Geometry unit 1.2
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This document provides an introduction to points, lines, and planes in geometry. It defines key terms like point, line, segment, ray, and plane. It provides examples of naming and drawing different geometric shapes involving these concepts, like naming coplanar points, drawing segments between two endpoints, and representing the intersection of lines and planes. The document emphasizes that points, lines, and planes are basic undefined terms in geometry and introduces concepts like collinear points, coplanar points, and using dashed lines to represent hidden parts of figures.
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Geometry deals with properties and relations of lines, angles, surfaces, and solids. A point has no length, breadth or thickness and represents a position. A line extends indefinitely in both directions from a point. A plane has no boundaries and flat surfaces are plane surfaces. Three or more collinear points all lie on the same line.
lec-2 geometry1.pptx
Geometry deals with properties and relations of lines, angles, surfaces, and solids. A plane has no boundaries and flat surfaces are plane surfaces. A point is a mark of position with no length, breadth or thickness. A line segment has a definite length, while a line has no end points. A ray is a line segment extended endlessly in one direction. Intersecting lines cross and share a common point, while parallel lines remain at an equal distance and never meet. Collinear points all lie on the same line.
1 3 points, lines, planes
This document defines key geometric concepts including points, lines, planes and their relationships. It explains that points have no size, lines extend indefinitely and planes are flat surfaces that extend without limits. It also covers topics like collinear points that lie on the same line, determining if objects are coplanar by lying on the same plane, and the different types of intersections between lines and planes, which can be a point, line or no intersection.
PARALLEL and PERPENDICULAR LINES.pptx
1. The document defines and provides examples of different types of lines including perpendicular, intersecting, concurrent, parallel, and skew lines. It also defines perpendicularity.
2. Perpendicular lines intersect to form right angles. If two lines intersect to form right angles at a point, they are perpendicular. The perpendicular bisector of a line segment is the line perpendicular to the segment at its midpoint.
3. Parallel lines never intersect and have the same slope when graphed on a coordinate plane. They are shown with double lines or slanted lines symbols.
pointlineplanepp.ppt
This document defines key geometry concepts and provides examples of naming points, lines, and planes and sketching their intersections. It begins by defining undefined terms in geometry like point and line, and defined terms like segment and ray. Examples are then given of naming collinear points, coplanar points, and different names for lines and planes. Further examples demonstrate sketching intersections between lines and planes, including lines within a plane, lines that do not intersect a plane, and lines intersecting a plane at a point. The final examples show sketching two planes intersecting in a line. Guided practice questions reinforce naming intersections and sketching relationships between geometric entities.
Similar to Geometry unit 1.2 (20)
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Geometry 201 Unit 4.1
This document discusses identifying congruent figures and using congruence to find missing measures. It defines congruent figures as having the same shape and size. The Side-Side-Side Rule for triangles is explained - if the sides of two triangles are congruent, then the triangles are congruent. Examples are provided of identifying congruent triangles and polygons based on corresponding sides and angles. The document also provides examples of using known congruent figures to find missing side lengths or angle measures.
Algebra 302 unit 11.4
This document discusses conditional probability and how to calculate and interpret joint, marginal, and conditional relative frequencies using two-way frequency tables. It provides examples of finding conditional probabilities based on frequency tables of data involving variables like whether students are taking ballet or tap dancing classes. One example determines the probability a student is not taking tap dancing given they are taking ballet class is about 69%.
Algebra 2 unit 10.6
This document discusses double-angle and half-angle identities in trigonometry. It provides examples of using these identities to evaluate expressions and prove trigonometric identities. Key double-angle identities covered are sin(2θ), cos(2θ), and tan(2θ). The document also derives the half-angle identities and provides examples of using them to find exact trigonometric values.
Algebra 2 unit 10.7
This document discusses double-angle and half-angle identities in trigonometry. It provides examples of using these identities to evaluate expressions and prove trigonometric identities. Key double-angle identities covered are sin(2θ), cos(2θ), and tan(2θ). The document also derives the half-angle identities and provides examples of using them to find exact trigonometric values.
Algebra 2 unit 10.5
The document discusses the Law of Cosines and Heron's Formula. It begins by explaining that the Law of Cosines can be used to find side lengths and angle measures of a triangle when given side-angle-side or side-side-side information. It then provides examples of using the Law of Cosines to solve for missing values of triangles. It also introduces Heron's Formula, which can be used to find the area of a triangle based on its side lengths. Examples are given of using both the Law of Cosines and Heron's Formula to solve multi-step word problems involving triangles.
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 5.3
This document discusses bisectors in triangles. It defines perpendicular bisectors as lines that bisect and are perpendicular to a side of a triangle. Angle bisectors bisect an angle of a triangle. The three perpendicular bisectors and angle bisectors of a triangle are both concurrent, meaning they intersect at a single point.
The point of concurrency of the perpendicular bisectors is called the circumcenter and is equidistant from the triangle's vertices. The point of concurrency of the angle bisectors is called the incenter and is always inside the triangle, equidistant from its sides. Examples show using bisectors to find distances and angles in triangles, as well as applications like placing a building or monument equidistant from three
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.
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Geometry unit 1.2
- 1. UUNNIITT 11..22 PPOOIINNTTSS,, LLIINNEESS AANNDD PPLLAANNEESS
- 2. Warm Up Graph each inequality. 1. x ≥ 3 -2 0 2 4 2. 2 ≤ x ≤ 6 0 2 4 6 3. x < 1 OR x > 0 0 1
- 3. Objective Identify, name, and draw points, lines, segments, rays, and planes and apply basic facts about them.
- 4. Vocabulary undefined term point line plane collinear coplanar segment endpoint ray opposite rays postulate
- 5. The most basic figures in geometry are undefined terms, which cannot be defined by using other figures. The undefined terms point, line, and plane are the building blocks of geometry.
- 7. Points that lie on the same line are collinear. K, L, and M are collinear. K, L, and N are noncollinear. Points that lie on the same plane are coplanar. Otherwise they are noncoplanar. M K L N
- 8. Example 1: Naming Points, Lines, and Planes A. Name four coplanar points. A, B, C, D B. Name three lines. Possible answer: AE, BE, CE
- 9. Check It Out! Example 1 Use the diagram to name two planes. Possible answer: Plane R and plane ABC.
- 11. Example 2: Drawing Segments and Rays Draw and label each of the following. A. a segment with endpoints M and N. M N B. opposite rays with a common endpoint T. T
- 12. Check It Out! Example 2 Draw and label a ray with endpoint M that contains N. M N
- 13. A postulate is a statement that is accepted as true without proof.
- 15. Example 3: Identifying Points and Lines in a Plane Name a line that passes through two points. XY
- 16. Check It Out! Example 3 Name a plane that contains three noncollinear points. Possible answer: plane GHF
- 17. Recall that a system of equations is a set of two or more equations containing two or more of the same variables. The coordinates of the solution of the system satisfy all equations in the system. These coordinates also locate the point where all the graphs of the equations in the system intersect. An intersection is the set of all points that two or more figures have in common. The next two postulates describe intersections involving lines and planes.
- 19. Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen.
- 20. Example 4: Representing Intersections A. Sketch two lines intersecting in exactly one point. B. Sketch a figure that shows a line that lies in a plane.
- 21. Check It Out! Example 4 Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane.
- 22. Lesson Quiz: Part I 1. Two opposite rays. CB and CD 2. A point on BC. Possible answer: D 3. The intersection of plane N and plane T. Possible answer: BD 4. A plane containing E, D, and B. Plane T
- 23. Lesson Quiz: Part II Draw each of the following. 5. a line intersecting a plane at one point 6. a ray with endpoint P that passes through Q
- 24. All rights belong to their respective owners. Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.