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This document covers key concepts about segments and congruence in geometry. It defines segments as parts of lines between two points and introduces the ruler postulate for assigning coordinates to points. The segment addition postulate states that if a point B lies between points A and C, then the sum of segments AB and BC equals segment AC. It also defines congruent segments as those with the same length and provides examples of finding lengths using the segment addition postulate and determining congruence. Homework problems at the end reinforce these concepts.
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This document provides instructions to determine if pairs of triangles are congruent using several postulates: Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), Angle-Angle-Side (AAS), or Hypotenuse-Leg (HL). If congruence cannot be proven with one of these postulates, the response should be "not necessarily congruent". It then lists homework problems from page 259 numbers 4-16, 22, and 24 to practice applying these postulates.
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3.2 notes
1) The document discusses parallel lines and transversals. It defines the corresponding angles postulate, the alternate interior angles theorem, and the alternate exterior angles theorem.
2) These state that if two parallel lines are cut by a transversal, then the corresponding angles are congruent, the alternate interior angles are congruent, and the alternate exterior angles are congruent.
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4.2 notes
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This document appears to be notes from a math class on identifying quadrilaterals. It includes examples of solving problems about the measures of angles and lengths of sides in different quadrilaterals. The class was split into groups to fill in properties of different quadrilaterals like rectangles, rhombi, squares, kites and trapezoids from their textbook. The document ends by listing the homework assignment.
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The document defines and provides examples of different types of symmetry:
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The document summarizes key properties of triangles, including:
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3) The acute angles of a right triangle are complementary, meaning their measures sum to 90 degrees.
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The document discusses two theorems for proving triangles are congruent:
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5.3 5.4 notes
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8.1 notes
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11.7 notes
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8.2 notes
The document discusses properties of parallelograms. It states that a parallelogram is a quadrilateral with both pairs of opposite sides parallel. It presents several theorems: if a quadrilateral is a parallelogram, then its opposite sides are congruent; its opposite angles are congruent; its consecutive angles are supplementary, summing to 180 degrees; and its diagonals bisect each other. The document appears to be notes from a geometry class discussing key properties and theorems related to parallelograms.
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1.2 notes
This document defines key concepts about line segments and congruence. It introduces coplanar points, ways to write line segments, and the segment addition postulate which states that if point B lies between points A and C on a line, then the length of AB plus BC equals the length of AC. It also defines congruent segments as line segments with the same length and provides examples of using the segment addition postulate and defining congruent segments. The document assigns homework problems applying these concepts.
1.2 notes
This document contains notes on segments, congruence, and homework problems from a geometry class. It defines key terms like coplanar points, line segments, and opposite rays. It outlines postulates for rulers, distance as the absolute value of the difference in coordinates, and segment addition. Examples are given for finding lengths using the segment addition postulate and determining congruent segments. The homework assigned covers problems 6-11, 13-17, and 24-27 defining segments and congruence in the textbook.
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1.1 notes
The document defines key geometric concepts such as points, lines, planes, and their relationships. Points have no dimension, lines have one dimension and extend indefinitely, and planes have two dimensions and extend indefinitely. Collinear points lie on the same line, and coplanar points lie in the same plane. Other concepts defined include line segments consisting of endpoints, rays with one endpoint, opposite rays, and intersections which contain one or more points in common. Homework problems are assigned from pages 6 problems 3-6 and 8-11, 17-22.
1.1 notes
The document defines key geometric concepts such as points, lines, planes, and their relationships. Points have no dimension, lines are one-dimensional, and planes are two-dimensional. Collinear points lie on the same line, and coplanar points lie in the same plane. Other concepts defined include line segments consisting of endpoints, rays with one endpoint, opposite rays, and intersections which contain one or more points in common. Homework problems are assigned from pages 6 problems 3-6 and 8-11, and 17-22.
1.1 notes
The document defines key geometric concepts such as points, lines, and planes. It explains that a point has no dimensions, a line is one-dimensional and extends without end, and a plane is two-dimensional and contains at least three non-collinear points. The document also defines related terms like collinear points, which lie on the same line, and coplanar points, which lie in the same plane. It provides examples and illustrations of these concepts.
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1) The document discusses parallel lines and transversals. It defines the corresponding angles postulate, the alternate interior angles theorem, and the alternate exterior angles theorem.
2) These state that if two parallel lines are cut by a transversal, then the corresponding angles are congruent, the alternate interior angles are congruent, and the alternate exterior angles are congruent.
3) It also presents the consecutive interior angles theorem, which states that if two parallel lines are cut by a transversal, then the sum of any pair of consecutive interior angles is 180 degrees.
4.2 notes
This document defines congruent figures as those where all corresponding parts are equal in measure. It introduces the Third Angle Theorem, which states that if two angles in one triangle are equal to two angles in another triangle, then the third angles are also equal. It also presents Theorem 4.4 on the properties of congruent triangles: reflexive, symmetric, and transitive properties. Homework problems are assigned.
8.6 notes
This document appears to be notes from a math class on identifying quadrilaterals. It includes examples of solving problems about the measures of angles and lengths of sides in different quadrilaterals. The class was split into groups to fill in properties of different quadrilaterals like rectangles, rhombi, squares, kites and trapezoids from their textbook. The document ends by listing the homework assignment.
9.6 notes
The document defines and provides examples of different types of symmetry:
1) A line of symmetry is a line where a figure can be mapped onto itself by a reflection.
2) Rotational symmetry occurs when a figure can be mapped onto itself by a rotation of 180 degrees or less.
3) A center of symmetry is the center point of a figure that displays symmetry.
4) Point symmetry is when every part of a figure has a matching part.
4.1 notes
The document summarizes key properties of triangles, including:
1) The sum of the measures of the interior angles of any triangle is always 180 degrees.
2) The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
3) The acute angles of a right triangle are complementary, meaning their measures sum to 90 degrees.
4.4 notes
The document discusses two theorems for proving triangles are congruent:
1) Side-Angle-Side (SAS) theorem - If two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, then the triangles are congruent.
2) Hypotenuse-Leg (HL) theorem - If two right triangles have one hypotenuse and one leg of one equal to the hypotenuse and corresponding leg of the other, then the triangles are congruent. The document also notes that the Side-Side-Angle (SSA) relationship is not sufficient alone to prove triangles congruent.
5.3 5.4 notes
The document discusses theorems and properties related to angle bisectors, medians, and altitudes of triangles. It provides definitions of an angle bisector, median, altitude, and centroid. It states the Angle Bisector Theorem, Converse of the Angle Bisector Theorem, Concurrency of Medians of a Triangle Theorem, and Concurrency of Altitudes Theorem. The document assigns math problems from sections 5.1, 5.2, 5.3, and 5.4 and lists bellwork tasks.
8.1 notes
The document discusses polygons and their interior and exterior angles. It states that the sum of the interior angles of any polygon is (n-2)180 degrees, where n is the number of sides. It also notes that the sum of the interior angles of any quadrilateral is always 360 degrees. Additionally, the sum of the exterior angles of any polygon is also 360 degrees.
11.7 notes
This document discusses geometric probability and provides examples of calculating probabilities based on ratios of lengths and areas. It gives the example of calculating the probability that a driver will be pulled over in front of a school based on the length of road in front of the school compared to the total length of the road. Another example calculates the probability that a fly will land on a beach ball based on the area of the beach ball relative to the total area of a blanket. The document concludes with assigning homework problems from page 774 #4-10 evens, 20-23, 30-33.
8.2 notes
The document discusses properties of parallelograms. It states that a parallelogram is a quadrilateral with both pairs of opposite sides parallel. It presents several theorems: if a quadrilateral is a parallelogram, then its opposite sides are congruent; its opposite angles are congruent; its consecutive angles are supplementary, summing to 180 degrees; and its diagonals bisect each other. The document appears to be notes from a geometry class discussing key properties and theorems related to parallelograms.
3.6 notes
This document contains information about distances between lines. It defines the distance from a point to a line as the length of the perpendicular segment from the point to the line. It defines the distance between two parallel lines as the length of any perpendicular segment joining the two lines. It provides a formula for calculating the distance between two points using their x- and y-coordinates. An example problem is presented for finding the distance between two parallel lines by first finding the equations of the lines and their point of intersection. Homework questions from page 195, problems 23 through 27 are assigned.
2.5 notes
The document discusses algebraic properties of equality that can be used for proofs, including:
1) The properties of addition, subtraction, multiplication, and division of equality.
2) The distributive property.
3) The reflexive, symmetric, and transitive properties of equality applied to real numbers, segment lengths, and angle measures.
4) The homework assignment is problems #3-11 odds, 21-25 odds, and 28-30 on page 108.
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This document contains a chapter 10 review with 15 math problems. It provides the answers to finding various values for x, y, determining if lines are tangent to circles, finding lengths, and writing the standard equation of a circle. The review is in preparation for a test on Monday.
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This semester review discusses the author's progress over the past few months, noting they completed 3 major projects, aced their statistics and computer science classes with A's, and landed an internship at a tech startup for the summer. While busy, the author feels satisfied with their accomplishments and is looking forward to the break before the next semester begins in January.
8.1 notes
The document discusses measures of angles in polygons. It states the Polygon Interior Angles Theorem, which is that the sum of the interior angles of an n-sided polygon is (n-2)180 degrees. It also notes that the sum of the interior angles of a quadrilateral is 360 degrees. Additionally, it provides the Polygon Exterior Angles Theorem that the sum of the exterior angles of any polygon is 360 degrees.
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1.2 notes
This document defines key concepts about line segments and congruence. It introduces coplanar points, ways to write line segments, and the segment addition postulate which states that if point B lies between points A and C on a line, then the length of AB plus BC equals the length of AC. It also defines congruent segments as line segments with the same length and provides examples of using the segment addition postulate and defining congruent segments. The document assigns homework problems applying these concepts.
1.2 notes
This document contains notes on segments, congruence, and homework problems from a geometry class. It defines key terms like coplanar points, line segments, and opposite rays. It outlines postulates for rulers, distance as the absolute value of the difference in coordinates, and segment addition. Examples are given for finding lengths using the segment addition postulate and determining congruent segments. The homework assigned covers problems 6-11, 13-17, and 24-27 defining segments and congruence in the textbook.
1.1 notes
The document defines key geometric concepts such as points, lines, and planes. It explains that a point has no dimensions, a line is one-dimensional, and a plane is two-dimensional. The document also defines related terms like collinear points that lie on the same line and coplanar points that lie in the same plane. It provides examples of line segments with endpoints and rays with one endpoint. The homework assignments are to identify these concepts and do problems on pages 6 and 8-11, 17-22.
1.1 notes
The document defines key geometric concepts such as points, lines, planes, and their relationships. Points have no dimension, lines have one dimension and extend indefinitely, and planes have two dimensions and extend indefinitely. Collinear points lie on the same line, and coplanar points lie in the same plane. Other concepts defined include line segments consisting of endpoints, rays with one endpoint, opposite rays, and intersections which contain one or more points in common. Homework problems are assigned from pages 6 problems 3-6 and 8-11, 17-22.
1.1 notes
The document defines key geometric concepts such as points, lines, planes, and their relationships. Points have no dimension, lines are one-dimensional, and planes are two-dimensional. Collinear points lie on the same line, and coplanar points lie in the same plane. Other concepts defined include line segments consisting of endpoints, rays with one endpoint, opposite rays, and intersections which contain one or more points in common. Homework problems are assigned from pages 6 problems 3-6 and 8-11, and 17-22.
1.1 notes
The document defines key geometric concepts such as points, lines, and planes. It explains that a point has no dimensions, a line is one-dimensional and extends without end, and a plane is two-dimensional and contains at least three non-collinear points. The document also defines related terms like collinear points, which lie on the same line, and coplanar points, which lie in the same plane. It provides examples and illustrations of these concepts.
6.3 notes
The document discusses using proportions with similar polygons. It defines similar polygons as those where corresponding angles are congruent and corresponding lengths are proportional. It introduces the scale factor as the ratio of lengths of corresponding sides in similar polygons. It states that the ratio of perimeters of similar polygons equals the ratios of corresponding side lengths. Finally, it notes that if two figures are congruent, then they are similar, and if two figures are similar, then the ratio of corresponding sides is the scale factor.
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This document contains notes from a math lesson on finding the volumes of prisms and cylinders. It defines volume as the amount of space inside an object. It explains that to find the volume of a prism or cylinder, you calculate the area of the base and multiply it by the height. For a prism, the base can be any shape and its area formula is used. For a cylinder, the base is a circle so the volume formula is πr^2h, with r as the radius and h as the height. Examples of volume problems are shown and homework problems are assigned from the textbook.
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12.3 notes
This document contains notes and examples for calculating the surface area of pyramids and cones. It provides the formulas for finding the surface area of regular pyramids and cones. For pyramids, the surface area equals one-half the perimeter of the base times the sum of the slant height and apothem. For cones, the surface area equals pi times the radius squared plus pi times the radius times the slant height. The document includes practice problems applying these formulas to find the surface area of various pyramids and cones.
12.2 notes
The document discusses surface area formulas for prisms and cylinders. It defines surface area as the sum of the areas of all surfaces that enclose a solid. For prisms, surface area is calculated as the perimeter of the base multiplied by the height, plus twice the area of the base. For cylinders, surface area is the circumference of the base multiplied by the height, plus twice the area of the circular base. The document provides examples of applying these formulas to prisms and cylinders.
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11.7 notes
This document contains notes from an 11th grade math class on the topic of geometric probability. It includes examples of calculating the probability of events based on ratios of lengths and areas, such as the probability of a driver speeding in front of a school based on the length of road in front of the school over the total road length. The homework problems assigned were problems 4-10 of the even numbers, 20-23, and 30-33 on page 774 of the textbook.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
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Chapter 5
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Chapter 6
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Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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1.2 notes
- 1. 1.2 Segments and Congruence August 15, 2011 Bellwork 1. Give definition of coplanar points. points that are on the same plane 2. What is another way to write line segment AB? BA or AB 3. Sketch a plane and two intersecting lines that lie in the plane. # 4 & 5 Y W V X 4. What is another name for ZY? YZ Z 5. A student says the VW and VZ are opposite rays because they have the same endpoint. Describe the error. Point V must lie between points W and Z, which means the 3 points must be collinear. HW pg. 12 #611, 1317, 2427 1
- 2. 1.2 Segments and Congruence August 15, 2011 1.2 Segments and Congruence Postulate: A rule that is accepted w/o proof Theorem: A rule that can be proved Ruler Postulate: Points on a line can be matched with real numbers (AKA coordinate) Names of Points A B x1 x2 Coordinates of Points Distance = absolute value of the difference AB A B x1 x2 AB = x2 - x1 HW pg. 12 #611, 1317, 2427 2
- 3. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 3
- 4. 1.2 Segments and Congruence August 15, 2011 Segment Addition Postulate: If B is between A & C, then AB + BC = AC. *between: the point in the middle of 2 collinear points on the same line A B C AC A B C AB BC HW pg. 12 #611, 1317, 2427 4
- 5. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 5
- 6. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 6
- 7. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 7
- 8. 1.2 Segments and Congruence August 15, 2011 Find Length of GH 36 F G H 21 FH = FG + GH (Segment Addition Postulate) 36 = 21 + GH 36-21 = GH 15 = GH Congruent Segements: line segments w/ same length Equal Congruent A B C D AB = CD AB ≅ CD HW pg. 12 #611, 1317, 2427 8
- 9. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 9
- 10. 1.2 Segments and Congruence August 15, 2011 HW pg. 12 #611, 1317, 2427 10
- 11. 1.2 Segments and Congruence August 15, 2011 HW: p 12 # 6-11 13-17 24-27 HW pg. 12 #611, 1317, 2427 11