Geometry Exam Notes

Unit 1

• Inequality Theorem (Side-Side-Side Inequality Theorem or SSS Inequality Theorem): If side 'a' > side 'b', then angle A > angle B. If not, then not A.
• Triangle Inequality Theorem: States that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
• Pythagorean Inequality Theorem:
  • If c^2 > b^2 + a^2, then the triangle is obtuse.
  • If c^2 < b^2 + a^2, then the triangle is acute.
  • If $c^2 = b^2 + a^2$, then the triangle is right.
• Hinge Theorem: Compares two triangles with two corresponding congruent sides.
• Natural Angle Restrictions:
  • Smallest angle: No restriction (0).
  • Largest angle: 0 to 180 degrees.

Unit 8

• 45-45-90 Triangle:
  • Legs are equal (a).
  • Hypotenuse = $a\sqrt{2}$.
• 30-60-90 Triangle: Side lengths are in a specific ratio.
• Rationalizing: Example: $4/\sqrt{2}$ (implies rationalizing the denominator).
• Area of a Triangle: $A = (1/2)bh$, $A = (1/2)ab\sin(C)$
• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.

Unit 11

• Definitions of a polygon:
  • Convex Polygon: All vertices lie on the exterior of a circle.
  • Concave Polygon: At least one vertex lies in the interior of the polygon.
  • Equilateral Polygon: All sides are congruent.
  • Equiangular Polygon: All angles are congruent.
  • Regular Polygon: All sides and all angles are congruent.
• Sum of Interior Angles: $180(n-2)$, where n is the number of sides.
• Each Interior Angle (for a regular polygon): $\frac{180(n-2)}{n}$.
• Sum of Exterior Angles: $360^\circ$.
• Each Exterior Angle (for a regular polygon): $\frac{360}{n}$.
• Areas of Parallelograms:
  • Parallelogram: $A = bh$.
  • Square: $A = bh$.
  • Trapezoid: $A = \frac{1}{2}h(b<em>1 + b</em>2)$.
• Prism:
  • Volume: $V = Bh$.
  • Surface Area: $SA = 2B + Ph$.
• Cylinder:
  • Volume: $V = \pi r^2 h$.
  • Surface Area: $SA = 2\pi r^2 + 2\pi rh$.
• Pyramid:
  • Volume: $V = \frac{1}{3} Bh$.
  • Surface Area: $SA = B + \frac{1}{2}PL$. (Here, L seems to refer to slant height and the second part of the SA formula).
• Cone:
  • SA = Area of base + area of sector = $SA = \pi r^2 + \pi r L$ with L being slant height.
• Sphere:
  • Volume: $V = \frac{4}{3}\pi r^3$.
  • Surface Area: $SA = 4\pi r^2$.
• Polygon Area Given Apothem: $A = \frac{1}{2}Pa$. (P = Perimeter, a = apothem length).
  • Pentagon: 5 sides.
  • Hexagon: 6 sides, $A = k^2$, k might relate to side length in specific hexagon formulas.
  • Heptagon: 7 sides.

Proofs

• Coordinate Proofs
• Triangle Congruence Postulates:
  • SAS (Side-Angle-Side).
  • ASA (Angle-Side-Angle).
  • AAS (Angle-Angle-Side).

Unit 12

• Trigonometric Ratios:
  • $\sin = \frac{opposite}{hypotenuse}$
  • $\cos = \frac{adjacent}{hypotenuse}$
  • $\tan = \frac{opposite}{adjacent}$
  • $\cot = \frac{adjacent}{opposite}$
• Law of Sines: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$.
• Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A$.
  • Can only switch from Law of Cosines to Law of Sines if you have the largest angle.
• Angle of Depression: Angle from the horizontal.
• Ambiguous Case (Law of Sines):
  • Acute Triangle:
    • Opposite < Adjacent: No solution or 2 solutions
    • Opposite = Adjacent: 1 solution
    • Opposite > Adjacent: 1 solution
  • Obtuse Triangle
    • Opposite ≤ Adjacent: No Solution
    • Opposite > Adjacent: 1 solution

Unit 9

• Relationships between angles: new angle = 180 - (sound angle + given angle)
• The Law of Sines can also be used in the form: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
• Properties of Quadrilaterals:
  • Parallelogram:
    1. Opposite sides are parallel.
    2. Opposite sides are congruent.
    3. Diagonals bisect each other.
    4. Opposite angles are congruent.
    5. Consecutive angles are supplementary.
  • Rectangle: Slopes
    1. Four right angles.
    2. Parallelogram properties.
    3. Diagonals are congruent.
  • Rhombus: Slopes
    1. Four congruent sides.
    2. Parallelogram properties.
    3. Diagonals bisect angles.
  • Square: Slopes
    1. All parallelogram, rectangle, and rhombus properties.
  • Trapezoid:
    1. One pair of parallel sides.
    2. Mid-segment is parallel to bases.
    3. Mid-segment length is half the sum of the lengths of the bases.
  • Isosceles Trapezoid: Distances
    1. Trapezoid properties.
    2. Legs are congruent.
    3. Base angles are congruent.
    4. Diagonals are congruent.
    5. Opposite angles are supplementary.
  • Kite: Slopes
    1. Two pairs of consecutive congruent sides.
    2. Diagonals are perpendicular.
    3. Angles formed by non-congruent sides are bisected by a diagonal.
    4. Diagonals connect Vertex
• Circle Equations and Properties:
  • Standard Equation: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
  • On the circle: $(x,y) = r$
  • Outside the circle: (x,y) > r
    • Inside the circle: (x,y) < r
• Arcs and Angles:
  • Minor Arc: < 180°
  • Major Arc: > 180° (measure = 360 - minor arc measure)
  • Arc Length: $L = \frac{x}{360} * 2\pi r$
  • Central Angle = measure of intercepted arc. ($m\angle ACB = m\stackrel{\frown}{AB}$)
  • Inscribed Angle = vertex on the circle, measure is half the intercepted arc. ($\angle ACB = \frac{1}{2} m\stackrel{\frown}{AB}$)
  • Angle Inside = vertex inside the circle, $m\angle AXB = \frac{1}{2} (Sum of arcs)$
  • Angle Outside = vertex outside the circle, $m\angle AXB = \frac{1}{2} (difference of arcs)$
  • Quadrilateral Inscribed in Circle: opposite angles are supplementary.
• Lines Intersecting Circles:
  • Chord: endpoints both on the circle.
  • Secant: line intersecting at 2 points, extends out of the circle.
  • Tangent: line intersecting at 1 point.
• Chord Theorems:
  • Chord Theorem 1: minor arcs are congruent if and only if corresponding chords are congruent.
  • Chord Theorem 2: 2 chords are congruent if they are equidistant from the center.
  • Chord Theorem 3: A diameter perpendicular to a chord bisects the chord and its arc.
• Segment Theorems:
  • Segments of Chord Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord: $a<em>b = c</em>d$.
  • Segments of Secants Theorem: If two secant segments are drawn to a circle from an exterior point, then the product of the length of one secant segment and its external segment equals the product of the length of the other secant segment and its external segment: $a(a+b) = c(c+d)$.
  • Corollary - 2 segments tangent: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the length of the secant segment and its external segment, or $a(a) = b(b+c)$. *Semicircle Right Angle (Thales)
    • only do this if it is an arc
• Statistical Measures:
  • Mean: add all data, divide by number of data points.
  • Median: middle value of data (when ordered).
  • Mode: most common number(s) (can be none).
  • Range: max - min.
  • Outliers affect the mean. Median is more reliable when outliers are present.
• Multiply if and, add if or
• Probability:
  • Sample Space: all possible outcomes.
  • Theoretical Probability: probability of an event occurring if all outcomes are equally likely.
  • Complement: outcomes not in the event: $P(A') = 1 - P(A)$.
  • Experimental Probability: probability based on repeated trials.
  • Geometric Probability: ratios of lengths, areas, or volumes.
  • Conditional Probability: $P(A|B)$.
• Frequency:
  • Joint Frequency: entry in a table.
  • Marginal Frequency: sums of rows or columns in a table.
• Probability Rules:
  • General Equation: $P(E \cup F) = P(E) + P(F) - P(E \cap F)$.
  • Mutually Exclusive Events: $P(A \cup B) = P(A) + P(B)$.
  • Independent Events: $P(A \cap B) = P(A) * P(B)$.
• Inequality EX
  • There are 3 segments (Subtract)
    • S+M>B
    • S+MB