Geometry End-of-Year Study Guide

Foundations of Geometry

• Inductive Reasoning: The process of making a conjecture (an educated guess) based on the observation of specific patterns or sequences.
• Deductive Reasoning: The process of proving a statement is true based on established facts, including definitions, theorems, and postulates.
• Counterexample: A specific case or example that disproves a general statement or conjecture.
• Undefined Terms: Concepts that are used to define other terms but are not themselves defined by more fundamental terms in geometry:     * Point: Indicated with a dot and labeled with a capital letter (e.g., $A$).     * Line: Goes on forever in both directions.     * Plane: A flat surface extending infinitely in all directions.
• Ray: A part of a line that begins at a single endpoint and extends infinitely in one direction.
• Line Segment: A part of a line that consists of two specific endpoints.
• Collinear: Points that lie on the exact same line.
• Coplanar: Points or lines that lie within the same plane.
• Intersecting Lines: Two lines that meet at exactly one point.
• 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 has been proven true using logic, postulates, and other theorems.

Reasoning, Proof, and Logic

• Hypothesis: The "if" part of a conditional statement, represented symbolically by the letter $p$.
• Conclusion: The "then" part of a conditional statement, represented symbolically by the letter $q$.
• Conditional Statement: An "if-then" statement written as $p \rightarrow q$.
• Related Conditionals:     * Converse: Formed by switching the hypothesis and the conclusion ($q \rightarrow p$).     * Inverse: Formed by negating both the hypothesis and the conclusion ($\neg p \rightarrow \neg q$).     * Contrapositive: Formed by switching and negating both parts of the original statement ($\neg q \rightarrow \neg p$). The contrapositive always has the same truth value as the original conditional statement.
• Biconditional Statement: Formed when a conditional and its converse are both true. It uses the phrase "if and only if" ($p \leftrightarrow q$).
• Counterexample (in Conditionals): A specific instance where the hypothesis ($p$) is true, but the conclusion ($q$) is false.

Angle Relationships and Segment Properties

• Angle Bisector: A figure (ray, line, or segment) that divides an angle into two congruent angles.
• Midpoint of a Segment: A point that divides a segment into two congruent segments.
• Distance and Measure Postulates:     * Segment Addition Postulate: If point $B$ lies between points $A$ and $C$, then $AB + BC = AC$.     * Angle Addition Postulate: If point $B$ is in the interior of $\angle AOC$, then $m\angle AOB + m\angle BOC = m\angle AOC$.
• Angle Pair Definitions:     * Adjacent Angles: Two angles that lie next to each other (e.g., $\angle 3$ and $\angle 4$).     * Vertical Angles: Two non-adjacent angles formed by intersecting lines; they are always congruent ($\angle 2 \cong ∠ 3$).     * Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays; their sum is always $180^{\circ}$.     * Complementary Angles: Two angles whose measures sum to $90^{\circ}$.     * Supplementary Angles: Two angles whose measures sum to $180^{\circ}$.

Properties of Equality and Congruence

• Reflexive Property:     * Equality: $a = a$     * Congruence: $\angle A \cong \angle A$
• Symmetric Property:     * Equality: If $a = b$, then $b = a$.     * Congruence: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.
• Transitive Property:     * Equality: If $a = b$ and $b = c$, then $a = c$.     * Congruence: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.
• Substitution Property: If $a = b$, then $a$ can be replaced by $b$ in any expression or equation.

Parallel and Perpendicular Lines

• Parallel Lines ($\|$): Two lines that will never intersect.
• Perpendicular Lines ($\perp$): Two lines that intersect to form right angles ($90^{\circ}$).
• Transversal and Parallel Line Angles: When a transversal intersects two parallel lines, specific angle pairs are created:     * Corresponding Angles: These are congruent (e.g., $\angle 1 \cong \angle 5$, $\angle 2 \cong \angle 6$, $\angle 7 \cong \angle 3$, $\angle 8 \cong \angle 4$).     * Alternate Interior Angles: These are congruent (e.g., $\angle 3 \cong \angle 6$, $\angle 5 \cong \angle 4$).     * Alternate Exterior Angles: These are congruent (e.g., $\angle 1 \cong \angle 8$, $\angle 7 \cong \angle 2$).     * Consecutive Interior Angles: These are supplementary ($\text{sum} = 180^{\circ}$, e.g., $m\angle 5 + m\angle 3 = 180^{\circ}$, $m\angle 6 + m\angle 4 = 180^{\circ}$).     * Consecutive Exterior Angles: These are supplementary ($\text{sum} = 180^{\circ}$, e.g., $m\angle 1 + m\angle 7 = 180^{\circ}$, $m\angle 2 + m\angle 8 = 180^{\circ}$).
• Proof of Parallelism:     * Use properties of parallel lines to prove angle congruence.     * Use the converses of the above statements to prove lines are parallel.     * If two lines are parallel to a third line, they are parallel to each other.     * In a plane, if two lines are perpendicular to a third line, they are parallel to each other.

Polygons: Interior and Exterior Angle Sums

• Interior Angle Sums: For any polygon with $n$ sides, the sum of the measures of the interior angles is given by:     * $\text{Sum} = (n - 2) \times 180^{\circ}$
• Regular Polygons (Interior): The measure of a single interior angle in a regular polygon (where all sides and angles are equal) is:     * $\text{Measure} = \frac{(n - 2) \times 180^{\circ}}{n}$
• Exterior Angle Sums: The sum of the measures of the exterior angles (one at each vertex) of any convex polygon is always $360^{\circ}$.
• Regular Polygons (Exterior): The measure of a single exterior angle in a regular polygon is:     * $\text{Measure} = \frac{360^{\circ}}{n}$

Triangle fundamentals and Relationships

• Angle Sum Theorem: The sum of the internal angles of any triangle is exactly $180^{\circ}$.
• Exterior Angle Theorem: Each exterior angle of a triangle is equal to the sum of the two remote (non-adjacent) interior angles.
• Triangle Classifications (by Sides):     * Scalene: No sides are congruent.     * Isosceles: At least two sides are congruent.     * Equilateral: All three sides are congruent.
• Triangle Classifications (by Angles):     * Acute: Three acute angles (less than $90^{\circ}$).     * Right: One right angle ($90^{\circ}$).     * Obtuse: One obtuse angle (greater than $90^{\circ}$).     * Equiangular: Three congruent angles (each is $60^{\circ}$).
• Isosceles and Equilateral Triangle Theorems:     * Isosceles Triangle Theorem: If two sides are congruent, then the angles opposite those sides are congruent.     * The bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector of the base.     * If a triangle is equilateral, then it is equiangular.
• Midsegment of a Triangle: A segment connecting the midpoints of two sides. It is parallel to the third side and its length is exactly half ($\frac{1}{2}$) of the third side.
• Points of Concurrency:     * Circumcenter: Point where the perpendicular bisectors intersect; it is equidistant from the triangle's vertices.     * Incenter: Point where the angle bisectors intersect; it is equidistant from the triangle's sides.     * Centroid: Point where the medians intersect; it is located at a point on each median two-thirds of the distance from the vertex to the opposite side's midpoint.     * Orthocenter: Point where the altitudes intersect.
• Triangle Inequality Rules:     * The sum of any two side lengths must be greater than the third side length.     * The measure of the third side must be less than the sum and greater than the difference of the other two sides.     * Side-Angle Relationship: The longest side is opposite the largest angle, and the smallest side is opposite the smallest angle.     * The exterior angle of a triangle is greater than either of the two non-adjacent interior angles.

Congruent and Similar Triangles

• Corresponding Parts of Congruent Triangles (CPCTC): In congruent figures, all corresponding components are congruent. This is used after proving triangle congruence.
• Third Angle Theorem: If two angles of one triangle are congruent to two angles of another, the third angles are also congruent.
• Triangle Congruence Postulates/Theorems:     * SSS (Side-Side-Side): All three sides are congruent.     * SAS (Side-Angle-Side): Two sides and the included angle are congruent.     * ASA (Angle-Side-Angle): Two angles and the included side are congruent.     * AAS (Angle-Angle-Side): Two angles and a non-included side are congruent.     * HL (Hypotenuse-Leg): Used for right triangles only; the hypotenuse and one leg are congruent.     * Note: "SSA" or "ASS" (Side-Side-Angle) is not a valid congruence postulate ("No Donkeys").
• Triangle Similarity Theorems:     * AA (Angle-Angle) Similarity Postulate: Two angles are congruent.     * SSS Similarity Theorem: All corresponding side lengths are proportional.     * SAS Similarity Theorem: One pair of congruent angles and the surrounding sides are proportional.
• Proportionality:     * If a line is parallel to one side of a triangle and intersects the other two, it divides those sides proportionally.     * Geometric Mean: In a right triangle, the altitude to the hypotenuse forms two triangles similar to each other and the original. The geometric mean of two numbers is $\sqrt{x \times y}$.     * Altitude Rule: $\frac{\text{part hyp}}{\text{altitude}} = \frac{\text{altitude}}{\text{other part hyp}}$     * Leg Rule: $\frac{\text{hyp}}{\text{leg}} = \frac{\text{leg}}{\text{projection}}$

Right Triangle Geometry: Pythagorean Theorem and Trigonometry

• Pythagorean Theorem: In a right triangle, $a^2 + b^2 = c^2$.     * Converse: If $c^2 = a^2 + b^2$, the triangle is right.     * Acute Triangle: If $c^2 < a^2 + b^2$.     * Obtuse Triangle: If $c^2 > a^2 + b^2$.
• Pythagorean Triples: Sets of integers that satisfy the theorem, such as $(3, 4, 5)$, $(5, 12, 13)$, $(7, 24, 25)$, and $(9, 40, 41)$.
• Special Right Triangles:     * 30-60-90: Hypotenuse = $2 \times \text{short leg}$; Longer leg = $\text{short leg} \times \sqrt{3}$.     * 45-45-90: Hypotenuse = $\text{leg} \times \sqrt{2}$.
• Trigonometric Ratios (SOH-CAH-TOA):     * $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$     * $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$     * $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$     * Reciprocal Ratios: Cosecant ($\csc$) is the reciprocal of sine; Secant ($\sec$) is the reciprocal of cosine; Cotangent ($\cot$) is the reciprocal of tangent.

Quadrilaterals

• Parallelograms: Opposite sides are parallel and congruent; opposite angles are congruent; consecutive angles are supplementary; diagonals bisect each other.
• Special Parallelograms:     * Rectangle: Has four right angles; diagonals are congruent.     * Rhombus: Has four congruent sides; diagonals are perpendicular and bisect opposite angles.     * Square: Is both a rhombus and a rectangle (four congruent sides and four right angles).
• Trapezoids:     * Isosceles Trapezoid: Legs are congruent; base angles are congruent; diagonals are congruent; opposite angles are supplementary.     * Midsegment of a Trapezoid: Parallel to bases; length = $\frac{1}{2}(\text{base}_1 + \text{base}_2)$.
• Kites: Diagonals are perpendicular; exactly one pair of opposite angles is congruent.

Circle Geometry

• Equations:     * At origin: $x^2 + y^2 = r^2$     * Center $(h, k)$: $(x - h)^2 + (y - k)^2 = r^2$
• Segment Rules:     * Radius perpendicular to a chord bisects the chord.     * Intersecting Chords: $\text{part}_1 \times \text{part}_2 = \text{part}_3 \times \text{part}_4$     * Secant-Secant Rule: $\text{whole secant} \times \text{external part} = \text{whole secant} \times \text{external part}$     * Secant-Tangent Rule: $\text{whole secant} \times \text{external part} = (\text{tangent})^2$     * Hat Rule: Two tangents drawn from the same external point to a circle are equal.
• Angle/Arc Rules:     * Central Angle: Equals the intercepted arc.     * Inscribed Angle: Equals $\frac{1}{2}$ the intercepted arc.     * Angle by tangent/chord: Equals $\frac{1}{2}$ the intercepted arc.     * Angle formed by 2 chords: Equals $\frac{1}{2}$ the sum of intercepted arcs.     * Angle formed by 2 tangents/2 secants: Equals $\frac{1}{2}$ the difference of intercepted arcs.

Transformations and Isometry

• Isometry: A transformation that preserves length (rigid motion).
• Translations: $(x, y) \rightarrow (x + a, y + b)$. Preserves orientation.
• Reflections (Rigid Motion):     * Over $x$-axis: $(x, y) \rightarrow (x, -y)$     * Over $y$-axis: $(x, y) \rightarrow (-x, y)$     * Over $y = x$: $(x, y) \rightarrow (y, x)$     * Over $y = -x$: $(x, y) \rightarrow (-y, -x)$     * Glide Reflection: Composition of a reflection and a translation.
• Rotations (Rigid Motion):     * $90^{\circ}$: $(x, y) \rightarrow (-y, x)$     * $180^{\circ}$: $(x, y) \rightarrow (-x, -y)$     * $270^{\circ}$: $(x, y) \rightarrow (y, -x)$
• Dilation (Not Rigid): $(x, y) \rightarrow (kx, ky)$. Preserves angle measures and collinearity but not length.
• Composite Transformations: Operations performed in sequence, often read right to left in notation.

Measurements, Precision, and Significant Figures

• Significant Figure Rules:     * Leading zeros never count ($0.00001$ has 1 sig fig).     * Trailing zeros before a decimal do not count ($40000$ has 1 sig fig).     * Zeros between non-zero digits always count ($40001$ has 5 sig figs).     * Trailing zeros after a decimal count ($0.3500$ has 4 sig figs).
• Operational Rules:     * Addition/Subtraction: Round to the least number of decimal places.     * Multiplication/Division: Round to the least number of significant figures.
• Unit Rates and Conversions:     * Standard rates: $12\,in = 1\,ft$, $1\,hr = 60\,min$, $5280\,ft = 1\,mi$.     * Ratio Rule for Conversions:         * Linear (Perimeter): Use ratio as is.         * Square (Area): Square the ratio.         * Cubic (Volume): Cube the ratio.     * Example: How many $ft^3$ in $2\,yd^3$? Given $3\,ft = 1\,yd$, the volume ratio is $3^3 : 1^3$, which is $27 : 1$. Therefore, $2 \times 27 = 54\,ft^3$.

Coordinate Geometry and Analytical Formulas

• Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$     * Parallel lines have equal slopes.     * Perpendicular lines have negative reciprocal slopes ($m_1 \times m_2 = -1$).
• Equations of Lines:     * Slope-Intercept: $y = mx + b$     * Point-Slope: $y - y_1 = m(x - x_1)$
• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Ratio Problems (4:5:9):     * Method 1: Sum the ratio ($4 + 5 + 9 = 18$). Each part is $\frac{\text{part}}{18} \times 180^{\circ}$.     * Method 2: Use an algebraic scale factor $x$: $4x + 5x + 9x = 180$. Solve for $x$ and multiply back.

Area and Volume Formulas

• 2D Shapes:     * Square: $A = s^2$     * Rectangle: $A = lw$     * Parallelogram: $A = bh$     * Triangle: $A = \frac{1}{2}bh$     * Trapezoid: $A = \frac{h(b_1 + b_2)}{2}$     * Circle: $C = 2\pi r = \pi d$; $A = \pi r^2$     * Regular Polygon: $A = \frac{1}{2}ap$ (where $a$ is apothem and $p$ is perimeter).     * Arc/Sector: Arc Length = $\frac{\theta}{360} \times 2\pi r$; Sector Area = $\frac{\theta}{360} \times \pi r^2$
• 3D Surfaces and Volumes (B = area of base):     * Prism: $V = Bh$; $LA = PH$; $SA = PH + 2B$     * Cylinder: $V = \pi r^2h$; $LA = 2\pi rh$; $SA = 2\pi rh + 2\pi r^2$     * Pyramid: $V = \frac{1}{3}Bh$; $LA = \frac{1}{2}pL$; $SA = \frac{1}{2}pL + B$ ($L$ = slant height).     * Cone: $V = \frac{1}{3}\pi r^2h$; $LA = \pi rL$; $SA = \pi rL + \pi r^2$     * Sphere: $V = \frac{4}{3}\pi r^3$; $SA = 4\pi r^2$
• Regular Solids (Faces): Tetrahedron (4), Cube (6), Octahedron (8), Dodecahedron (12), Icosahedron (20).

Geometric Constructions and Miscellaneous

• Constructions: Perpendicular bisector, Angle bisector, Line parallel to a given line, Copying an angle.
• Locus Theorems:     * Fixed distance from a point: A circle ($x^2 + y^2 = r^2$).     * Fixed distance from a line: Two parallel lines.     * Equidistant from 2 points: Perpendicular bisector.     * Equidistant from 2 parallel lines: One parallel line between them.     * Equidistant from 2 intersecting lines: Two angle bisectors.
• Radicals: Rationalize denominators by multiplying by $\frac{\sqrt{x}}{\sqrt{x}}$. Simplified form Example: $\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$.