Geometry Comprehensive End of the Year Study Guide

Corresponding Parts and Congruent Figures

• Definition of Congruence in Figures: In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure.

• Triangle Congruence Example: Given $\Delta ABC \cong \Delta FED$   - Corresponding Angles: $\angle A \cong \angle F$, $\angle B \cong \angle E$, $\angle C \cong \angle D$.   - Corresponding Sides: $AB \cong FE$, $BC \cong ED$, $AC \cong FD$.

• Congruence Statement Rule: When writing a congruence statement, always list the corresponding vertices in the same order.

Congruent Triangles Postulates and Theorems

• Third Angle Theorem: If two angles of two triangles are congruent, then the third angles are also congruent.

• Triangle Congruence Postulates/Theorems:   - SSS: Side-Side-Side.   - SAS: Side-Angle-Side.   - AAS: Angle-Angle-Side.   - ASA: Angle-Side-Angle.   - HL: Hypotenuse-Leg (specifically for right triangles).

• CPCTC Application: Use "Corresponding Parts of Congruent Triangles are Congruent" (CPCTC) 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.

• Vertex Angle Bisector: In an isosceles triangle, the bisector of the vertex angle is the perpendicular bisector of the base.

• Equilateral Triangle Property: If a triangle is equilateral, then the triangle is equiangular.

Relationships Within Triangles and Points of Concurrency

• Point of Concurrency: The point where 3 or more lines intersect.

• Circumcenter:   - Definition: The point of concurrency of the perpendicular bisectors of a triangle.   - Property: The circumcenter of a triangle is equidistant from the vertices.

• Incenter:   - Definition: The point of concurrency of the angle bisectors.   - Property: The incenter of a triangle is equidistant from the sides.

• Centroid:   - Definition: The point of concurrency of the medians.   - Location Property: The centroid is 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.

• Midsegment:   - Definition: The segment that connects the midpoints of two sides of a triangle.   - Length Property: The midsegment is $\frac{1}{2}$ the length of the 3rd side.   - Parallel Property: The midsegment is parallel ($\parallel$) to the 3rd side.

• Perpendicular Bisector Theorem: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints.

• 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 Dimensional Relationships

• Side Length Sum Property: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

• Third Side Range Property: 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.

• Size Relationships:   - The longest side of a triangle is opposite the largest angle.   - The smallest side of a triangle is opposite the smallest angle.

Similarity Postulates and Theorems

• Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

• Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

• Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

• Parallel Line Proportionality: If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally.   - Ratio: $\frac{AE}{EB} = \frac{AF}{FC}$

• Altitude to the Hypotenuse: 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 and Parallelograms

• Properties of 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 Paracelelograms:   - Rectangle: A quadrilateral is a rectangle if and only if it has four right angles. A parallelogram is a rectangle if and only if its diagonals are congruent.   - Rhombus: A quadrilateral is a rhombus if and only if it has four congruent sides. A parallelogram is a rhombus if and only if its diagonals are perpendicular. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.   - Square: A quadrilateral is a square if and only if it is both a rhombus and a rectangle.

Trapezoids and Kites

• Isosceles Trapezoid Properties:   - Each pair of base angles is congruent.   - A trapezoid is isosceles if and only if its diagonals are congruent.

• Trapezoid Midsegment: The midsegment of a trapezoid is parallel to each base, and its length is one-half the sum of the lengths of the bases.

• Kite Properties:   - Diagonals are perpendicular.   - Exactly one pair of opposite angles are congruent.

Right Triangle Trigonometry and The Pythagorean Theorem

• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.   - Formula: $a^2 + b^2 = c^2$

• Converse of the Pythagorean Theorem: If $c^2 = a^2 + b^2$, then it is a right triangle.

• Triangle Classification via Side Lengths:   - Acute triangle: If c^2 < a^2 + b^2, then it is an acute triangle.   - Obtuse triangle: If c^2 > a^2 + b^2, then it is an obtuse triangle.

• Common Pythagorean Triples: (Integers that form right triangles)   - $3, 4, 5$   - $5, 12, 13$   - $7, 24, 25$   - $9, 40, 41$

Special Right Triangles and Trigonometric Ratios

• 30-60-90 Triangle:   - Ratio of sides: $x$ (shorter leg) : $x\times\sqrt{3}$ (longer leg) : $2x$ (hypotenuse).   - Hypotenuse = $2 \times \text{shorter leg}$.   - Longer leg = $\text{shorter leg} \times \sqrt{3}$.

• 45-45-90 Triangle:   - Ratio of sides: $x$ (leg) : $x$ (leg) : $x\times\sqrt{2}$ (hypotenuse).   - Hypotenuse = $\text{leg} \times \sqrt{2}$.

• Trigonometric Ratios (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

• Degree of Precision Rules:   - Leading Zeros: Do not count. Example: $000.01$ is precise to $1$ digit.   - Trailing Zeros before decimal: Do not count. Example: $40000$ is precise to $1$ digit.   - Zeros between nonzeros: Do count. Example: $40001$ is precise to $5$ digits.   - Trailing Zeros after decimal: Do count. Example: $0.3500$ is precise to $4$ digits.

• Arithmetic with Significant Figures:   - Adding or Subtracting: Round to the least decimal place of the components.     - Example: $4.113 + 1000.44 = 1004.553$, round to $1004.55$.   - Multiplication and Division: Round to the least number of significant digits.     - Example: $4.01 \times 3.1 = 12.431$, round to $12$ (since $3.1$ has only $2$ significant digits).

Simplifying Radicals for Geometry

• Factoring Perfect Squares:   - 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 the Denominator: No radicals allowed in denominators. Multiply by a fraction made of the radical over itself.   - Example 1: $\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$   - Example 2: $\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$

Unit Rates and Conversions

• Definition: A unit rate describes how many units of the first quantity correspond to one unit of the second quantity.

• Standard Conversions:   - $\frac{12\,in}{1\,ft}$ and $\frac{1\,ft}{12\,in}$   - $\frac{1\,hr}{60\,min}$ and $\frac{60\,min}{1\,hr}$   - $\frac{5,280\,ft}{1\,mi}$ and $\frac{1\,mi}{5,280\,ft}$

• Dimensional Analysis Examples:   - Convert $60\,mi/hr$ to $ft/sec$:     - $\frac{60\,mi}{1\,hr} \times \frac{5,280\,ft}{1\,mi} \times \frac{1\,hr}{60\,min} \times \frac{1\,min}{60\,sec} = \frac{316,800\,ft}{3,600\,sec} = 88\,ft/sec$

• Converting Multi-Dimensional Units:   - Linear (Perimeter): Use ratio as is.   - Square (Area): Square the ratio first.   - Cubic (Volume): Cube the ratio first.   - Example: How many feet are in $2\,yd^3$ if $1\,yd = 3\,ft$?     - Ratio is $3/1$. For volume, cube the ratio: $3^3 = 27$.     - The ratio for cubic yards to cubic feet is $1:27$.     - $2\,yd^3 \times 27 = 54\,ft^3$.

Coordinate Geometry Formulas

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Slopes are equal.

• Perpendicular Lines: Slopes are negative reciprocals. The product of their slopes equals $-1$ ($m_1 \times m_2 = -1$).

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Solving Ratio Type Problems (e.g., 4:5:9)

• Problem: Three angles in a triangle have the ratio $4:5:9$. What is each measure?

• Method #1 (Sum as Denominator):   - Sum of ratio: $4+5+9 = 18$.   - Small Angle: $\frac{4}{18} \times 180 = 40^{\circ}$   - Medium Angle: $\frac{5}{18} \times 180 = 50^{\circ}$   - Largest Angle: $\frac{9}{18} \times 180 = 90^{\circ}$

• Method #2 (Algebraic Scale Factor):   - Create equation: $4x + 5x + 9x = 180$.   - Simplify: $18x = 180$, therefore $x = 10$.   - Multiply: $4(10) = 40^{\circ}$, $5(10) = 50^{\circ}$, $9(10) = 90^{\circ}$.

Geometry End-of-Course (EOC) Test Map

• Logical Arguments and Proof: Items: $5-8$, Points: $6-8$.

• Proving/Applying Properties of 2-D Figures: Items: $15-19$, Points: $21-24$.

• Figures in a Coordinate Plane: Items: $5-8$, Points: $7-9$.

• Course-Specific Content: Items: $3-5$, Points: $6$.

• Total Items Scale Score: $43$.

• Symbols Guide:   - $\sim$ : Similar   - $\cong$ : Congruent   - $\approx$ : Approximate (used when rounding)   - $\perp$ : Perpendicular   - $\parallel$ : Parallel