The document summarizes key concepts about parallel and perpendicular lines including:
1) Parallel lines have the same slope and never intersect, while perpendicular lines have slopes that are negative reciprocals of each other.
2) When lines are cut by a transversal, corresponding angles and alternate exterior angles are congruent for parallel lines, and alternate interior angles are congruent.
3) The slope formula is used to determine if lines are parallel or perpendicular based on their slopes.
The document discusses different types of polygons including simple, convex, and star-shaped polygons. It then presents the convexification algorithm, which takes a star-shaped polygon and transforms it into a convex polygon through a series of edge swaps. The algorithm works by traversing the polygon and swapping any edges that form a left-hand turn. A proof is provided that this algorithm will always terminate with a convex polygon after at most n(n-1)/2 swaps, where n is the number of sides in the original polygon.
This presentation provides an overview of using LinkedIn for personal and business purposes. It discusses maintaining professional relationships, networking for opportunities, and finding industry news for personal pages. For business pages, it outlines increasing brand visibility, showcasing products/services, generating leads, targeted advertising, and recruiting talent. LinkedIn is positioned as the world's largest professional network with over 200 million members for networking and furthering careers.
The document summarizes key concepts about parallel and perpendicular lines including:
1) Parallel lines have the same slope and never intersect, while perpendicular lines have slopes that are negative reciprocals of each other.
2) When lines are cut by a transversal, corresponding angles and alternate exterior angles are congruent for parallel lines, and alternate interior angles are congruent.
3) The slope formula is used to determine if lines are parallel or perpendicular based on their slopes.
OPTIMIZATION OF FREIGHT TRANSPORTATION WITHIN NAFTA COUNTRIES: FRAMING THE IS...Josué Isaac Hernández Díaz
This document discusses optimization of freight transportation within NAFTA countries. It provides background on freight transportation in the US, including key statistics on tonnage and value shipped by various modes of transportation annually. It then discusses the establishment of NAFTA and its impact on increasing land freight transportation between the US, Canada, and Mexico.
The document discusses different types of polygons including simple, convex, and star-shaped polygons. It then presents the convexification algorithm, which takes a star-shaped polygon and transforms it into a convex polygon through a series of edge swaps. The algorithm works by traversing the polygon and swapping any edges that form a left-hand turn. A proof is provided that this algorithm will always terminate with a convex polygon after at most n(n-1)/2 swaps, where n is the number of sides in the original polygon.
This presentation provides an overview of using LinkedIn for personal and business purposes. It discusses maintaining professional relationships, networking for opportunities, and finding industry news for personal pages. For business pages, it outlines increasing brand visibility, showcasing products/services, generating leads, targeted advertising, and recruiting talent. LinkedIn is positioned as the world's largest professional network with over 200 million members for networking and furthering careers.
The document summarizes key concepts about parallel and perpendicular lines including:
1) Parallel lines have the same slope and never intersect, while perpendicular lines have slopes that are negative reciprocals of each other.
2) When lines are cut by a transversal, corresponding angles and alternate exterior angles are congruent for parallel lines, and alternate interior angles are congruent.
3) The slope formula is used to determine if lines are parallel or perpendicular based on their slopes.
OPTIMIZATION OF FREIGHT TRANSPORTATION WITHIN NAFTA COUNTRIES: FRAMING THE IS...Josué Isaac Hernández Díaz
This document discusses optimization of freight transportation within NAFTA countries. It provides background on freight transportation in the US, including key statistics on tonnage and value shipped by various modes of transportation annually. It then discusses the establishment of NAFTA and its impact on increasing land freight transportation between the US, Canada, and Mexico.
72.9% of Mexico's water withdrawals in 2009 were for agricultural purposes, with 12.0% for industrial uses and 8.0% for cooling thermoelectric plants. From 2003 to 2011, over 43% of Mexico's water-related government budget was spent on water supply and sanitation, while close to 95% of water-related official development assistance went to water supply and sanitation projects. Mexico faces water challenges including water pollution, overexploitation of groundwater sources, and effects of water-related natural disasters.
The document defines different types of conditional statements such as conditional, converse, inverse, contrapositive and bi-conditional statements. It provides examples of each. It also discusses counterexamples, deductive reasoning, symbolic notation, the law of detachment, law of syllogism, algebraic proofs, segment and angle properties, the linear pair postulate, congruent complement and supplement theorem, vertical angles theorem, and the common segments theorem. Examples are given for most concepts.
Q3-Module 3 Parallel Lines Cut by a Transversal 2024.pptxPhil Acuña
This document discusses parallel lines and transversals. It defines a transversal as a line that intersects two or more parallel lines at different points. It then explains the four properties of parallel lines cut by a transversal:
1) Alternate exterior angles are congruent
2) Alternate interior angles are congruent
3) Interior angles on the same side of the transversal are supplementary
4) Corresponding angles are congruent
Examples are given to illustrate each property. The document also discusses how transversals can connect parallel aspects of life, like people.
The document discusses different types of angles formed when a line (called a transversal) intersects two or more other lines. It defines corresponding angles, alternate interior angles, same-side interior angles, vertical angles, and linear pairs. Examples are given to demonstrate how to identify and name these different angle types in diagrams. Key points covered are that corresponding angles are congruent, alternate interior angles are congruent if the lines are parallel, and same-side interior angles are supplementary.
The document defines and identifies different types of lines and their relationships when intersected by a transversal line. It defines parallel lines, perpendicular lines, skew lines, and parallel planes. It also defines transversals and identifies the angle relationships that exist between corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles when two lines are cut by a transversal. Examples are provided and properties such as corresponding angles and alternate angles being congruent are stated.
This document defines and categorizes the different types of angles formed when parallel lines are intersected by a transversal line. There are 7 types of angles: 1) interior angles, 2) exterior angles, 3) alternate interior angles, 4) alternate exterior angles, 5) corresponding angles, 6) interior angles on the same side of the transversal, and 7) exterior angles on the same side of the transversal. Each type is defined by its position relative to the parallel lines and transversal.
The document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
The document discusses parallel lines and transversals. It defines parallel lines and introduces the symbol || to represent parallel lines. It defines a transversal as a line that intersects two or more other lines. It identifies and defines exterior angles, interior angles, alternate interior angles, alternate exterior angles, and corresponding angles that are formed when a transversal intersects two lines. It provides examples of determining whether statements about angle relationships formed by lines and transversals are true or false.
The document discusses various topics related to graphs:
- It defines directed and undirected graphs, paths, connected graphs, trees, degree, isomorphic graphs, cut sets, and labeled graphs.
- Key aspects include paths being sequences of vertices with edges connecting them, connected graphs having paths between all vertex pairs, trees being connected and acyclic graphs, and isomorphic graphs having the same structure.
- It also covers graph concepts such as degrees measuring incident edges, cut sets separating graphs, and labeling providing additional data to graphs' vertices or edges.
This document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
Parallel Lines Initial Definitions and Theoremsk3smith_ODU
The document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
When two parallel lines are cut by a transversal, several pairs of angles are formed with specific relationships. Corresponding angles are angles that have the same relative position on either side of the transversal. Alternate interior angles are interior angles on opposite sides of the transversal. Alternate exterior angles are exterior angles on opposite sides of the transversal. Consecutive interior angles are interior angles next to each other on one side of the transversal.
72.9% of Mexico's water withdrawals in 2009 were for agricultural purposes, with 12.0% for industrial uses and 8.0% for cooling thermoelectric plants. From 2003 to 2011, over 43% of Mexico's water-related government budget was spent on water supply and sanitation, while close to 95% of water-related official development assistance went to water supply and sanitation projects. Mexico faces water challenges including water pollution, overexploitation of groundwater sources, and effects of water-related natural disasters.
The document defines different types of conditional statements such as conditional, converse, inverse, contrapositive and bi-conditional statements. It provides examples of each. It also discusses counterexamples, deductive reasoning, symbolic notation, the law of detachment, law of syllogism, algebraic proofs, segment and angle properties, the linear pair postulate, congruent complement and supplement theorem, vertical angles theorem, and the common segments theorem. Examples are given for most concepts.
Q3-Module 3 Parallel Lines Cut by a Transversal 2024.pptxPhil Acuña
This document discusses parallel lines and transversals. It defines a transversal as a line that intersects two or more parallel lines at different points. It then explains the four properties of parallel lines cut by a transversal:
1) Alternate exterior angles are congruent
2) Alternate interior angles are congruent
3) Interior angles on the same side of the transversal are supplementary
4) Corresponding angles are congruent
Examples are given to illustrate each property. The document also discusses how transversals can connect parallel aspects of life, like people.
The document discusses different types of angles formed when a line (called a transversal) intersects two or more other lines. It defines corresponding angles, alternate interior angles, same-side interior angles, vertical angles, and linear pairs. Examples are given to demonstrate how to identify and name these different angle types in diagrams. Key points covered are that corresponding angles are congruent, alternate interior angles are congruent if the lines are parallel, and same-side interior angles are supplementary.
The document defines and identifies different types of lines and their relationships when intersected by a transversal line. It defines parallel lines, perpendicular lines, skew lines, and parallel planes. It also defines transversals and identifies the angle relationships that exist between corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles when two lines are cut by a transversal. Examples are provided and properties such as corresponding angles and alternate angles being congruent are stated.
This document defines and categorizes the different types of angles formed when parallel lines are intersected by a transversal line. There are 7 types of angles: 1) interior angles, 2) exterior angles, 3) alternate interior angles, 4) alternate exterior angles, 5) corresponding angles, 6) interior angles on the same side of the transversal, and 7) exterior angles on the same side of the transversal. Each type is defined by its position relative to the parallel lines and transversal.
The document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
The document discusses parallel lines and transversals. It defines parallel lines and introduces the symbol || to represent parallel lines. It defines a transversal as a line that intersects two or more other lines. It identifies and defines exterior angles, interior angles, alternate interior angles, alternate exterior angles, and corresponding angles that are formed when a transversal intersects two lines. It provides examples of determining whether statements about angle relationships formed by lines and transversals are true or false.
The document discusses various topics related to graphs:
- It defines directed and undirected graphs, paths, connected graphs, trees, degree, isomorphic graphs, cut sets, and labeled graphs.
- Key aspects include paths being sequences of vertices with edges connecting them, connected graphs having paths between all vertex pairs, trees being connected and acyclic graphs, and isomorphic graphs having the same structure.
- It also covers graph concepts such as degrees measuring incident edges, cut sets separating graphs, and labeling providing additional data to graphs' vertices or edges.
This document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
Parallel Lines Initial Definitions and Theoremsk3smith_ODU
The document defines and describes different types of angles formed when two lines are cut by a transversal line, including interior angles, exterior angles, corresponding angles, alternate interior angles, and alternate exterior angles. It also explains properties and relationships between these angles, such as corresponding angles being congruent and alternate angles being congruent, that hold true when the two lines are parallel but not necessarily when the lines are not parallel.
When two parallel lines are cut by a transversal, several pairs of angles are formed with specific relationships. Corresponding angles are angles that have the same relative position on either side of the transversal. Alternate interior angles are interior angles on opposite sides of the transversal. Alternate exterior angles are exterior angles on opposite sides of the transversal. Consecutive interior angles are interior angles next to each other on one side of the transversal.
This document discusses geometry concepts related to parallel lines and transversals. It defines key terms like transversal, alternate interior angles, same side interior angles, and corresponding angles. It presents examples and problems involving using properties of parallel lines cut by a transversal to determine angle congruencies and values. The goal is for students to understand properties of angles formed by parallel lines and transversals to solve related problems.
This document defines key terms and concepts related to parallel lines and transversals, including:
- A transversal is a line that intersects two other lines at two different points.
- Interior angles are formed between the two lines, while exterior angles are outside the lines.
- Alternate interior angles, corresponding angles, and other angle relationships are defined.
- Examples demonstrate identifying angle relationships and using properties of parallel lines to calculate unknown angle measures.
This document provides information about parallel lines and transversals. It defines key terms like parallel lines, transversals, interior and exterior angles. It describes angle relationships that exist between parallel lines cut by a transversal, such as corresponding angles being congruent, alternate interior angles being congruent, same side interior angles being supplementary. Examples are provided to illustrate these concepts and properties. The document also discusses using these properties to find missing angle measures.
This document discusses various geometry concepts related to parallel and perpendicular lines and planes including: parallel lines never intersect and exist on the same plane, parallel planes never touch; skew lines exist in different planes and do not intersect; a transversal is a line that crosses two parallel lines; corresponding, alternate interior/exterior, and same-side interior angles are defined; the corresponding angle, alternate interior angle, and same-side interior angle theorems state relationships between these angles for parallel lines cut by a transversal; and the transitive property applies to parallel and perpendicular lines and their relationships with a third line.
This document discusses properties of parallel lines cut by a transversal. It defines key terms like parallel lines, transversal, corresponding angles, alternate interior angles, and alternate exterior angles. It then presents conjectures that corresponding angles, alternate interior angles, and alternate exterior angles are congruent when two parallel lines are cut by a transversal. Examples are given to illustrate these properties and how determining the measure of one angle allows calculating the measures of all other angles formed.
This document discusses properties of parallel lines cut by a transversal. It defines key terms like parallel lines, transversal, corresponding angles, alternate interior angles, and alternate exterior angles. It then presents conjectures that corresponding angles, alternate interior angles, and alternate exterior angles are congruent when two parallel lines are cut by a transversal. Examples are given to illustrate these properties and how determining the measure of one angle allows calculating the measures of all other angles formed.
This document provides information about parallel lines and transversals. It defines parallel lines and transversals, and describes the different types of angles formed when parallel lines are cut by a transversal, including corresponding angles, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, linear pairs, and vertical angles. It also provides examples of using these angle properties to solve for unknown angle measures.
1. The document discusses angles formed when lines are intersected by a transversal line.
2. It defines types of angles formed, including corresponding angles, alternate interior/exterior angles, and consecutive interior/exterior angles.
3. Examples are provided to demonstrate identifying angle types and using angle properties involving parallel lines cut by a transversal.
2. Parallellines/planes and skew lines Parallel lines: two coplanar lines that never intersect. Ex: AB and CD, AC and EG, FH and EG Parallel planes: two planes that never intersect. Ex:[] ABDC and [] EFHG, [] EACG and [] FBDH, [] ABFE and [] CDHG Skew lines: two lines that have no relationship whatsoever. Ex: AC and EF, GH and AE, BD and CG A B E F ([] means plane) C D G H
4. AnglesFormed by the Transversal Corresponding: angles that lie in the same side of the transversal. EX: <1and<5, <4and<8, etc. 1 2 Alternate exterior: angles in the 3 4 opposite side of the transversal but in the outside. Ex: <1and<8 and <2and<7 5 6 7 8 Alternate interior: angles in the opposite side of the transversal but in the interior. Ex: <3and<6, <4and <5 Same-side interior: same side of the transversal in the interior. Ex: <3and<5, <4and<6
5. Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Converse: if the pairs of corresponding angles are congruent, then two parallel lines have to be cut by a transversal. Corresponding angles: <1and<5 1 2 <2and<6 3 4 <3and<7 <4and<8 5 6 7 8
6. Alternate Exterior Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Exterior angles are congruent. Converse: If the pairs of Alternate Exterior angles are congruent, then two parallel lines were cut by a transversal. Alternate exterior angles: <1and<8 1 2 <2and<7 3 4 5 6 7 8
7. Alternate Interior Postulate: If two parallel lines are cut by a transversal, then the pairs of Alternate Interior angles are congruent. Converse: If the pairs of Alternate Interior angles are congruent, then two parallel lines were cut by a transversal. Alternate exterior angles: <3and<6 1 2 <4and<5 3 4 5 6 7 8
8. Same-Side Interior Postulate: If two parallel lines are cut by a transversal, then the pairs of Same-Side Interior angles are supplementary. Converse: If the pairs of Same-Side Interior angles are Supplementary, then two parallel lines were cut by a transversal. Alternate exterior angles: <3and<5 1 2 <4and<6 3 4 5 6 7 8
9. Perpendicular Transversal Theorem Theorem: If a line is perpendicular to one of the parallel lines, then it must be perpendicular to the other line too. Ex: A _|_ B A _|_ C I _|_ G I _|_ Y M _|_ J M _|_ E A G Y I B I J C E
10. Howdoes the Transative property Apply in Parallel and Perpendicular lines? We know that parallel lines never touch so if line A is parallel to line B and line B is parallal to line C, then line A is parallel to line C. In perpendicular lines this is not possible because if line A is perpendicular to line B and B is perpendicular to line C then line A and line C mudt be parallel. Ex: B B A B C C A C A B C A
11. Slope Slope is the rise of a line over the run of that same line (rise/run) In many equations slope is represented by the lower-case letter m.} Formula: Y¹ –Y² (X,Y) (X,Y) X¹- X² 1 no -1/3 slope 0
12. Slope´srelation With Parallel and Perpendicular lines Parallel: All parallel lines have the same slope as its complementing pair. slopes: line1=1 line1= -1/3 line2=1 line2= -1/3 Perpendicular: All perpendicular lines have the negative reciprocal slope of its complementing pair. slopes: line1= -1/3 line1=1/6 line2= 3/1 line2= -6/1
13. Slope/Intercept Form Formula: Y=mX+b You would use it when the slope and interceps are given. Ex: Y=1X+2 Y=1/2+1 Y=-2/3-2
14. Point/Intercept Form Formula: Y-Y¹= m(X-X¹) You would use it when points are given. Ex: Y-3=1(X+2) Y-0=1/2(X+3) Y+1=(X+0)