Geometry End of the Year Study Guide

Vocabulary-style flashcards covering major geometry concepts including precision rules, logical reasoning, line and angle relationships, triangle properties, quadrilaterals, and trigonometry based on the study guide transcript.

Inductive reasoning

Making a conjecture or guess based on the observation of patterns.

Deductive reasoning

Proving a statement based on facts such as definitions, theorems, and postulates.

Counterexample

A specific example that disproves a statement or shows the hypothesis is true while the conclusion is false.

Undefined terms

In geometry, these are point, line, and plane.

Collinear

Points that lie on the same line.

Coplanar

Points or lines that lie in the same plane.

Skew lines

Lines that are non-coplanar and never intersect.

Postulate (Axiom)

A statement that is assumed to be true without proof.

Theorem

A statement that must be proven true.

Conditional statement

An "if-then" statement where "if" is the hypothesis ($p$) and "then" is the conclusion ($q$).

Converse

A logic statement formed by switching the hypothesis and the conclusion of a conditional.

Inverse

A logic statement formed by negating both the hypothesis and the conclusion of a conditional.

Contrapositive

A logic statement formed by switching and negating both the hypothesis and the conclusion.

Biconditional

A statement that combines a true conditional and its true converse using the phrase "if and only if".

Angle Bisector

Any figure that divides an angle into two congruent angles.

Midpoint of a Segment

A point that divides a segment into two congruent segments.

Segment Addition Postulate

If B is between A and C, then $AB + BC = AC$.

Angle Addition Postulate

If B is in the interior of $ext{AOC}$, then $m ext{∠AOB} + m ext{∠BOC} = m ext{∠AOC}$.

Vertical angles

A pair of non-adjacent angles formed by intersecting lines that are congruent.

Linear pair

A pair of adjacent angles whose non-common sides are opposite rays, having a sum of $180^∘$.

Complementary Angles

Two angles whose measures sum to $90^∘$.

Supplementary Angles

Two angles whose measures sum to $180^∘$.

Reflexive Property of Equality

A property stating that $a = a$.

Symmetric Property of Equality

If $a = b$, then $b = a$.

Transitive Property of Equality

If $a = b$ and $b = c$, then $a = c$.

Substitution Property of Equality

If $a = b$, then $a$ can be substituted for $b$ in any expression.

Corresponding Angles

Angles in the same relative position at each intersection where a straight line crosses two others; they are congruent if the two lines are parallel.

Alternate Interior Angles

Angles between two parallel lines and on opposite sides of a transversal; they are congruent.

Consecutive Interior Angles

Angles between two parallel lines and on the same side of a transversal; they are supplementary (sum to $180^∘$).

Polygon angle sum formula

For a polygon with $n$ sides, the sum of the interior angles is $(n-2) imes 180^∘$.

CPCTC

An acronym for "Corresponding Parts of Congruent Triangles are Congruent," used after proving triangles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those two sides are congruent.

Circumcenter

The point of concurrency of the perpendicular bisectors of a triangle, which is equidistant from the vertices.

Incenter

The point of concurrency of the angle bisectors of a triangle, which is equidistant from the sides.

Centroid

The point of concurrency of the medians of a triangle, located two-thirds of the distance from the vertex to the midpoint of the opposite side.

Orthocenter

The point of concurrency of the altitudes of a triangle.

Triangle Midsegment

A segment connecting the midpoints of two sides of a triangle; it is half the length of the third side and parallel to it.

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Geometric mean

The positive square root of the product of two positive numbers.

Parallelogram

A quadrilateral with opposite sides congruent, opposite angles congruent, consecutive angles supplementary, and diagonals that bisect each other.

Rhombus

A parallelogram with four congruent sides and diagonals that are perpendicular and bisect opposite angles.

Midsegment of a trapezoid

A segment parallel to each base with a length equal to one-half the sum of the lengths of the bases.

Kite

A quadrilateral with perpendicular diagonals and exactly one pair of opposite angles congruent.

Pythagorean Theorem

In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Obtuse triangle inequality

A triangle is obtuse if c^2 > a^2 + b^2, where $c$ is the longest side.

SOH CAH TOA

A mnemonic for trigonometric ratios: ext{sin}( heta) = rac{ ext{opposite}}{ ext{hypotenuse}}, ext{cos}( heta) = rac{ ext{adjacent}}{ ext{hypotenuse}}, and ext{tan}( heta) = rac{ ext{opposite}}{ ext{adjacent}}.

Unit Rate

Describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.

Precision (Significant Figures)

A measure of accuracy where leading zeros and trailing zeros before a decimal do not count, while zeros between nonzeros and trailing zeros after a decimal do count.

Rationalizing the Denominator

The process of removing a radical from the denominator by multiplying the fraction by a form of 1 ($rac{ ext{radical}}{ ext{radical}}$).

Perpendicular line slopes

The slopes of these lines are negative reciprocals, and their product equals $-1$ ($m_1 imes m_2 = -1$).