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Questions and Answers
In right triangle $\triangle ABC$, with right angle at $C$, what additional information is sufficient to prove that it is congruent to right triangle $\triangle DEF$, with a right angle at $F$, using the Hypotenuse-Angle (HA) Theorem?
In right triangle $\triangle ABC$, with right angle at $C$, what additional information is sufficient to prove that it is congruent to right triangle $\triangle DEF$, with a right angle at $F$, using the Hypotenuse-Angle (HA) Theorem?
- Knowing the measures of $\angle A$ and $\angle D$.
- Knowing the length of hypotenuse $AB$ and the measure of $\angle A$, and that the length of hypotenuse $DE$ and the measure of $\angle D$ are the same. (correct)
- Knowing the lengths of leg $AC$ and hypotenuse $DE$.
- Knowing the lengths of both legs $AC$ and $BC$.
Which condition does not guarantee that two lines are parallel?
Which condition does not guarantee that two lines are parallel?
- Alternate exterior angles are supplementary. (correct)
- Corresponding angles are congruent.
- Consecutive interior angles are supplementary.
- Alternate interior angles are congruent.
Line $l$ and line $m$ are intersected by a transversal $t$. Which of the following conditions is sufficient to prove that line $l$ is parallel to line $m$?
Line $l$ and line $m$ are intersected by a transversal $t$. Which of the following conditions is sufficient to prove that line $l$ is parallel to line $m$?
- Two pairs of corresponding angles are supplementary.
- A pair of alternate interior angles are congruent. (correct)
- A pair of same-side interior angles are congruent.
- Two pairs of vertical angles are congruent.
A fair six-sided die is rolled. What is the probability of rolling an even number or a number less than 3?
A fair six-sided die is rolled. What is the probability of rolling an even number or a number less than 3?
A bag contains 4 red marbles, 5 blue marbles, and 3 green marbles. If one marble is drawn at random, what is the probability that it is not blue?
A bag contains 4 red marbles, 5 blue marbles, and 3 green marbles. If one marble is drawn at random, what is the probability that it is not blue?
Two lines intersect, forming angles $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$. Given that $\angle 1$ and $\angle 3$ are vertical angles, and $\angle 2$ and $\angle 4$ are vertical angles, which statement must be true if the two lines are perpendicular?
Two lines intersect, forming angles $\angle 1$, $\angle 2$, $\angle 3$, and $\angle 4$. Given that $\angle 1$ and $\angle 3$ are vertical angles, and $\angle 2$ and $\angle 4$ are vertical angles, which statement must be true if the two lines are perpendicular?
What is the probability of drawing a heart or a face card (Jack, Queen, or King) from a standard 52-card deck?
What is the probability of drawing a heart or a face card (Jack, Queen, or King) from a standard 52-card deck?
In $\triangle ABC$ and $\triangle XYZ$, $\angle A \cong \angle X$, $\angle B \cong \angle Y$, and $AB \cong XY$. Which postulate or theorem can be used to prove that the triangles are congruent?
In $\triangle ABC$ and $\triangle XYZ$, $\angle A \cong \angle X$, $\angle B \cong \angle Y$, and $AB \cong XY$. Which postulate or theorem can be used to prove that the triangles are congruent?
A spinner is divided into 5 equal sections, numbered 1 through 5. What is the probability of spinning a 3, followed by spinning an even number?
A spinner is divided into 5 equal sections, numbered 1 through 5. What is the probability of spinning a 3, followed by spinning an even number?
Given right triangles $\triangle PQR$ and $\triangle STU$ with right angles at $Q$ and $T$ respectively, if $PR \cong SU$ and $PQ \cong ST$, which theorem proves that $\triangle PQR \cong \triangle STU$?
Given right triangles $\triangle PQR$ and $\triangle STU$ with right angles at $Q$ and $T$ respectively, if $PR \cong SU$ and $PQ \cong ST$, which theorem proves that $\triangle PQR \cong \triangle STU$?
Flashcards
Hypotenuse-Leg (HL) Theorem
Hypotenuse-Leg (HL) Theorem
If the hypotenuse and one leg of a right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
Angle-Angle (AA) Similarity
Angle-Angle (AA) Similarity
If the acute angles of two right triangles are congruent, then the triangles are similar.
Corresponding Angles Postulate
Corresponding Angles Postulate
If two lines are cut by a transversal such that the corresponding angles are congruent, then the lines are parallel.
Alternate Interior Angles Theorem
Alternate Interior Angles Theorem
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Probability
Probability
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Simple Event
Simple Event
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Probability of a Simple Event
Probability of a Simple Event
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Study Notes
- Right Triangle Congruency Theorems and Postulates
Hypotenuse-Leg (HL) Theorem
- If the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent
- Only applicable to right triangles
- Hypotenuse must be congruent
- One corresponding leg must be congruent
Leg-Leg (LL) Theorem
- If the two legs of one right triangle are congruent to the corresponding two legs of another right triangle, then the two triangles are congruent
- Essentially a specific case of the Side-Angle-Side (SAS) congruence postulate, since the right angles are congruent
Angle-Leg (AL) Theorem
- If one acute angle and a leg of one right triangle are congruent to the corresponding acute angle and leg of another right triangle, then the two triangles are congruent
- Combination of Angle-Angle-Side AAS and ASA congruence postulates
- The leg can be either adjacent to the acute angle (ASA) or opposite the acute angle (AAS)
Hypotenuse-Angle (HA) Theorem
- If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the two triangles are congruent
- It's a specific adaptation of the Angle-Angle-Side (AAS) congruence postulate for right triangles
Important Considerations
- The right angles are congruent by definition
- Pythagorean Theorem can be used to find missing side lengths to prove congruency
- Theorems and Postulates Used to Guarantee Parallelism
Corresponding Angles Postulate Converse
- If two lines are cut by a transversal such that the corresponding angles are congruent, then the lines are parallel
- Establishes parallelism based on congruent corresponding angles
- Corresponding angles are on the same side of the transversal and in corresponding positions relative to the two lines
Alternate Interior Angles Theorem Converse
- If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel
- Alternate interior angles are on opposite sides of the transversal and between the two lines
Alternate Exterior Angles Theorem Converse
- If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel
- Alternate exterior angles are on opposite sides of the transversal and outside the two lines
Consecutive Interior Angles Theorem Converse
- If two lines are cut by a transversal such that the consecutive interior angles are supplementary (add up to 180 degrees), then the lines are parallel
- Also known as Same-Side Interior Angles Theorem
- Consecutive interior angles are on the same side of the transversal and between the two lines
Transitive Property of Parallel Lines
- If two lines are parallel to the same line, then they are parallel to each other
- If a || b and b || c, then a || c
Lines Perpendicular to a Transversal Theorem
- In a plane, if two lines are perpendicular to the same line, then they are parallel to each other
- If a ⊥ c and b ⊥ c, then a || b
- Introduction to Probability
Definition of Probability
- Probability is a measure of the likelihood that an event will occur
- It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty
Basic Formula for Probability of an Event
- P(E) = Number of favorable outcomes / Total number of possible outcomes
- P(E) represents the probability of event E occurring
Sample Space
- Sample space (S) is the set of all possible outcomes of a random experiment
- Each outcome in the sample space is called a sample point
Event
- An event is a subset of the sample space
- It is a specific outcome or a set of outcomes that you are interested in
Types of Events
- Simple Event: An event consisting of only one outcome
- Compound Event: An event consisting of more than one outcome
Complement of an Event
- The complement of an event E (denoted as E' or Eᶜ) includes all outcomes in the sample space that are not in E
- P(E') = 1 - P(E)
Basic Properties of Probability
- 0 ≤ P(E) ≤ 1 for any event E
- P(S) = 1, where S is the sample space (the probability of the entire sample space is 1)
- P(∅) = 0, where ∅ is the empty set (the probability of an impossible event is 0)
Equally Likely Outcomes
- If all outcomes in the sample space are equally likely, the probability of an event E is the number of outcomes in E divided by the total number of outcomes in the sample space
- P(E) = n(E) / n(S), where n(E) is the number of outcomes in E and n(S) is the number of outcomes in S
Probability as a Ratio
- Probability can be expressed as a fraction, decimal, or percentage
- Converting between these forms is straightforward
Law of Large Numbers
- As the number of trials in a probability experiment increases, the experimental probability (relative frequency) of an event gets closer to the theoretical probability of the event
- Probability of Simple Events
Definition of a Simple Event
- A simple event is an event that consists of only one outcome in the sample space
- It represents a single, specific result of an experiment
Calculating Probability of a Simple Event
- P(E) = 1 / Total number of possible outcomes, when all outcomes are equally likely
- If the outcomes are not equally likely, P(E) is determined by the inherent probability associated with that specific outcome
Examples of Simple Events
- Rolling a specific number on a die (e.g., rolling a 4)
- Drawing a specific card from a deck (e.g., drawing the Ace of Spades)
- Selecting a specific item from a collection (e.g., picking a particular marble from a bag)
Equally Likely Outcomes
- When all outcomes in the sample space are equally likely, the probability of a simple event is simply the reciprocal of the number of possible outcomes
Unequally Likely Outcomes
- In cases where outcomes are not equally likely, the probability of a simple event is given by the probability assigned to that specific outcome
- For example, if a weighted die is used, the probability of rolling a particular number may not be 1/6
Probability Assignment
- The probabilities of all simple events in a sample space must add up to 1
- Σ P(Ei) = 1, where Ei represents each simple event in the sample space
Applications
- Simple events form the building blocks for calculating probabilities of more complex (compound) events
- Understanding the probability of simple events is essential for analyzing and predicting outcomes in various scenarios
Complementary Events
- The complement of a simple event E consists of all other simple events in the sample space
- P(E') = 1 - P(E), where E' is the complement of E
Relative Frequency
- The probability of a simple event refers to the theoretical or expected likelihood of that event occurring
- The relative frequency is the proportion of times the event actually occurs in a series of trials
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