6.3 notes
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The document defines similar polygons as polygons 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 the theorem that if two polygons are similar, then the ratio of their perimeters is equal to the ratio of their 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 their corresponding sides is the scale factor.
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6.3 Similar Figures and Scale Drawings
The document discusses similar figures and scale drawings. Similar figures have the same shape but not necessarily the same size, and have equal corresponding angles and proportional corresponding sides. Scale drawings are enlarged or reduced drawings that are similar to an actual object, with the scale ratio representing the ratio between a distance on the drawing and the actual distance. Scale drawings and similar figures can be used to indirectly measure quantities through setting up proportional relationships between known and unknown values.
6.5 notes
This document discusses two theorems for proving triangle similarity: the SSS Similarity Theorem and the SAS Similarity Theorem. The SSS Similarity Theorem states that if three corresponding sides of two triangles are proportional, then the triangles are similar. The SAS Similarity Theorem states that if one angle of one triangle is congruent to one angle of another triangle and their included sides are proportional, then the triangles are similar. Examples are provided to illustrate each theorem.
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.
6.5 notes
This document discusses two theorems for proving triangle similarity:
1) The SSS Similarity Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
2) The SAS Similarity Theorem states that if one angle of a triangle is congruent to an angle of another triangle and the included sides are proportional, then the triangles are similar. Diagrams are provided to illustrate each theorem. The document appears to be notes on triangle similarity theorems for a math class.
6.7 notes
The document discusses dilations, which are transformations that stretch or shrink shapes by a scale factor relative to a fixed center point. A dilation is notated as (x,y) -> (kx, ky), where k is the scale factor and (x,y) are the original coordinates being dilated. If k is between 0 and 1, it represents a reduction, while if k is greater than 1, it represents an enlargement. Points on a figure and their images under a dilation lie on the same ray from the center of dilation.
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.
2.7 notes
This document discusses several angle pair theorems: the right angle congruence theorem which states that all right angles are congruent; the congruent supplements theorem which states that if two angles are supplementary to the same angle then they are congruent; the congruent complements theorem which states that if two angles are complementary to the same angle then they are congruent; the linear pair postulate which states that if two angles form a linear pair then they are supplementary; and the vertical angles theorem which states that vertical angles are congruent. The document provides proofs for some of these theorems and examples of how to prove the congruent complements theorem. It also assigns homework problems related to these angle pair theorems.
6.1 notes
The document discusses ratios, proportions, and geometric mean. It defines a ratio as a/b or a:b and an equivalent ratio as having the same simplified form. It states that proportions exist when two ratios are equal, and can be solved using cross multiplication. The geometric mean is defined as the square root of the product of two numbers.
Recommended
6.3 Similar Figures and Scale Drawings
The document discusses similar figures and scale drawings. Similar figures have the same shape but not necessarily the same size, and have equal corresponding angles and proportional corresponding sides. Scale drawings are enlarged or reduced drawings that are similar to an actual object, with the scale ratio representing the ratio between a distance on the drawing and the actual distance. Scale drawings and similar figures can be used to indirectly measure quantities through setting up proportional relationships between known and unknown values.
6.5 notes
This document discusses two theorems for proving triangle similarity: the SSS Similarity Theorem and the SAS Similarity Theorem. The SSS Similarity Theorem states that if three corresponding sides of two triangles are proportional, then the triangles are similar. The SAS Similarity Theorem states that if one angle of one triangle is congruent to one angle of another triangle and their included sides are proportional, then the triangles are similar. Examples are provided to illustrate each theorem.
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.
6.5 notes
This document discusses two theorems for proving triangle similarity:
1) The SSS Similarity Theorem states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
2) The SAS Similarity Theorem states that if one angle of a triangle is congruent to an angle of another triangle and the included sides are proportional, then the triangles are similar. Diagrams are provided to illustrate each theorem. The document appears to be notes on triangle similarity theorems for a math class.
6.7 notes
The document discusses dilations, which are transformations that stretch or shrink shapes by a scale factor relative to a fixed center point. A dilation is notated as (x,y) -> (kx, ky), where k is the scale factor and (x,y) are the original coordinates being dilated. If k is between 0 and 1, it represents a reduction, while if k is greater than 1, it represents an enlargement. Points on a figure and their images under a dilation lie on the same ray from the center of dilation.
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.
2.7 notes
This document discusses several angle pair theorems: the right angle congruence theorem which states that all right angles are congruent; the congruent supplements theorem which states that if two angles are supplementary to the same angle then they are congruent; the congruent complements theorem which states that if two angles are complementary to the same angle then they are congruent; the linear pair postulate which states that if two angles form a linear pair then they are supplementary; and the vertical angles theorem which states that vertical angles are congruent. The document provides proofs for some of these theorems and examples of how to prove the congruent complements theorem. It also assigns homework problems related to these angle pair theorems.
6.1 notes
The document discusses ratios, proportions, and geometric mean. It defines a ratio as a/b or a:b and an equivalent ratio as having the same simplified form. It states that proportions exist when two ratios are equal, and can be solved using cross multiplication. The geometric mean is defined as the square root of the product of two numbers.
6.4 notes
The document discusses the AA Similarity Postulate in geometry. It states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. The postulate is explained and examples are provided of proving triangles similar using the AA criterion. Worksheets and homework assignments related to applying the AA Similarity Postulate are also referenced.
6.3 notes
The document discusses similar polygons. It defines similar polygons as polygons where corresponding angles are congruent and corresponding lengths are proportional. The scale factor is introduced as the ratio of lengths of corresponding sides in similar polygons. It is stated 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.
6.6 notes
1) The document discusses proportionality theorems in geometry including the Triangle Proportionality Theorem and its converse.
2) It states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
3) It also presents Theorem 6.6 which says that if three parallel lines intersect two transversals, they divide the transversals proportionally.
6.6 notes
1) The document discusses proportionality theorems in geometry including the Triangle Proportionality Theorem and its converse. The theorems state that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
2) It also presents Theorem 6.6 which says that if three parallel lines intersect two transversals, they divide the transversals proportionally.
3) Theorem 6.7 is described as stating that if a ray bisects an angle of a triangle, it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
4.5 notes
The document discusses two triangle congruence theorems:
1) Angle-Side-Angle (ASA): If two angles and the included side of one triangle are congruent to those of another triangle, then the triangles are congruent.
2) Angle-Angle-Side (AAS): If two angles and the non-included side of one triangle are congruent to those of another triangle, then the triangles are congruent.
It also notes that Angle-Angle-Angle (AAA) only proves similarity, not congruence, and Side-Side-Angle (SSA) does not provide a conclusion.
6.2 notes
This document discusses proportion properties including:
1) The reciprocal property which states if two ratios are equal, then their reciprocals are equal.
2) Interchanging the means - if a/c then a/b and c/d.
3) Adding the denominator to the numerator - if a/c then (a+b)/(c+d).
It also defines scale drawings as having the same shape as the object represented and explains scale as the ratio describing how drawing dimensions relate to actual object dimensions.
6.3 notes
This document discusses similar polygons and their properties. It defines similar polygons as polygons where corresponding angles are congruent and corresponding side lengths are proportional. The scale factor is introduced as the ratio of lengths of corresponding sides in similar polygons. It is stated that if two polygons are similar, then the ratio of their perimeters will equal the ratio of corresponding side lengths. Congruence is described as a special case of similarity where the scale factor is 1.
4.3 notes
The document discusses the Side-Side-Side (SSS) congruence postulate for proving that two triangles are congruent. It states that if three pairs of corresponding sides of two triangles are congruent, then the two triangles are congruent. It provides an example of two triangles ABC and DEF where the three sides of triangle ABC are congruent to the three sides of triangle DEF, and therefore the two triangles are congruent.
4.7 notes
The document defines and discusses properties of isosceles and equilateral triangles. It states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent (the base angles theorem), and vice versa. It also notes that if a triangle is equilateral, then it is also equiangular, and if a triangle is equiangular, then it is equilateral. Examples are given to demonstrate finding side lengths and angle measures in isosceles and equilateral triangles.
6.5 notes
This document discusses two theorems for proving triangle similarity: the SSS Similarity Theorem and the SAS Similarity Theorem. The SSS Similarity Theorem states that if three corresponding sides of two triangles are proportional, then the triangles are similar. The SAS Similarity Theorem states that if one angle of one triangle is congruent to one angle of another triangle and the two included sides are proportional, then the triangles are similar. The document provides examples and explanations of how to use these theorems to prove triangle similarity.
4.2 notes
The document discusses congruence and triangles. It defines congruent figures as those where all corresponding parts are equal in measure. It presents 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 outlines three properties of congruent triangles: reflexive, symmetric, and transitive properties. The homework assigned is problems #3-15 odd, 19-21, and 26 on page 228.
3.2 notes
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12.7 notes
This document contains notes from a math class on similar solids. It defines similar solids as two solids of the same type that have equal ratios, known as a scale factor. It then states the Similar Solids Theorem: if two similar solids have a scale factor of a:b, their corresponding areas have a ratio of a2:b2 and their corresponding volumes have a ratio of a3:b3. Examples of homework problems involving similar solids are listed.
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Similar to 6.3 notes
6.4 notes
The document discusses the AA Similarity Postulate in geometry. It states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. The postulate is explained and examples are provided of proving triangles similar using the AA criterion. Worksheets and homework assignments related to applying the AA Similarity Postulate are also referenced.
6.3 notes
The document discusses similar polygons. It defines similar polygons as polygons where corresponding angles are congruent and corresponding lengths are proportional. The scale factor is introduced as the ratio of lengths of corresponding sides in similar polygons. It is stated 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.
6.6 notes
1) The document discusses proportionality theorems in geometry including the Triangle Proportionality Theorem and its converse.
2) It states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
3) It also presents Theorem 6.6 which says that if three parallel lines intersect two transversals, they divide the transversals proportionally.
6.6 notes
1) The document discusses proportionality theorems in geometry including the Triangle Proportionality Theorem and its converse. The theorems state that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
2) It also presents Theorem 6.6 which says that if three parallel lines intersect two transversals, they divide the transversals proportionally.
3) Theorem 6.7 is described as stating that if a ray bisects an angle of a triangle, it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
4.5 notes
The document discusses two triangle congruence theorems:
1) Angle-Side-Angle (ASA): If two angles and the included side of one triangle are congruent to those of another triangle, then the triangles are congruent.
2) Angle-Angle-Side (AAS): If two angles and the non-included side of one triangle are congruent to those of another triangle, then the triangles are congruent.
It also notes that Angle-Angle-Angle (AAA) only proves similarity, not congruence, and Side-Side-Angle (SSA) does not provide a conclusion.
6.2 notes
This document discusses proportion properties including:
1) The reciprocal property which states if two ratios are equal, then their reciprocals are equal.
2) Interchanging the means - if a/c then a/b and c/d.
3) Adding the denominator to the numerator - if a/c then (a+b)/(c+d).
It also defines scale drawings as having the same shape as the object represented and explains scale as the ratio describing how drawing dimensions relate to actual object dimensions.
6.3 notes
This document discusses similar polygons and their properties. It defines similar polygons as polygons where corresponding angles are congruent and corresponding side lengths are proportional. The scale factor is introduced as the ratio of lengths of corresponding sides in similar polygons. It is stated that if two polygons are similar, then the ratio of their perimeters will equal the ratio of corresponding side lengths. Congruence is described as a special case of similarity where the scale factor is 1.
4.3 notes
The document discusses the Side-Side-Side (SSS) congruence postulate for proving that two triangles are congruent. It states that if three pairs of corresponding sides of two triangles are congruent, then the two triangles are congruent. It provides an example of two triangles ABC and DEF where the three sides of triangle ABC are congruent to the three sides of triangle DEF, and therefore the two triangles are congruent.
4.7 notes
The document defines and discusses properties of isosceles and equilateral triangles. It states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent (the base angles theorem), and vice versa. It also notes that if a triangle is equilateral, then it is also equiangular, and if a triangle is equiangular, then it is equilateral. Examples are given to demonstrate finding side lengths and angle measures in isosceles and equilateral triangles.
6.5 notes
This document discusses two theorems for proving triangle similarity: the SSS Similarity Theorem and the SAS Similarity Theorem. The SSS Similarity Theorem states that if three corresponding sides of two triangles are proportional, then the triangles are similar. The SAS Similarity Theorem states that if one angle of one triangle is congruent to one angle of another triangle and the two included sides are proportional, then the triangles are similar. The document provides examples and explanations of how to use these theorems to prove triangle similarity.
4.2 notes
The document discusses congruence and triangles. It defines congruent figures as those where all corresponding parts are equal in measure. It presents 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 outlines three properties of congruent triangles: reflexive, symmetric, and transitive properties. The homework assigned is problems #3-15 odd, 19-21, and 26 on page 228.
3.2 notes
This document discusses parallel lines and transversals. It introduces the corresponding angles postulate, which states that if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. It also presents three theorems: 1) the alternate interior angles theorem, which states that the pairs of alternate interior angles are congruent, 2) the alternate exterior angles theorem, which states that the pairs of alternate exterior angles are congruent, and 3) the consecutive interior angles theorem, which states that the pairs of consecutive interior angles are supplementary. Students are instructed to complete guided practice problems with a partner.
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Volleyfest Schedule 2015
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12.7 notes
This document contains notes from a math class on similar solids. It defines similar solids as two solids of the same type that have equal ratios, known as a scale factor. It then states the Similar Solids Theorem: if two similar solids have a scale factor of a:b, their corresponding areas have a ratio of a2:b2 and their corresponding volumes have a ratio of a3:b3. Examples of homework problems involving similar solids are listed.
12.6 notes
The document discusses surface area and volume formulas for spheres. It provides examples of calculating the surface area of spheres with given diameters, as well as the volume of spheres and other objects like snow cones that can be modeled as spheres. Students are assigned homework problems applying these formulas to calculate surface areas and volumes.
12.5 notes
The document is notes from a math class lesson on finding the volumes of cones and pyramids. It includes the objectives of learning the formulas for these volumes. The formulas given are that the volume of a cone is 1/3πr^2h and the volume of a pyramid is 1/3Bh, where B is the area of the base and h is the height. There are examples provided of applying these formulas to find volumes.
12.4 notes
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Indian volleyball tournament
A volleyball tournament will be held on May 2nd at 9am at the Max Evans Gym to benefit the Strafford Indian Volleyball Program. The tournament will feature fun rules, games and prizes and cost $10 per person to register, either as part of a team of 6-8 players or individually. People can register their teams on Facebook or by contacting the coaches Ashley Bough or Angelina Hedrick by May 1st.
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.
12.5 notes
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12.4 notes
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12.1 notes
This document contains notes from a math class where students were working on homework problems involving trigonometry (SOHCAHTOA) and multiples of 3. The notes are dated April 08, 2015 and reference homework page 648 problems 12 through 36 dealing with multiples of 3.
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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.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
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
Text Books:
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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6.3 notes
- 1. 6.3 Similar Figures November 16, 2012 Bellwork Nov 298:47 AM HW pg. 376 #412 evens, 2028 evens 1
- 2. 6.3 Similar Figures November 16, 2012 6.3 Similar Polygons Similar polygons: corresponding angles are congruent and B C corresponding lengths are proportional F G <A ≅ <E ... A D E H AB BC -- = -- ∴ ABCD ~ EFGH EF FG ... Nov 298:57 AM HW pg. 376 #412 evens, 2028 evens 2
- 3. 6.3 Similar Figures November 16, 2012 Scale Factor: ratio of lengths of corr. sides in similar polygons Nov 299:37 AM HW pg. 376 #412 evens, 2028 evens 3
- 4. 6.3 Similar Figures November 16, 2012 Nov 299:41 AM HW pg. 376 #412 evens, 2028 evens 4
- 5. 6.3 Similar Figures November 16, 2012 Perimeters of Similar Polygons Theorem: If 2 polys are ~, then the ratio of perimeters = the ratios of corr. side lengths If KLMN ~ PQRS, then KL+LM+MN+NK = KL = LM = MN= NK K L PQ+QR+RS+SP PQ QR RS SP P Q N M S R Nov 299:45 AM HW pg. 376 #412 evens, 2028 evens 5
- 6. 6.3 Similar Figures November 16, 2012 Similarity & Congruence: If 2 figures are ≅, then they are ~. If 2 figures are ~, then the ratio of corr. sides = scale factor Nov 2910:26 AM HW pg. 376 #412 evens, 2028 evens 6
- 7. 6.3 Similar Figures November 16, 2012 HW pg. 376 #4-12 evens, 20-28 evens Nov 2910:25 AM HW pg. 376 #412 evens, 2028 evens 7