example, a pencil may be 80 mm long, meaning that it is 80 times as long as a unit of 1 mm. However, the same pencil is 8 cm long, so that it is the same length as 8 cm laid end-to-end. (This is because 1 cm is the same length as 10 mm.)

Conditional Statement

A statement written in "If ..., then ..." form. For example, "If a rectangle has four congruent sides, then it is a square."

Right Triangle

A triangle that has one right angle. The side of this opposite the right angle is called the hypotenuse, and the two sides adjacent to the right angle are called legs. Note that legs of this are always shorter than its hypotenuse.

Alternate Interior Angles

Angles between a pair of lines that switch sides of a third intersecting line (called a transversal). These angles are not adjacent.

Triangle Inequality

In a triangle with side lengths a, b, and c, c must be less than the sum of a and b and greater than the difference of a and b.

Line

One-dimensional and extends without end in two directions. It is made up of points and has no thickness. It can be named with a letter (such as l), but also can be labeled using two points on it, such as AB. It is a mathematically undefined term in geometry.

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 this.

Pythagorean Theorem

The statement relating the lengths of the legs of a right triangle to the length of the hypotenuse: leg #1 squared + leg#2 squared = hypotenuse squared. This theorem is powerful because if you know the lengths of any two sides of a right triangle, you can use this relationship to find the length of the third side.

Triangle Angle Sum Theorem

The sum of the measures of the interior angles in any triangle is 180°.

Vertical Angles

The two opposite (that is, non-adjacent) angles formed by two intersecting lines. This is a relationship between pairs of angles, so one angle cannot be called this. Angles that form this are always congruent.

Legs

The two sides of a right triangle that form the right angle. Note that these are always shorter than its hypotenuse.

System of Linear Equations

This is a set of linear equations with the same variables. Solving this means finding one or more solutions that make each of the equations in the system true. A solution to this gives a point of intersection of the graphs of the equations. There may be zero, one, or several solutions to this.

Supplementary Angles

Two angles a and b for which a + b = 180°. Each angle is called the supplement of the other.

Same-Side Interior Angles

Two angles between two lines and on the same side of a third line that intersects them (called a transversal). Note that if the two lines that are cut by the transversal are parallel, then the two angles are supplementary (add up to 180°). Conversely, if the two angles are supplementary, then the two lines that are cut by the transversal are parallel.

Complementary Angles

Two angles whose measures add up to 90°.

Equivalent

Two expressions are this if they have the same value.

Congruent

Two shapes are this if there is a sequence of rigid transformations that carries one onto the other. The corresponding angles and sides of polygons have equal measures. These shapes are similar and have a scale factor of 1. The symbol is ≅.