Two distinct lines in a plane that never intersect, maintaining a constant distance from each other.
Parallel lines have the same slope in coordinate geometry and are fundamental to understanding angle relationships with transversals.
Card 2
Transversal
Answer
A line that intersects two or more other lines at distinct points.
Transversals create specific angle pairs (e.g., corresponding, alternate interior) whose relationships are determined by whether the intersected lines are parallel.
Card 3
Alternate Interior Angles
Answer
A pair of angles formed by a transversal intersecting two lines, located between the two lines and on opposite sides of the transversal.
If the two lines are parallel, alternate interior angles are congruent; students often confuse them with consecutive interior angles.
Card 4
Corresponding Angles
Answer
A pair of angles formed by a transversal intersecting two lines, located in the same relative position at each intersection.
If the two lines are parallel, corresponding angles are congruent, often visualized as angles that 'slide' into each other's position.
Card 5
Vertical Angles
Answer
A pair of non-adjacent angles formed by the intersection of two lines.
Vertical angles are always congruent, regardless of whether the intersecting lines are perpendicular or parallel to other lines.
Card 6
Linear Pair
Answer
Two adjacent angles that share a common vertex and a common side, and whose non-common sides form a straight line.
Angles in a linear pair are always supplementary, meaning their measures sum to 180 degrees.
Card 7
Triangle Angle Sum Theorem
Answer
The sum of the measures of the interior angles of any triangle is always 180 degrees.
This fundamental theorem allows unknown angle measures in a triangle to be calculated if the other two are known.
Card 8
Congruent Triangles
Answer
Two triangles are congruent if all three pairs of corresponding sides and all three pairs of corresponding angles are equal in measure.
Congruence means the triangles are exact copies of each other and can be perfectly superimposed; students often confuse this with similarity.
Card 9
CPCTC
Answer
An acronym for 'Corresponding Parts of Congruent Triangles are Congruent,' used to prove that specific sides or angles are congruent after two triangles have been proven congruent.
CPCTC is the justification used in two-column proofs to establish the congruence of individual parts once triangle congruence is established.
Card 10
SAS Congruence Postulate
Answer
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
The 'included angle' is crucial; it must be the angle *between* the two sides, not just any angle.
Card 11
Similar Triangles
Answer
Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.
Similar triangles have the same shape but not necessarily the same size, meaning one is a scaled version of the other.
Card 12
AA Similarity Postulate
Answer
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
This is the simplest criterion for proving triangle similarity because the third angles must also be congruent by the Triangle Angle Sum Theorem.
Card 13
Pythagorean Theorem
Answer
In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
This theorem applies *only* to right triangles and is used to find unknown side lengths or to prove if a triangle is right-angled.
Card 14
Isosceles Triangle Theorem
Answer
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Its converse is also true.
This theorem establishes a direct link between side lengths and angle measures in isosceles triangles.
Card 15
Radius of a Circle
Answer
A line segment connecting the center of a circle to any point on its circumference, or the length of that segment.
The radius is half the length of the diameter and is fundamental for calculating circumference and area.
Card 16
Chord of a Circle
Answer
A line segment whose endpoints both lie on the circumference of a circle.
The longest chord in any circle is its diameter, which passes through the center.
Card 17
Tangent Line to a Circle
Answer
A line that intersects a circle at exactly one point, called the point of tangency.
A tangent line is always perpendicular to the radius drawn to the point of tangency.
Card 18
Central Angle
Answer
An angle whose vertex is the center of a circle and whose sides are two radii.
The measure of a central angle is equal to the measure of its intercepted arc.
Card 19
Inscribed Angle
Answer
An angle whose vertex is on the circumference of a circle and whose sides are chords of the circle.
The measure of an inscribed angle is half the measure of its intercepted arc, a common point of confusion with central angles.
Card 20
Circumference of a Circle
Answer
The distance around the outside of a circle, calculated using the formula C = 2πr or C = πd.
Circumference is a one-dimensional measure, analogous to the perimeter of a polygon.
Card 21
Area of a Circle
Answer
The amount of two-dimensional space enclosed within the boundary of a circle, calculated using the formula A = πr².
Area is a two-dimensional measure, representing the surface covered by the circle.
Card 22
Volume of a Cylinder
Answer
The amount of three-dimensional space occupied by a cylinder, calculated by multiplying the area of its circular base by its height (V = πr²h).
This formula is an extension of the general volume formula for prisms (Base Area × Height), where the base is a circle.
Card 23
Volume of a Pyramid
Answer
The amount of three-dimensional space occupied by a pyramid, calculated as one-third of the area of its base multiplied by its height (V = (1/3)Bh).
The (1/3) factor distinguishes pyramid volume from prism volume with the same base and height.
Card 24
Distance Formula
Answer
A formula used to find the length of a line segment between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane: d = √((x₂ - x₁)² + (y₂ - y₁)²).
This formula is derived directly from the Pythagorean Theorem by forming a right triangle with the segment as its hypotenuse.
Card 25
Midpoint Formula
Answer
A formula used to find the coordinates of the midpoint of a line segment connecting two points (x₁, y₁) and (x₂, y₂) in a coordinate plane: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
The midpoint's coordinates are simply the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Card 26
Slope of a Line
Answer
A measure of the steepness and direction of a line, calculated as the ratio of the change in y-coordinates to the change in x-coordinates between any two points on the line (m = (y₂ - y₁)/(x₂ - x₁)).
Parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.
Card 27
Perpendicular Lines (Coordinate Geometry)
Answer
Two lines that intersect to form a right angle (90 degrees), and whose slopes are negative reciprocals of each other (unless one is vertical and the other horizontal).
The product of the slopes of two non-vertical perpendicular lines is -1; students often forget the special case of vertical/horizontal lines.
Card 28
Postulate
Answer
A statement that is assumed to be true without proof, serving as a basic building block for a mathematical system.
Postulates are foundational truths from which theorems are logically deduced, like Euclid's postulates.
Card 29
Theorem
Answer
A statement that has been proven to be true based on previously established definitions, postulates, and other proven theorems.
Unlike postulates, theorems require a formal proof to establish their validity.
Card 30
Deductive Reasoning
Answer
A logical process where a conclusion is reached by applying general rules or principles to specific cases, moving from the general to the specific.
Mathematical proofs primarily rely on deductive reasoning to establish the truth of theorems, ensuring the conclusion is certain if the premises are true.