Comprehensive Geometry End of the Year Study Guide

Foundations of Geometry

• Inductive reasoning: The process of making a conjecture (guess) based on the observation of patterns.
• Deductive reasoning: The process of proving a statement based on facts, such as definitions, theorems, and postulates.
• Counterexample: A specific example that disproves a statement.
• Undefined terms: There are three undefined terms in geometry: 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: A statement that is assumed to be true without proof. It is also referred to as an "axiom."
• Theorem: A statement that must be proven true.

Reasoning and Proof

• Hypothesis: Represented by the variable $p$, this is the "if" part of a conditional statement.
• Conclusion: Represented by the variable $q$, this is the "then" part of a conditional statement.
• Conditional statement: An "if-then" statement commonly written as $p \rightarrow q$.
• Converse: Formed by switching the "if" (hypothesis) and the "then" (conclusion) parts of the original conditional statement ($q \rightarrow p$).
• Inverse: Formed by negating both the "if" and the "then" parts of the conditional statement ($\neg p \rightarrow \neg q$).
• Contrapositive: Formed by both switching and negating both parts of the original conditional statement ($\neg q \rightarrow \neg p$).
• Biconditional: A statement formed when a conditional and its converse are both true, combined using the phrase "if and only if."
• Counterexample: A specific example where the hypothesis of a conditional is true, but the conclusion is false.

Angle Relationships

• Angle Bisector: Any figure (such as a ray or segment) 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 $\angle AOC$, then $m\angle AOB + m\angle BOC = m\angle AOC$.
• Adjacent angles: Angles that are next to each other, such as $\angle 23$ and $\angle 24$.
• Vertical angles: Opposite angles formed by intersecting lines, such as $\angle 22$ and $\angle 23$, which are equal (congruent).
• Linear pair: A pair of adjacent angles, such as $\angle 41$ and $\angle 23$, whose sum is $180$ degrees.
• Complementary Angles: Two angles, such as $\angle 22$ and $\angle 25$, whose measures sum to $90$ degrees.
• Supplementary Angles: Two angles, such as $\angle 21$ and $\angle 23$, whose measures sum to $180$ degrees.

Properties of Equality and Congruence

• Reflexive Property of Equality: $a = a$.
• 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.
• Reflexive Property of Congruence: $\angle A \cong \angle A$.
• Symmetric Property of Congruence: If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.
• Transitive Property of Congruence: If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Parallel and Perpendicular Lines

• Parallel Lines and Transversals: If a transversal intersects parallel lines, the following relationships exist:     * Corresponding Angles: These are congruent. Examples: $\angle 1 \cong \angle 5$, $\angle 2 \cong \angle 6$, $\angle 7 \cong \angle 3$, $\angle 8 \cong \angle 4$.     * Alternate Interior Angles: These are congruent. Examples: $\angle 3 \cong \angle 6$, $\angle 5 \cong \angle 4$.     * Alternate Exterior Angles: These are congruent. Examples: $\angle 1 \cong \angle 8$, $\angle 7 \cong \angle 2$.     * Consecutive Interior Angles: These are supplementary (sum to $180$ degrees). Examples: $m\angle 5 + m\angle 3 = 180^{\circ}$, $m\angle 6 + m\angle 4 = 180^{\circ}$.     * Consecutive Exterior Angles: These are supplementary (sum to $180$ degrees). Examples: $m\angle 1 + m\angle 7 = 180^{\circ}$, $m\angle 2 + m\angle 8 = 180^{\circ}$.
• Methods of Proof:     * Use Properties of Parallel Lines to prove angle congruence.     * Use Converses of the above rules to prove lines are parallel.     * If two lines are parallel to a third line, then they are parallel to each other.     * In a plane, if two lines are perpendicular to a third line, they are parallel to each other.

Triangles and Polygons

• Triangle Angle Sum: The sum of the interior angles in a triangle is $180$ degrees.
• Triangle Exterior Angles: Each exterior angle is equal to the sum of the two remote interior angles.
• Polygon Angle Sum: For a polygon with $n$ sides, the interior angles add up to $(n - 2) \times 180$.
• Regular Polygons:     * The measure of a single interior angle of a regular polygon is $\frac{(n - 2) \times 180}{n}$.     * The sum of the measures of the exterior angles is always $360$ degrees.     * The measure of a single exterior angle is $\frac{360}{n}$.

Congruent Triangles and Corresponding Parts

• Corresponding Parts: In two congruent figures, all parts of one figure are congruent to the corresponding parts of the other figure. For example, if $\triangle ABC \cong \triangle FED$:     * Corresponding angles: $\angle A \cong \angle F$, $\angle B \cong \angle E$, $\angle C \cong \angle D$.     * Corresponding sides: $AB = FE$, $BC = ED$, $AC = FD$.     * Writing Congruence Statements: Always list the corresponding vertices in the same order.
• Third Angle Theorem: If two angles of two triangles are congruent to two angles of another triangle, then the third angles are also congruent.
• Triangle Congruence Postulates/Theorems:     * SSS: Side-Side-Side.     * SAS: Side-Angle-Side.     * ASA: Angle-Side-Angle.     * AAS: Angle-Angle-Side.     * HL: Hypotenuse-Leg (used for right triangles).
• CPCTC: Stands for "Corresponding Parts of Congruent Triangles are Congruent." This is used after proving triangles are congruent to prove that specific parts of the triangles are congruent.

Isosceles and Equilateral Triangles

• Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those two sides are congruent.
• Isosceles Triangle Bisector: In an isosceles triangle, the bisector of the vertex angle is the perpendicular bisector of the base.
• Equilateral Triangles: If a triangle is equilateral (all sides congruent), then the triangle is equiangular (all angles congruent).

Relationships Within Triangles

• Point of concurrency: The single point where three or more lines intersect.
• Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle. It is equidistant from the vertices of the triangle.
• Incenter: The point of concurrency of the angle bisectors. It is equidistant from the sides of the triangle.
• Centroid: The point of concurrency of the medians. It is located at a point on each median 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.
• Midsegment: The segment connecting the midpoints of two sides of a triangle.     * The midsegment is half the length of the third side.     * The midsegment is parallel to the third side.
• Perpendicular Bisector Theorem: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
• Angle Bisector Theorem: If a point lies on the angle bisector of an angle, then it is equidistant from the sides of the angle.

Triangle Inequality and Similarity

• Triangle Inequality Rules:     * The sum of the lengths of any two sides of a triangle is greater than the length of the third side.     * The measure of the third side of a triangle must be less than the sum of the other two sides and greater than their difference.     * The longest side of a triangle is opposite the largest angle.     * The smallest side of a triangle is opposite the smallest angle.
• Similarity Postulates/Theorems:     * Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another, the triangles are similar.     * Side-Side-Side (SSS) Similarity: If the corresponding side lengths of two triangles are proportional, the triangles are similar.     * Side-Angle-Side (SAS) Similarity: If an angle of one triangle is congruent to an angle of a second triangle and the sides including these angles are proportional, the triangles are similar.
• Proportionality: If a line parallel to a side of a triangle intersects the other two sides, it divides those sides proportionally.
• Right Triangles: The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
• Geometric Mean: The geometric mean of two positive numbers is the positive square root of their product.

Quadrilaterals

• Parallelograms:     * Opposite sides and opposite angles are congruent.     * Consecutive angles are supplementary.     * Diagonals bisect each other.     * If one pair of opposite sides is both congruent and parallel, the quadrilateral is a parallelogram.
• Special Parallelograms:     * Rectangle: A quadrilateral with four right angles. A parallelogram is a rectangle if and only if its diagonals are congruent.     * Rhombus: A quadrilateral with four congruent sides. A parallelogram is a rhombus if and only if its diagonals are perpendicular and if each diagonal bisects a pair of opposite angles.     * Square: A quadrilateral that is both a rhombus and a rectangle.
• Trapezoids and Kites:     * Isosceles Trapezoid: Each pair of base angles is congruent. Diagonals are congruent.     * Trapezoid Midsegment: Parallel to each base and its length is one-half the sum of the lengths of the bases: $\frac{b_1 + b_2}{2}$.     * Kite: Diagonals are perpendicular. Exactly one pair of opposite angles are congruent.

Right Triangle Trigonometry

• Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs ($a^2 + b^2 = c^2$).
• Classifying Triangles by Sides:     * Right: If $c^2 = a^2 + b^2$.     * Acute: If $c^2 < a^2 + b^2$.     * Obtuse: If $c^2 > a^2 + b^2$.
• Special Right Triangles:     * 30-60-90 Triangle: Sides are $x$ (shorter leg), $x\sqrt{3}$ (longer leg), and $2x$ (hypotenuse). Hypotenuse = $2 \times \text{shorter leg}$. Longer leg = $\text{shorter leg} \times \sqrt{3}$.     * 45-45-90 Triangle: Sides are $x$, $x$, and $x\sqrt{2}$. Hypotenuse = $\text{leg} \times \sqrt{2}$.
• Trigonometry Functions (SOH CAH TOA):     * Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$     * Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$     * Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Precision and Significant Figures

• Leading zeros: Do not count (e.g., $000.01$ has $1$ significant digit).
• Trailing zeros before decimal: Do not count (e.g., $40000$ has $1$ significant digit).
• Zeros between non-zeros: Do count (e.g., $40001$ has $5$ significant digits).
• Trailing zeros after decimal: Do count (e.g., $0.3500$ has $4$ significant digits).
• Calculations:     * Adding/Subtracting: Round to the least number of decimal places. Example: $4.113 + 1000.44 = 1004.553$, round to $1004.55$.     * Multiplication/Division: Round to the least number of significant digits. Example: $4.01 \times (3.1) = 12.431$, round to $12$ (since $3.1$ has $2$ digits).

Radicals and Unit Rates

• Simplifying Radicals: Factor out perfect square factors.     * Example 1: $\sqrt{40} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$.     * Example 2: $\sqrt{125} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$.
• Rationalizing Denominators: Eliminating radicals from the denominator by multiplying by a fraction of the radical over itself.     * Example: $\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$.     * Example: $\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$.
• Unit Rates and Conversions: A unit rate describes how many units of one type correspond to one unit of another (e.g., $\frac{12\,\text{in.}}{1\,\text{ft.}}$, $\frac{5,280\,\text{ft.}}{1\,\text{mi.}}$).
• Dimensional Conversions:     * Linear (Perimeter): Use the ratio as is.     * Square (Area): Square the ratio first.     * Cubic (Volume): Cube the ratio first. Example: Finding feet in $2\,\text{yd}^3$ given $3\,\text{ft} = 1\,\text{yd}$. Since it is volume, cube the ratio $3^3/1^3 = 27$. Therefore, $2 \times 27 = 54\,\text{ft}^3$.

Coordinate Geometry and Ratios

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.     * Parallel lines have equal slopes.     * Perpendicular lines have negative reciprocal slopes ($m_1 \times m_2 = -1$).
• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Ratio Type Problems (e.g., 4:5:9):     * Method 1: Add the numbers (sum = $18$). Use the sum as the denominator. Angles = $\frac{4}{18} \times 180 = 40^{\circ}$, $\frac{5}{18} \times 180 = 50^{\circ}$, $\frac{9}{18} \times 180 = 90^{\circ}$.     * Method 2: Use common scale factor $x$. $4x + 5x + 9x = 180$. Solve $18x = 180$, so$x = 10$. Multiply ratio values by$10$to get$40$,$50$, and$90.\n\n# Symbols and Pythagorean Triples\n\n* **Symbols**:\n    * \sim: Similar\n    * \cong: Congruent\n    * \approx: Approximate\n    * \perp: Perpendicular\n    * \parallel: Parallel\n* **Pythagorean Triples**: Sets of integers that satisfy a^2 + b^2 = c^2.\n    * 3, 4, 5\n    * 5, 12, 13\n    * 7, 24, 25\n    * 9, 40, 41$$

Geometry End-of-Course Test Matrix

• Logical arguments and proof: 5-8 multiple choice items, 0-1 technology-enhanced items; 6-8 total points.
• Proving and applying properties of 2-dimensional figures: 15-19 multiple choice, 2-4 technology-enhanced, 1-3 other items; 24-26 total points.
• Figures in a coordinate plane and measurement: 5-8 multiple choice, 1-3 technology-enhanced, 0-1 other items; 7-9 total points.
• Course-Specific content: 3-5 multiple choice, 1-3 technology-enhanced items; 6 total points.
• Total Number of Items: 43 items (includes 6 try-out questions).