Geometry End of the Year Study Guide

Foundations of Geometry\n\n* Inductive reasoning: making a conjecture (guess) based on observation of patterns.\n* Deductive reasoning: proving a statement based on facts (definitions, theorems, postulates, …).\n* Counterexample: an example that disproves a statement. It is a specific example where the hypothesis of a conditional is true but the conclusion is false.\n* Undefined terms: point, line, plane.\n* Collinear: points that lie on the same line.\n* Coplanar: points or lines that lie in the same plane.\n* Skew lines: non-coplanar lines that never intersect.\n* Postulate: a statement that is assumed to be true (also called an \"axiom\").\n* Theorem: a statement that must be proven true.\n\n# Reasoning and Proof\n\n* Hypothesis: the \"if\" part of a conditional statement (represented by $p$).\n* Conclusion: the \"then\" part of a conditional statement (represented by $q$).\n* Conditional statement: an \"if-then\" statement.\n* Converse: switch the \"if\" and the \"then\" parts of the conditional ($q \\rightarrow p$).\n* Inverse: negate both the \"if\" and the \"then\" parts ($\\neg p \\rightarrow \\neg q$).\n* Contrapositive: switch and negate both parts ($\\neg q \\rightarrow \\neg p$).\n* Biconditional: a statement created when a conditional and its converse are both true, combined using the phrase \"if and only if\".\n\n# Angle and Segment Relationships\n\n* Angle Bisector: any figure (like a ray or line) that divides an angle into two congruent angles.\n* Midpoint of a Segment: a point that divides the segment into two congruent segments.\n* Segment Addition Postulate: if point $B$ is between points $A$ and $C$, then $AB + BC = AC$.\n* Angle Addition Postulate: if point $B$ is in the interior of $\\angle AOC$, then $m\\angle AOB + m\\angle BOC = m\\angle AOC$.\n* Adjacent angles: angles that are next to each other (e.g., $\\∠ 3$ and $\\∠ 4$).\n* Vertical angles: angles opposite each other when two lines intersect; they are congruent ($\\∠ 2 \\cong \\∠ 3$).\n* Linear pair: two adjacent angles whose non-common sides are opposite rays; they have a sum of $180^∘$.\n* Complementary Angles: two angles whose measures sum to $90^∘$ (e.g., $\\∠ 2$ and $\\∠ 5$).\n* Supplementary Angles: two angles whose measures sum to $180^∘$ (e.g., $\\∠ 1$ and $\\∠ 3$).\n\n# Properties of Equality and Congruence\n\n* Reflexive Property of Equality: $a = a$.\n* Symmetric Property of Equality: if $a = b$, then $b = a$.\n* Transitive Property of Equality: if $a = b$ and $b = c$, then $a = c$.\n* Substitution Property of Equality: if $a = b$, then $a$ can be substituted for $b$ in any expression.\n* Reflexive Property of Congruence: $\\∠ A \\cong \\∠ A$.\n* Symmetric Property of Congruence: if $\\∠ A \\cong \\∠ B$, then $\\∠ B \\cong \\∠ A$.\n* Transitive Property of Congruence: if $\\∠ A \\cong \\∠ B$ and $\\∠ B \\cong \\∠ C$, then $\\∠ A \\cong \\∠ C$.\n\n# Parallel and Perpendicular Lines\n\nWhen a transversal intersects parallel lines, the following angle relationships apply:\n* Corresponding Angles are congruent: $\\∠ 1 \\cong \\∠ 5$, $\\∠ 2 \\cong \\∠ 6$, $\\∠ 7 \\cong \\∠ 3$, $\\∠ 8 \\cong \\∠ 4$.\n* Alternate Interior Angles are congruent: $\\∠ 3 \\cong \\∠ 6$, $\\∠ 5 \\cong \\∠ 4$.\n* Alternate Exterior Angles are congruent: $\\∠ 1 \\cong \\∠ 8$, $\\∠ 7 \\cong \\∠ 2$.\n* Consecutive Interior Angles are supplementary: $\\∠ 5 + \\∠ 3 = 180^∘$, $\\∠ 6 + \\∠ 4 = 180^∘$.\n* Consecutive Exterior Angles are supplementary: $\\∠ 1 + \\∠ 7 = 180^∘$, $\\∠ 2 + \\∠ 8 = 180^∘$.\n* Notes on Proofs:\n * Use properties of parallel lines to prove angle congruence.\n * Use converses of these theorems to prove lines are parallel.\n * If two lines are parallel to a third line, they are parallel to each other.\n * In a plane, if two lines are perpendicular to a third line, they are parallel to each other.\n\n# Triangle and Polygon Angle Sums\n\n* Triangle Angle Sum: the sum of the interior angles of a triangle is always $180^∘$.\n* Triangle Exterior Angles: each exterior angle is equal to the sum of the two remote interior angles.\n* Polygon Angle Sum: for a polygon with $n$ sides, interior angles sum to $(n - 2) \\times 180^∘$.\n* Measure of an interior angle of a regular polygon: $\frac{(n - 2) \\times 180^∘}{n}$.\n* Sum of exterior angles: always $360^∘$ for any convex polygon.\n* Measure of a single exterior angle of a regular polygon: $\frac{360^∘}{n}$.\n\n# Congruent Figures and Triangles\n\n* Corresponding Parts: in two congruent figures, all parts of one figure are congruent to the corresponding parts of the other figure. If $\\triangle ABC \\cong \\∆ FED$, then:\n * Corresponding angles: $\\∠ A \\cong \\∠ F$, $\\∠ B \\cong \\∠ E$, $\\∠ C \\cong \\∠ D$.\n * Corresponding sides: $AB \\cong FE$, $BC \\cong ED$, $AC \\cong FD$.\n * Rule: always list corresponding vertices in the same order in a congruence statement.\n* Third Angle Theorem: if two angles of two triangles are congruent, then the third angles are also congruent.\n* Triangle Congruence Postulates/Theorems: SSS, SAS, ASA, AAS, and HL (Hypotenuse-Leg for right triangles).\n* CPCTC: stands for \"Corresponding Parts of Congruent Triangles are Congruent.\" Used after proving triangles are congruent to prove specific parts are congruent.\n\n# Isosceles and Equilateral Triangles\n\n* Isosceles Triangle Theorem: if two sides of a triangle are congruent, then the angles opposite those sides are congruent.\n* Vertex Angle Bisector: in an isosceles triangle, the bisector of the vertex angle is the perpendicular bisector of the base.\n* Equilateral/Equiangular link: if a triangle is equilateral (all sides congruent), then it is equiangular (all angles congruent at $60^∘$).\n\n# Relationships Within Triangles\n\n* Point of Concurrency: the intersection point of 3 or more lines.\n* Circumcenter: point of concurrency of perpendicular bisectors. Equidistant from the vertices.\n* Incenter: point of concurrency of angle bisectors. Equidistant from the sides.\n* Centroid: point of concurrency of medians. It lies two-thirds of the distance from each vertex to the midpoint of the opposite side.\n* Orthocenter: point of concurrency of altitudes.\n* Midsegment: connects the midpoints of two sides. It is parallel to the third side and its length is half the third side.\n* Perpendicular Bisector Theorem: if a point lies on the perpendicular bisector of a segment, it is equidistant from the segment's endpoints.\n* Angle Bisector Theorem: if a point lies on the angle bisector of an angle, it is equidistant from the sides of the angle.\n\n# Triangle Inequality\n\n* The sum of the lengths of any two sides of a triangle is greater than the length of the third side.\n* The measure of the third side ($x$) must satisfy: $\text{difference} < x < \text{sum}$.\n* Side-Angle relationship: the longest side is opposite the largest angle; the smallest side is opposite the smallest angle.\n\n# Similarity\n\n* Angle-Angle (AA) Similarity Postulate: if two angles of one triangle are congruent to two angles of another, the triangles are similar.\n* Side-Side-Side (SSS) Similarity Theorem: if corresponding side lengths are proportional, the triangles are similar.\n* Side-Angle-Side (SAS) Similarity Theorem: if an angle is congruent and the including sides are proportional, the triangles are similar.\n* Triangle Proportionality: if a line parallel to one side intersects the other two, it divides them proportionally ($\frac{AE}{EB} = \frac{AF}{FC}$).\n* Right Triangle Altitude: the altitude to the hypotenuse of a right triangle forms two triangles similar to each other and the original.\n* Geometric Mean: for two positive numbers $a$ and $b$, the geometric mean is $\\sqrt{a \\times b}$.\n\n# Quadrilaterals\n\n* Parallelograms:\n * Opposite sides and opposite angles are congruent.\n * Consecutive angles are supplementary.\n * Diagonals bisect each other.\n * Special case: if one pair of opposite sides is both congruent and parallel, the figure is a parallelogram.\n* Special Parallelograms:\n * Rectangle: four right angles; diagonals are congruent.\n * Rhombus: four congruent sides; diagonals are perpendicular; diagonals bisect opposite angles.\n * Square: both a rhombus and a rectangle.\n* Trapezoids and Kites:\n * Isosceles Trapezoid: base angles are congruent; diagonals are congruent.\n * Trapezoid Midsegment: parallel to both bases; length is $\frac{1}{2}(\text{base}_1 + \text{base}_2)$.\n * Kite: diagonals are perpendicular; exactly one pair of opposite angles is congruent.\n\n# Right Triangle Trigonometry\n\n* Pythagorean Theorem: $a^2 + b^2 = c^2$.\n* Classification using $c^2$:\n * $c^2 = a^2 + b^2 \\rightarrow$ Right triangle.\n * $c^2 < a^2 + b^2 \\rightarrow$ Acute triangle.\n * $c^2 > a^2 + b^2 \\rightarrow$ Obtuse triangle.\n* Special Right Triangles:\n * 30-60-90: hypotenuse = $2 \\times \\text{short leg}$; longer leg = $\text{short leg} \\times \\sqrt{3}$.\n * 45-45-90: hypotenuse = $\text{leg} \\times \\sqrt{2}$.\n* Trig Ratios (SOH CAH TOA):\n * $\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}$\n * $\\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}$\n * $\\tan \\theta = \\frac{\\text{opposite}}{\\text{adjacent}}$\n\n# Precision and Significant Figures\n\n* Counting Rules:\n * Leading zeros do not count ($0.0001$ has 1 sig fig).\n * Trailing zeros before decimal point do not count ($40000$ has 1 sig fig).\n * Zeros between non-zeros count ($40001$ has 5 sig figs).\n * Trailing zeros after decimal sign count ($0.3500$ has 4 sig figs).\n* Calculations:\n * Addition/Subtraction: round to the least decimal place (Example: $4.113 + 1000.44 = 1004.553 \\rightarrow 1004.55$).\n * Multiplication/Division: round to the fewest significant figures (Example: $4.01 (3.1) = 12.431 \\rightarrow 12$ because $3.1$ has 2 sig figs).\n\n# Radicals and Coordinate Geometry\n\n* Simplifying Radicals: factor out perfect squares (Example: $\\√ 40 = \\√ 4 \\times \\√ 10 = 2\\sqrt{10}$; $\\√ 125 = \\√ 25 \\times \\√ 5 = 5\\sqrt{5}$).\n* Rationalizing: no radicals in denominators. Multiply numerator and denominator by the radical (Example: $\frac{5}{\\sqrt{3}} = \frac{5\\sqrt{3}}{3}$; $\frac{6}{\\sqrt{2}} = \frac{6\\sqrt{2}}{2} = 3\\sqrt{2}$).\n* Coordinate Formulas:\n * Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$.\n * Parallel slopes are equal; perpendicular slopes are negative reciprocals ($m_1 \\times m_2 = -1$).\n * Distance: $d = \\√((x_2 - x_1)^2 + (y_2 - y_1)^2)$.\n * Midpoint: $M = (\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2})$.\n * Partitioning a Segment: finding a point between $(x_1, y_1)$ and $(x_2, y_2)$ given a ratio $k:n$. Example: partitioning $(1, 1)$ and $(9, 5)$ in ratio 3:1 results in $(7, 4)$.\n\n# Measurement Conversions\n\n* Linear: use scale ratio as is.\n* Square (Area): square the ratio.\n* Cubic (Volume): cube the ratio.\n * Example: if 3 feet = 1 yard, then 1 cubic yard = $3^3 = 27$ cubic feet. Thus, 2 cubic yards = 54 cubic feet.\n\n# Circles and Advanced Topics\n\n* Angles in Circles:\n * Central angle = arc measure.\n * Inscribed angle = $\frac{1}{2}$ arc measure.\n* Formulas:\n * Arc Length: $AL = \\pi \\times d \\times (\\frac{\\theta}{360})$.\n * Sector Area: $SA = \\pi \\times r^2 \\times (\\frac{\\theta}{360})$.\n* Power of a Point Theorems:\n * Chord-Chord: $\text{segment}_1 \\times \text{segment}_2 = \text{segment}_1 \\times \text{segment}_2$.\n * Secant-Secant: $\text{outside} \\times \text{whole} = \text{outside} \\times \text{whole}$.\n * Tangent-Secant: $\text{tangent}^2 = \text{outside} \\times \text{whole}$.\n* Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius. Example: $(x + 2)^2 + (y - 3)^2 = 16$ has center $(-2, 3)$ and $r = 4$.", "title": "Geometry End of Year Study Guide"}