Comprehensive Geometry Notes

Chapter 11: Circumference, Area, and Volume

• Learning Goals:

  • Understand and use the formula for the circumference of a circle.

  • Use arc lengths to find measures.

  • Solve real-life problems involving circumference and arc lengths.

  • Measure angles in radians.

  • Understand and use the formula for the area of a circle.

  • Apply the formula for population density.

  • Find areas of sectors.

  • Use areas of sectors to solve problems.

  • Find areas of rhombuses and kites.

  • Find angle measures in regular polygons.

  • Calculate areas of regular polygons.

  • Classify solids.

  • Describe cross sections of solids.

  • Sketch and describe solids of revolution.

  • Find volumes of prisms and cylinders.

  • Apply the formula for density.

  • Use volumes of prisms and cylinders in problem-solving.

  • Find volumes of pyramids.

  • Use volumes of pyramids in applications.

  • Find surface areas of right cones.

  • Calculate volumes of cones.

  • Use volumes of cones to solve problems.

  • Find surface areas of spheres.

  • Calculate volumes of spheres.

Key Definitions

• Circumference of a Circle: The distance around the circle.

• Radian: For a circle with radius r, the measure of an angle in standard position whose terminal side intercepts an arc of length r is one radian.

• Sector of a Circle: The region bounded by two radii of the circle and their intercepted arc.

• Population Density: A measure of how many people live within a given area (city, county, or state).

• Center of a Regular Polygon: The center of its circumscribed circle.

• Apothem of a Regular Polygon: The distance from the center to any side of a regular polygon.

• Radius of a Regular Polygon: The radius of its circumscribed circle.

• Central Angle of a Regular Polygon: An angle formed by two radii drawn to consecutive vertices of the polygon.

• Polyhedron: A solid that is bounded by polygons.

• Face: A flat surface of a polyhedron.

• Edge: A line segment formed by the intersection of two faces of a polyhedron.

• Vertex: A point where three or more edges of a polyhedron meet.

• Arc Length: A portion of the circumference of a circle.

• Cross Section: The intersection of a plane and a solid.

• Solid of Revolution: A three-dimensional figure formed by rotating a two-dimensional shape around an axis.

• Axis of Revolution: The line around which a two-dimensional shape is rotated to form a three-dimensional figure.

• Volume of a Solid: The number of cubic units contained in its interior.

• Cavalieri’s Principle: If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

• Density: The amount of matter that an object has in a given unit of volume.

• Similar Solids: Two solids of the same type with equal ratios of corresponding linear measures.

• Lateral Surface of a Cone: Consists of all segments that connect the vertex with points on the base edge.

• Chord of a Sphere: A segment whose endpoints are on the sphere.

• Great Circle of a Sphere: The intersection of a plane and a sphere such that the plane contains the center of the sphere.

Formulas and Equations

• Circumference of a Circle:
  C = \pi d where d is the diameter.
  C = 2\pi r where r is the radius.

• Arc Length:
  \frac{\text{Arc length of } AB}{2\pi} = \frac{mAB}{360^\circ}
  \text{Arc length of } AB = \frac{mAB}{360^\circ} \cdot 2\pi

• Area of a Sector:
  \frac{\text{Area of sector } APB}{\pi r^2} = \frac{mAB}{360^\circ}
  \text{Area of sector } APB = \frac{mAB}{360^\circ} \cdot \pi r^2

• Area of a Circle:
  A = \pi r^2

• Area of a Regular Polygon:
  A = \frac{1}{2}aP where a is the apothem and P is the perimeter.
  Also expressed as A = \frac{1}{2} a \cdot ns where n is the number of sides and s is the side length.

• Area of a Rhombus or Kite:
  A = \frac{1}{2} d1 d2 where d1 and d2 are the lengths of the diagonals.

• Volume of a Prism:
  V = Bh where B is the area of the base and h is the height.

• Volume of a Cylinder:
  V = Bh = \pi r^2 h

• Volume of a Pyramid:
  V = \frac{1}{3} Bh

• Surface Area of a Right Cone:
  S = \pi r^2 + \pi r \ell where r is the radius of the base and \ell is the slant height.

• Volume of a Cone:
  V = \frac{1}{3} Bh = \frac{1}{3} \pi r^2 h

• Surface Area of a Sphere:
  S = 4\pi r^2

• Volume of a Sphere:
  V = \frac{4}{3} \pi r^3

Conversions

• Degrees to Radians: Multiply degree measure by \frac{\pi \text{ radians}}{180^\circ}.

• Radians to Degrees: Multiply radian measure by \frac{180^\circ}{\pi \text{ radians}}.

Similar Solids

• Two solids are similar if they have equal ratios of corresponding linear measures (heights, radii).

• The ratio of corresponding linear measures is called the scale factor (k).

• If two similar solids have a scale factor of k, then the ratio of their volumes is k^3.

Chapter 12: Probability

• Learning Goals:

  • Find sample spaces.

  • Find theoretical probabilities.

  • Find experimental probabilities.

  • Determine whether events are independent events.

  • Find probabilities of independent and dependent events.

  • Find conditional probabilities.

  • Make two-way tables.

  • Find relative and conditional relative frequencies.

  • Use conditional relative frequencies to find conditional probabilities.

  • Find probabilities of compound events.

  • Use more than one probability rule to solve real-life problems.

  • Use the formula for the number of permutations.

  • Use the formula for the number of combinations.

  • Use combinations and the Binomial Theorem to expand binomials.

  • Construct and interpret probability distributions.

  • Construct and interpret binomial distributions.

Key Definitions

• Probability Experiment: An action, or trial, that has varying results.

• Outcomes: The possible results of a probability experiment.

• Event: A collection of one or more outcomes in a probability experiment.

• Sample Space: The set of all possible outcomes for an experiment.

• Probability of an Event: A measure of the likelihood, or chance, that the event will occur.

• Theoretical Probability: The ratio of the number of favorable outcomes to the total number of outcomes when all outcomes are equally likely.

• Geometric Probability: A probability found by calculating a ratio of two lengths, areas, or volumes.

• Experimental Probability: The ratio of the number of successes, or favorable outcomes, to the number of trials in a probability experiment.

• Independent Events: Two events are independent events when the occurrence of one event does not affect the occurrence of the other event.

• Dependent Events: Two events are dependent events when the occurrence of one event does affect the occurrence of the other event.

• Conditional Probability: The probability that event B occurs given that event A has occurred, written as P(B|A).

• Two-Way Table: A frequency table that displays data collected from one source that belong to two different categories.

• Joint Frequency: Each entry in a two-way table.

• Marginal Frequencies: The sums of the rows and columns in a two-way table.

• Joint Relative Frequency: The ratio of a frequency that is not in the total row or the total column to the total number of values or observations in a two-way table.

• Marginal Relative Frequency: The sum of the joint relative frequencies in a row or a column in a two-way table.

• Conditional Relative Frequency: The ratio of a joint relative frequency to the marginal relative frequency in a two-way table.

• Compound Event: The union or intersection of two events.

• Overlapping Events: Two events that have one or more outcomes in common.

• Disjoint (Mutually Exclusive) Events: Two events that have no outcomes in common.

• Permutation: An arrangement of objects in which order is important.

• n Factorial: The product of the integers from 1 to n, for any positive integer n.

• Combination: A selection of objects in which order is not important.

• Random Variable: A variable whose value is determined by the outcomes of a probability experiment.

• Probability Distribution: A function that gives the probability of each possible value of a random variable.

• Binomial Distribution: A type of probability distribution that shows the probabilities of the outcomes of a binomial experiment.

• Binomial Experiment: An experiment in which there are a fixed number of independent trials, exactly two possible outcomes for each trial, and the probability of success is the same for each trial.

Formulas and Equations

• Probability of the Complement of an Event:
  P(\overline{A}) = 1 - P(A)

• Probability of Independent Events:
  P(A \text{ and } B) = P(A) \cdot P(B)

• Probability of Dependent Events:
  P(A \text{ and } B) = P(A) \cdot P(B|A)

• Probability of Compound Events:

  • If A and B are any two events:
    P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

  • If A and B are disjoint events:
    P(A \text{ or } B) = P(A) + P(B)

• Permutations:

  • The number of permutations of n objects is: nPn = n!

  • The number of permutations of n objects taken r at a time is: nPr = \frac{n!}{(n-r)!}

• Combinations:

  • The number of combinations of n objects taken r at a time is: nCr = \frac{n!}{(n-r)! \cdot r!}

• Binomial Theorem:
  (a + b)^n = nC0 a^n b^0 + nC1 a^{n-1} b^1 + nC2 a^{n-2} b^2 + \dots + nCn a^0 b^n

• Binomial Experiments:
  P(k \text{ successes}) = nCk \cdot p^k (1 - p)^{n-k}

Relative and Conditional Relative Frequencies

• Joint Relative Frequency: Ratio of a frequency (not in total row/column) to the total number of values.

• Marginal Relative Frequency: Sum of joint relative frequencies in a row or column.

• Conditional Relative Frequency: Ratio of a joint relative frequency to the marginal relative frequency.

Chapter 7: Quadrilaterals and Other Polygons

• Learning Goals:

  • Use the interior angle measures of polygons.

  • Use the exterior angle measures of polygons.

  • Use properties to find side lengths and angles of parallelograms.

  • Use parallelograms in the coordinate plane.

  • Identify and verify parallelograms.

  • Show that a quadrilateral is a parallelogram in the coordinate plane.

  • Use properties of special parallelograms.

  • Use properties of diagonals of special parallelograms.

  • Use coordinate geometry to identify special types of parallelograms.

  • Use properties of trapezoids.

  • Use the Trapezoid Midsegment Theorem to find distances.

  • Use properties of kites.

  • Identify quadrilaterals.

Key Definitions

• Diagonal of a Polygon: A segment that joins two nonconsecutive vertices.

• Equilateral Polygon: A polygon in which all sides are congruent.

• Equiangular Polygon: A polygon in which all angles are congruent.

• Regular Polygon: A convex polygon that is both equilateral and equiangular.

• Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.

• Rhombus: A parallelogram with four congruent sides.

• Rectangle: A parallelogram with four right angles.

• Square: A parallelogram with four congruent sides and four right angles.

• Trapezoid: A quadrilateral with exactly one pair of parallel sides.

• Bases of a Trapezoid: The parallel sides of a trapezoid.

• Base Angles of a Trapezoid: Two consecutive angles whose common side is a base.

• Legs of a Trapezoid: The nonparallel sides of a trapezoid.

• Isosceles Trapezoid: A trapezoid with congruent legs.

• Midsegment of a Trapezoid: The segment that connects the midpoints of its legs.

• Kite: A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorems and Corollaries

• Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex n-gon is (n - 2) \cdot 180^\circ.

• Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360^\circ.

• Corollary 7.1: The sum of the measures of the interior angles of a quadrilateral is 360^\circ.

• Parallelogram Opposite Sides Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent.

• Parallelogram Opposite Angles Theorem: If a quadrilateral is a parallelogram, then its opposite angles are congruent.

• Parallelogram Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

• Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other.

• Parallelogram Opposite Sides Converse: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

• Parallelogram Opposite Angles Converse: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

• Opposite Sides Parallel and Congruent Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

• Parallelogram Diagonals Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

• Rhombus Diagonals Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular.

• Rhombus Opposite Angles Theorem: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

• Rectangle Diagonals Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent.

• Isosceles Trapezoid Base Angles Theorem: If a trapezoid is isosceles, then each pair of base angles is congruent.

• Isosceles Trapezoid Base Angles Converse: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

• Isosceles Trapezoid Diagonals Theorem: A trapezoid is isosceles if and only if its diagonals are congruent.

• Trapezoid Midsegment Theorem: 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 Diagonals Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular.

• Kite Opposite Angles Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

• Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides.

• Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles.

• Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Ways to Prove a Quadrilateral is a Parallelogram

1. Show that both pairs of opposite sides are parallel (Definition).

2. Show that both pairs of opposite sides are congruent (Parallelogram Opposite Sides Converse).

3. Show that both pairs of opposite angles are congruent (Parallelogram Opposite Angles Converse).

4. Show that one pair of opposite sides are congruent and parallel (Opposite Sides Parallel and Congruent Theorem).

5. Show that the diagonals bisect each other (Parallelogram Diagonals Converse).

Chapter 8: Similarity

• Learning Goals:

  • Use similarity statements.

  • Find corresponding lengths in similar polygons.

  • Find perimeters and areas of similar polygons.

  • Decide whether polygons are similar.

  • Use the Angle-Angle Similarity Theorem.

  • Solve real-life problems.

  • Use the Side-Side-Side Similarity Theorem.

  • Use the Side-Angle-Side Similarity Theorem.

  • Prove slope criteria using similar triangles.

  • Use the Triangle Proportionality Theorem and its converse.

  • Use other proportionality theorems.

Key Concepts

• Similar Polygons: Polygons with congruent corresponding angles and proportional corresponding side lengths.

• Similarity Transformation: Preserves angle measure and enlarges or reduces side lengths by a scale factor k.

Theorems

• Theorem 8.1 (Perimeters of Similar Polygons): If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

• Theorem 8.2 (Areas of Similar Polygons): If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths.

• Theorem 8.3 (Angle-Angle (AA) Similarity Theorem): If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

• Theorem 8.4 (Side-Side-Side (SSS) Similarity Theorem): If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

• Theorem 8.5 (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.

• Theorem 8.6 (Triangle Proportionality Theorem): If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

• Theorem 8.7 (Converse of the Triangle Proportionality Theorem): If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

• Theorem 8.8 (Three Parallel Lines Theorem): If three parallel lines intersect two transversals, then they divide the transversals proportionally.

• Theorem 8.9 (Triangle Angle Bisector Theorem): If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

Triangle Similarity Theorems

• AA Similarity Theorem: If \angle A \cong \angle D and \angle B \cong \angle E, then \triangle ABC \sim \triangle DEF.

• SSS Similarity Theorem: If for triangles ABC and DEF, \frac{DA}{DA} = \frac{AB}{DE} = \frac{AB}{DE}, then \triangle ABC \sim \triangle DEF.

• SAS Similarity Theorem: If \angle A \cong \angle D and \frac{AA}{DA} = \frac{AB}{DE}, then \triangle ABC \sim \triangle DEF.

Corresponding Parts of Similar Polygons

• Corresponding angles are congruent.

• Corresponding side lengths are proportional.

Chapter 10: Circles

• Learning Goals:

  • Identify special segments and lines related to circles.

  • Draw and identify common tangents.

  • Use properties of tangents.

  • Find arc measures.

  • Identify congruent arcs.

  • Prove that circles are similar.

  • Use chords of circles to find lengths and arc measures.

  • Use inscribed angles.

  • Use inscribed polygons.

  • Find angle and arc measures formed by tangents and chords.

  • Use circumscribed angles.

  • Use segments of chords, tangents, and secants to find lengths.

  • Write and graph equations of circles.

  • Write coordinate proofs involving circles.

  • Solve real-life problems using graphs and circles.

Key Definitions

• Circle: The set of all points in a plane that are equidistant from a given point.

• Center: The point from which all points on a circle are equidistant.

• Radius: A segment whose endpoints are the center and any point on the circle.

• Chord: A segment whose endpoints are on the circle.

• Diameter: A chord that contains the center of the circle.

• Secant: A line that intersects a circle in two points.

• Tangent: A line in the plane of a circle that intersects the circle in exactly one point.

• Point of Tangency: The point at which a tangent line intersects a circle.

• Tangent Circles: Coplanar circles that intersect in one point.

• Concentric Circles: Coplanar circles that have a common center.

• Common Tangent: A line or segment that is tangent to two coplanar circles.

• Central Angle: An angle whose vertex is the center of the circle.

• Minor Arc: An arc with a measure less than 180^\circ.

• Major Arc: An arc with a measure greater than 180^\circ.

• Semicircle: An arc with endpoints that are the endpoints of a diameter.

• Adjacent Arcs: Arcs of a circle that have exactly one point in common.

• Congruent Circles: Circles that can be mapped onto each other by a rigid motion or a composition of rigid motions; circles that have the same radius.

• Congruent Arcs: Arcs that have the same measure and are of the same circle or of congruent circles.

• Similar Arcs: Arcs that have the same measure.

• Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle.

• Intercepted Arc: An arc that lies between two lines, rays, or segments.

• Inscribed Polygon: A polygon when all its vertices lie on a circle.

• Subtend: If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to subtend the angle.

• Circumscribed Circle: A circle that contains all the vertices of an inscribed polygon.

• Circumscribed Angle: An angle whose sides are tangent to a circle.

• Segments of a Chord: The segments formed from two chords that intersect in the interior of a circle.

• Tangent Segment: A segment that is tangent to a circle at an endpoint.

• Secant Segment: A segment that contains a chord of a circle and has exactly one endpoint outside the circle.

• External Segment: The part of a secant segment that is outside the circle.

Theorems

• Theorem 10.1 (Tangent Line to Circle Theorem): In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

• Theorem 10.2 (External Tangent Congruence Theorem): Tangent segments from a common external point are congruent.

• Theorem 10.3 (Congruent Circles Theorem): Two circles are congruent circles if and only if they have the same radius.

• Theorem 10.4 (Congruent Central Angles Theorem): In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

• Theorem 10.5 (Similar Circles Theorem): All circles are similar.

• Theorem 10.6 (Congruent Corresponding Chords Theorem): In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

• Theorem 10.7 (Perpendicular Chord Bisector Theorem): If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

• Theorem 10.8 (Perpendicular Chord Bisector Converse): If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter.

• Theorem 10.9 (Equidistant Chords Theorem): In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

• Theorem 10.10 (Measure of an Inscribed Angle Theorem): The measure of an inscribed angle is one-half the measure of its intercepted arc.

• Theorem 10.11 (Inscribed Angles of a Circle Theorem): If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

• Theorem 10.12 (Inscribed Right Triangle Theorem): If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

• Theorem 10.13 (Inscribed Quadrilateral Theorem): A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

• Theorem 10.14 (Tangent and Intersected Chord Theorem): If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.

• Theorem 10.15 (Angles Inside the Circle Theorem): If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

• Theorem 10.16 (Angles Outside the Circle Theorem): If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

• Theorem 10.17 (Circumscribed Angle Theorem): The measure of a circumscribed angle is equal to 180^\circ minus the measure of the central angle that intercepts the same arc.

• Theorem 10.18 (Segments of Chords 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.

• Theorem 10.19 (Segments of Secants Theorem): If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

• Theorem 10.20 (Segments of Secants and Tangents Theorem): If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Postulates

• Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Standard Equation of a Circle

The standard equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2