Geometry unit 2.3
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The document defines and provides examples of conditional statements, converses, and biconditionals ("if and only if" or "iff" statements). It explains that a biconditional combines a conditional statement with its converse, making the statement true in both directions. Examples show biconditionals used to define mathematical and geometric relationships, such as a number being divisible by 5 iff it ends in 0 or 5. Definitions are also explained to be reversible, and can thus be written as biconditionals.
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1) Conditional statements relate two parts - a hypothesis (if part) and a conclusion (then part). If the hypothesis is true, then the conclusion must be true as well.
2) The converse of a conditional statement switches the hypothesis and conclusion. The inverse negates both parts. The contrapositive obtains the inverse and then switches parts.
3) A biconditional statement uses "if and only if" to join a conditional statement with its converse when both are true. This creates a single statement expressing their relationship.
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2. Other types of conditional statements include the converse, which switches the hypothesis and conclusion; the negation, which negates the original statement; and the contrapositive, which negates both the hypothesis and conclusion and switches their order.
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Converse: If two angles are supplementary, then they form a linear pair.
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Contrapositive: If two angles are not supplementary, then they do not form a linear pair.
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Converse: If a parallelogram is a rectangle, then it has a right angle.
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Algebra 2 unit 10.4
This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
Algebra 2 unit 10.3
This document provides instruction on right triangles and trigonometric ratios. It begins with examples of finding missing angles and side lengths in right triangles using trigonometric functions like sine, cosine and tangent. Special right triangles involving 30-60-90 and 45-45-90 triangles are discussed. Real world applications include finding the height a skier leaves a ramp and the height of a tree using an angle of elevation. The document also covers cosecant, secant and cotangent functions and has practice problems for students.
Algebra 2 unit 10.2
This document discusses inverse trigonometric functions and how to use them to solve problems. It defines the inverse sine, cosine, and tangent functions and explains that they find the measure of angles given the value of a trigonometric function. Examples are provided to demonstrate evaluating inverse trigonometric functions and using them to solve equations and application problems like finding the angle a ladder should make with the ground given its length. Restricting the domains of the trigonometric functions is necessary to define the inverse functions since more than one angle can produce the same output value.
11.1 combination and permutations
This document discusses permutations and combinations. It provides examples of using formulas and the fundamental counting principle to calculate the number of possible outcomes for different scenarios involving selecting items with or without regard for order. Examples include determining the number of possible sandwiches, voicemail passwords, bead necklace designs, and ways to select people or items from a group. The key concepts of permutations, combinations, and factorials are explained.
Unit 11.3 probability of multiple events
This document summarizes key concepts about probability of multiple events from Holt Algebra 1. It discusses independent and dependent events and provides examples of calculating probabilities. For independent events, the probability is calculated by multiplying the individual probabilities. For dependent events, the probability of the second event is based on the sample space after the first event occurs. Examples include flipping coins, drawing marbles from a bag with and without replacement, and selecting items randomly. Diagrams are used to illustrate how the sample space changes for dependent events.
Unit 11.2 experimental probability
This document provides an overview of experimental probability:
- It defines key terms like experiment, trial, outcome, and sample space.
- It explains how to calculate experimental probability by taking the number of times an event occurs over the total number of trials.
- It discusses how experimental probability can be used to make predictions about future experiments based on past results.
Unit 11.2 theoretical probability
This document provides an overview of theoretical probability concepts including:
- Theoretical probability is calculated as the number of favorable outcomes divided by the total number of equally likely outcomes.
- Experiments are considered "fair" if all outcomes are equally likely.
- The probability of an event occurring plus the probability of it not occurring equals 1.
- Complement probability refers to outcomes not included in the main event.
- Odds can be used to express likelihood and converted to and from probabilities using ratios of favorable to total outcomes.
11.1 11.1 combination and permutations
This document discusses permutations and combinations. It provides examples of using the fundamental counting principle and formulas for permutations and combinations to calculate the number of possible outcomes for various events and selections. Examples include determining the number of possible meal combinations, voicemail passwords, bead stringing patterns, sandwich ingredients, swimming race outcomes, people lining up, and selecting groups from a class. The key concepts of permutations versus combinations and using factorials are also explained.
Geometry 201 unit 5.7
1) The document discusses using inequalities to compare angles and side lengths in two triangles. It provides examples of applying the hinge theorem and its converse to determine relationships between angles and sides.
2) One example involves comparing the distances traveled by two people leaving school at different routes to determine who is farther from school.
3) The document also demonstrates how to write two-column proofs to prove triangle relationships based on given information about the triangles. It includes practice problems for the reader to check their understanding.
Geometry 201 unit 5.5
This document provides examples and explanations of indirect proofs. It begins with examples of writing indirect proofs to show that a triangle cannot have two right angles or that a number is greater than 0. It then discusses using inequalities in indirect proofs involving triangles. Examples demonstrate ordering triangle sides and angles, applying the triangle inequality theorem to determine if a triangle can exist with given side lengths, and using indirect proofs to find possible side lengths. The document concludes with practice problems applying these concepts.
Geometry 201 unit 5.4
1) A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The medians of any triangle are concurrent at the centroid.
2) An altitude of a triangle is a perpendicular segment from a vertex to the opposite side. The three altitudes of a triangle are concurrent at the orthocenter.
3) This document discusses properties of medians and altitudes of triangles, including using the centroid and orthocenter to solve problems involving lengths of segments and finding coordinates of special points like the centroid and orthocenter. Examples are provided to illustrate these concepts.
Geometry 201 unit 4.7
This document provides information about using the Hypotenuse-Leg Theorem to prove triangles congruent. It gives examples of when the theorem can and cannot be used. It also asks what additional information is needed to prove triangles congruent using the theorem. The document then discusses how to identify and separate overlapping triangles when proving congruence. It provides examples of writing paragraph proofs to show triangles are congruent using angle-angle-side, side-side-side, and angle-side-angle criteria.
Geometry 201 unit 4.4
1) The document discusses using corresponding parts of congruent triangles (CPCTC) to prove that parts of triangles are congruent. CPCTC states that if two triangles are congruent, then their corresponding parts are congruent.
2) It provides examples of using CPCTC to prove that corresponding angles and sides of congruent triangles are congruent. Proofs involve showing triangles are congruent using criteria like SSS, SAS, ASA etc. and then invoking CPCTC.
3) The document emphasizes working backwards in proofs - looking for congruent angles/sides and then identifying congruent triangles to apply CPCTC. Several practice examples are provided to illustrate this technique.
Geometry 201 unit 4.3
This document discusses triangle congruence using the ASA, AAS, and Hypotenuse-Leg (HL) theorems. It provides examples of applying these theorems to determine if triangles are congruent and to prove triangles congruent. The examples include real world applications involving directions and distances between locations. Key terms like included side are also defined. Practice problems are included for students to determine if a triangle congruence theorem can be used or additional information is needed.
Geometry 201 unit 4.2
This document provides instruction on proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It includes examples of using SSS and SAS to show that triangles are congruent, as well as practice problems for students to verify congruence. Key concepts covered are triangle rigidity, included angles, and applying SSS and SAS to real-world geometry problems.
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- 2. Contains the phrase “if & only if” Abbr. iff It is a conditional statement & its converse all in one. Ex: 3 points are collinear iff they are on the same line. Conditional: If 3 pts are collinear, then they are on the same line. Converse: If 3 pts are on the same line, then they are collinear.
- 4. x=3 if and only if x2=9. Yes, it’s a biconditional. Orig. Conditional statement? If x=3, then x2=9. Converse? If x2=9, then x=3. True or False? Biconditional is False True False, x could be -3
- 6. Conditional: If a # ends in 0 or 5, then it is evenly divisible by 5. TTrruuee Converse: If a # is evenly divisible by 5, then it ends in 0 or 5. True or False? TTrruuee Biconditional? A # ends in 0 or 5 iff it is divisible by 5.
- 7. •Definitions can be written in “if-then” form. •The converse of a definition is always true. •This means the defn. is true forwards & backwards. BICONDITIONAL!!!
- 8. Show that the definition of right angle is reversible. Then write it as a biconditional. •Definition: A right angle is an angle whose measure is 90.
- 9. CCoonnddiittiioonnaall:: An angle is a right angle then the angle measures 90 degrees CCoonnvveerrssee:: True An angle is 90 degrees then it is a right angle BBiiccoonnddiittiioonnaall An angle is a right angle iff it measures 90 degrees
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