Geometry Review: Part A – Flashcards
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Undefined terms
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Point, line and plane. They have no formal definitions but have an agreed meaning.
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Defined terms
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Segment, rays. They can be described using known words.
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Line segment
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Consists of 2 endpoints and a connecting line
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Ray
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Consists of 1 endpoint; travels in 1 direction
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Opposite ray
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Two rays that travel in opposite directions
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Postulate (Axiom)
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A rule that is accepted without proof
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Congruent segments
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Segments that have the same length
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Midpoint
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A point that divides a segment into 2 congruent segments.
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Segment bisector
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A point, ray, line, line segment or plane that divides a segment into 2 congruent segments
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Acute angle
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Angles that measure 0<x<90 degrees
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Right angle
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Angles that measure 90 degrees
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Obtuse angle
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Angles that measure 90<x<180 degrees
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Straight angle
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Angles that measure 180 degrees
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Congruent angles
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Angles that have the same degrees
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Angle bisector
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A ray that divides an angle into 2 congruent angles
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Linear pair
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Two adjacent angles that have a non-common side which are opposite rays
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Vertical angles
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Both sides of each angles are opposite rays. ><
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Polygon
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A closed plane figure that is formed by vertices and sides.
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Convex
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There is no indent in a polygon. █
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Concave
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There is an indent in a polygon. ⋈
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N-gon
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N is the number of the polygon's sides
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Equilateral
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All sides are congruent
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Equiangular
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All angles are congruent
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Regular
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A polygon that is both equilateral and equiangular.
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Conjecture
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An unproven statement based on an observation.
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Inductive reasoning
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Reasoning based on patterns and examples
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Counterexample
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An example to prove that a conjecture is false.
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Conditional Statement
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A statement that contains a hypothesis and a conclusion. Matches with contrapositive. If I defeat the Elite Four, then I will be entered in the Hall of fame. True
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Converse
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A statement where both the hypothesis and conclusion are negated. Matches with inverse. If I was entered in the Hall of Fame, then I defeated the Elite Four. True
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Inverse
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A statement where both the hypothesis and conclusion are negated. Matches with converse. If I did not defeat the Elite Four, then I was not entered in the Hall of Fame. True
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Contrapositive
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A statement where both the converse and inverse are applied. Matches with conditional statement. If I was not entered in the Hall of Fame, then I did not defeat the Elite Four. True
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If-then form
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The "if" part contains the hypothesis and the "then" part contains the conclusion.
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Negation
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To make a statement opposite of its original statement.
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Equivalent statements
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When 2 statements are both true or both false.
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Perpendicular lines
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When 2 lines intersect to form a right angle.
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Biconditional statement
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A statement that contains both the inverse and the converse. A biconditional can only be written when the conditional and converse are both true. I defeated the Elite Four if and only if I was entered in the Hall of Fame.
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Deductive reasoning
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Reasoning based on known facts.
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Proof
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The annoying part of geometry. (A logical statement that shows a statement is true)
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Two-column proof
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It makes you want to pull your hair out. (Numbered statements in corresponding order to show an argument in logical order)
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Theorem
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A statement that can be proven.
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Parallel lines
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Lines that do not intersect and are coplanar.
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Skew lines
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Lines that do not intersect and are not coplanar.
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Parallel planes
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Any 2 planes that do not intersect
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Transversal
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A line that intersects 2 or more coplanar lines at different points. (Those coplanar points do not have to be parallel)
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Corresponding angles
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Angles that have corresponding positions.
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Alternate interior angles
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Two angles that lie between 2 coplanar lines and are on opposite sides of a transversal.
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Alternate exterior angles
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Two angles that lie outside of 2 coplanar lines and are on opposite sides of a transversal.
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Consecutive interior angles
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Two angles that lie between 2 coplanar lines and are on the same sides of a transversal.
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Paragraph proof
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A type of proof that is written in sentences to form a paragraph. Arranged in logical order.
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Slope
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The ratio of vertical and horizontal change of a line.
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Slope-intercept form
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y = mx + b, where "m" is slope and "b" is the y-intercept.
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Standard form
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Ax + By = C, where A and B are not fractions, or zero. A is not negative.
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Distance from a point to a line
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The length of a perpendicular segment from a point to a line. (Since perpendicular segments are the shortest way there)
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Scalene triangle
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A type of triangle where no sides are equal in length
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Isosceles triangle
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A type triangle where at least 2 sides are equal in length
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Equilateral triangle
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A type of triangle where all 3 sides are equal in length
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Acute triangle
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A type of triangle where all 3 angles are acute
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Right triangle
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A type of triangle where there is 1 right angle
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Obtuse triangle
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A type of triangle where there is 1 obtuse triangle
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Equiangular triangle
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A type of triangle where all 3 angles are congruent (meaning that they all measure 60 degrees)
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Interior angles
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Angles that make up the inside of a triangle
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Exterior angles
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Angles that are linear to interior angles
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Corollary
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A statement that can be easily proved by a theorem. Ex: Acute angles of a right triangle are complementary. (m<1+m<2=180-90)
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Congruent figures
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Two figures that are congruent; meaning that they have corresponding parts
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Corresponding parts
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Angles or sides of 2 congruent figures
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Legs (of a right triangle)
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The 2 sides of a right triangle that are perpendicular
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Hypotenuse
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The side that is the opposite to the right angle of a right triangle
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Flow proof
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Uses arrows to show the flow of the logical argument
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Legs (of an isosceles triangle)
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The 2 sides of an isosceles triangle that are congruent
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Vertex angle (of an isosceles triangle)
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The angle formed by the 2 legs of an isosceles triangle
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Base (of an isosceles triangle)
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The length of an isosceles triangle that is opposite from the vertex angle.
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Base angles (of an isosceles triangle)
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The 2 angles that are adjacent to the base
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Transformation
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Moves or changes a geometric figure in someway to produce a new figure
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Translation
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Moves every point of a figure by the same distance in the same direction
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Reflection
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Uses a line of reflection to create a mirror image
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Rotation
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Turns a figure around a fixed point, called the center of rotation
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Midsegment of triangle
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A segment that connects the midpoints of 2 sides. Every triangle has 3
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Coordinate proof
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Involves placing geometric figures in a coordinate plane. when you use variables to represent a figure in a coordinate proof, the results are true for all figures of that type.
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Perpendicular bisector
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A segment, ray or line that is perpendicular to a segment at its midpoint
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Equidistant
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Equal distance
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Point of concurrency
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The point of intersection of rays, lines or segments
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Circumcenter
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The point of concurrency of the 3 perpendicular bisectors from a triangle. It is equidistant to all 3 vertices.
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Incenter
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The point of concurrency of the 3 angle bisectors from a triangle. It is equidistant to all 3 sides.
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Median of a triangle
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The segment that connects the midpoint of a side to the opposite vertex. Each triangle has 3.
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Centroid
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The point of concurrency where all 3 medians intersect.
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Altitude of a triangle
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The perpendicular segment that connects a vertex to its opposite side.
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Orthocenter
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The point of concurrency where all 3 altitudes intersect.
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Indirect proof
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Start by making a temporary assumption that the desired conclusion is false. Then show that it is a logical impossibility, you prove the original statement by contradiction.
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Ratio
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Written as x:y, it is the quantitative relation between two numbers.
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Proportion
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An equation that states that 2 ratios are equal
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Means
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The denominator of the first ratio and numerator of the second ratio
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Extremes
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The numerator of the first ratio and denominator of the second ratio
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Geometric mean
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x^2 = ab, so x = √ab
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Scale drawing
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A drawing that is in the same shape that it represents
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Scale
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A ratio that describes how 2 figures are related to each other
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Similar polygons
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Polygons that have congruent angles but not congruent sides
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Scale factor of 2 similar polygons
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The number that you multiply by to get from the original figure to the new one
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Dilation
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A transformation that stretches or shrinks the original figure to create a similar figure
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Center of dilation
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A point where a figure stretches or shrinks by.
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Scale factor of a dilation
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The ratio of a side length to the original corresponding side length
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Reduction
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0<k<1, where k is the scale factor.
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Enlargement
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k>1, where k is the scale factor.
