2.5.1 Triangle Angle Relationships
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Classify triangles by sides and by angles. Find the measures of missing angles of right and equiangular triangles Find the measures of missing remote interior and exterior angles.
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2.5.2 Isosceles and Equilateral Triangles
Identify isosceles and equilateral triangles by side length and angle measure.
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This document discusses two theorems for proving triangles are congruent: Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS). For ASA, if two angles and the included side of one triangle are equal to those of another triangle, then the triangles are congruent. For AAS, if two angles and a non-included side of one triangle are equal to those of another, and the non-included sides are corresponding, then the triangles are congruent. It also notes there is no Angle-Side-Side (ASS) congruence theorem.
9.4 Notes
Triangles are shapes with three line segments that intersect at endpoints to form three angles. The sum of the angles in any triangle is always 180 degrees. In triangle SUN, with angles of 29 degrees and 99 degrees for S and N, the remaining angle U must be 52 degrees. The measures of angles in a triangle are in a 3:4:5 ratio, so the angles would be 30, 40, and 60 degrees. Triangles can be classified as acute, obtuse, right, or equiangular by their angles, and as scalene, isosceles, or equilateral by the lengths of their sides.
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The document discusses triangles and their classification. It begins by asking questions to establish that triangles have 3 sides and 3 angles, while quadrilaterals have 4 sides and 4 angles. It then defines the different types of triangles based on their angles (right, acute, obtuse) and their sides (equilateral, isosceles, scalene). Several key points are made, such as triangles being the strongest shape and most commonly used in construction due to their ability to hold their shape and provide support. The document provides examples and diagrams to illustrate each type of triangle. It concludes by assigning an art project to create something using different types of triangles.
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This document provides instruction on factoring polynomials and quadratic equations. It begins by reviewing factoring techniques like finding the greatest common factor and factoring trinomials and binomials. Examples are provided to demonstrate the factoring methods. The document then discusses solving quadratic equations by factoring, putting the equation in standard form, and setting each factor equal to zero. An example problem demonstrates solving a quadratic equation through factoring. The document concludes by assigning homework and an optional reading for the next class.
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.1 Functions
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
2.5 Transformations of Functions
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
2.2 More on Functions and Their Graphs
This document discusses functions and their graphs. It defines increasing, decreasing and constant functions based on how the function values change as the input increases. Relative maxima and minima are points where a function changes from increasing to decreasing. Symmetry of functions is classified by the y-axis, x-axis and origin. Even functions are symmetric about the y-axis, odd functions are symmetric about the origin. Piecewise functions have different definitions over different intervals.
1.6 Other Types of Equations
This document provides examples and steps for solving various types of equations beyond linear equations, including:
1) Polynomial equations solved by factoring
2) Equations with radicals where radicals are eliminated by raising both sides to a power
3) Equations with rational exponents where both sides are raised to the reciprocal power
4) Equations quadratic in form where an algebraic substitution is made to transform into a quadratic equation
5) Absolute value equations where both positive and negative solutions must be considered.
1.5 Quadratic Equations (Review)
This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
9.6 Binomial Theorem
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.
13.3 Venn Diagrams & Two-Way Tables
The document discusses using Venn diagrams and two-way tables to organize data and calculate probabilities. It provides examples of completing Venn diagrams and two-way tables based on survey data about students' activities. It then uses the tables and diagrams to calculate probabilities of different outcomes. The examples illustrate how to set up and use these visual representations of categorical data.
13.2 Independent & Dependent Events
* Identify events as independent or dependent
* Calculate the probabilities of independent and dependent events
9.5 Counting Principles
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
13.1 Geometric Probability
* Calculate geometric probabilities
* Use geometric probability to predict results in real world situations.
9.4 Series and Their Notations
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
9.3 Geometric Sequences
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
9.2 Arithmetic Sequences
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
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2.5.1 Triangle Angle Relationships
- 1. Triangle Angle Relationships The student is able to (I can): • Classify triangles by sides and by angles • Find the measures of missing angles of right and equiangular triangles • Find the measures of missing remote interior and exterior angles
- 2. Classifying Triangles Recall that triangles are classified by their side lengths and their angle measures as follows: • By side length – equilateral – all sides congruent (equal) – isosceles – two or moreor moreor moreor more sides congruent – scalene – no sides congruent • By angle measure – acute – all acute angles – right – one right angle – obtuse – one obtuse angle – equiangular – all angles congruent
- 3. Practice Classify each triangle by its angles and sides. 1. 3. 2. 4. 90° 110°
- 4. Practice Classify each triangle by its angles and sides. 1. 3. 2. 4. 90° 110° right scalene equiangular equilateral acute isosceles obtuse isosceles
- 5. Triangle Angle SumTriangle Angle SumTriangle Angle SumTriangle Angle Sum TheoremTheoremTheoremTheorem – all angles of a triangle add up to 180°. Example: Find the measure of the missing angle 56˚ 29˚ 180 – (56 + 29) = 180 – 85 = 95˚
- 6. corollarycorollarycorollarycorollary – a theorem whose proof follows directly from another theorem. Right TriangleRight TriangleRight TriangleRight Triangle CorollaryCorollaryCorollaryCorollary – the acute angles of a right triangle are complementary. A B C m∠A+m∠B+m∠C=180˚ m∠A + 90˚ + m∠C = 180˚ m∠A + m∠C = 90˚
- 7. Equiangular TriangleEquiangular TriangleEquiangular TriangleEquiangular Triangle CorollaryCorollaryCorollaryCorollary – the measure of each angle of an equiangular triangle is 60˚. E Q U m∠E = m∠Q = m∠U m∠E + m∠Q + m∠U = 180˚ m∠E + m∠E + m∠E = 180˚ 3(m∠E) = 180˚ m∠E = 60˚
- 8. interiorinteriorinteriorinterior angleangleangleangle – the angle formed by two sides of a polygon exteriorexteriorexteriorexterior angleangleangleangle – the angle formed by one side of a polygon and the extension of an adjacent side remote interiorremote interiorremote interiorremote interior angleangleangleangle – an interior angle that is not adjacent to an exterior angle 1 2222 3333 4444 interiorinteriorinteriorinterior exterior
- 9. Exterior AngleExterior AngleExterior AngleExterior Angle TheoremTheoremTheoremTheorem – the measure of an exterior angle of a triangle is equal to the sum of its remote interior angles. m∠3 + m∠4 = m∠1 + m∠2 + m∠3 m∠4 = m∠1 + m∠2 1 2222 3333 4444 interiorinteriorinteriorinterior exterior
- 10. Third AnglesThird AnglesThird AnglesThird Angles TheoremTheoremTheoremTheorem – if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. X E T L R A ∠R ≅ ∠E
- 11. Practice 1. What is m∠1? 2. Solve for x 140° 105° 1 15° (2x+3)° (5x‒60)°
- 12. Practice 1. What is m∠1? 2. Solve for x 140° 105° 1 15° (2x+3)° (5x‒60)° 140 = 105 + m∠1 m∠1 = 35° 5x – 60 = 2x + 3 + 15 5x – 60 = 2x + 18 3x – 60 = 18 3x = 78 x = 26