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# Geometry

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by

Tweet## Haley Leavitt

on 11 January 2013#### Transcript of Geometry

Angles Term: The space between two lines extending from one common point or vertex (measured by degrees). Types Of Angles:

Zero Angle: An angle who's measure is zero degrees.

Acute Angle: An angle who's measure is less than ninety degrees.

Obtuse Angle: An angle who's measure is more than ninety degrees,

Right Angle: An angle who's measure is exactly ninety degrees.

Straight Angle: An angle who's measure is exactly one hundred and eighty degrees.

Reflex Angle: An angle who's measure is greater than one hundred and eighty degrees.

Complete Angle: A angle who's measure is exactly three hundred and sixty degrees. Real Life Examples: - One example of angle use is in a bike spoke. The angles on the bike spokes make it possible for them to evenly pull the rim toward the hub. The stiffness from the rim prevents it from collapsing. - Another example is a steel square. This is a tool carpenters usually use to create ninety degree angles. These can be used in creating stairs, roofs, or just for measurements. Web Link: http://www.mathopenref.com/angle.html Bisector Definition: A line that evenly divides a figure, angle or line segment into two equal parts and/or angles. 2 A B C - Angle C is bisected by line segment CD to create two equal angles. (angle C1 and angle C2). C1 c2 web link: http://www.wyzant.com/Help/Geometry/Triangles/Bise Real Life Examples: - One example of a bisector would be an intersection. This creates a perpendicular bisector with ninety degrees for every angle.

- Another example of a bisector is when you diagonally cut a sandwich in half. The cut down the center divided it into two equal parts and the angle that it was cut is now evenly divided into each part. 3 Midpoint Definition: A point on a line segment or arc that creates two equidistant parts when measured from both endpoints. web link: http://www.regentsprep.org/Regents/math/Geometry/GCG2/Lmidpoint.htm Real Life Examples: - One example of midpoint could be used when traveling.If you were planning on traveling a long distance, you might want to take a break in the middle somewhere. This break would make each driving distance equal.

- Another example would be a seesaw. In order for it to be balanced, each side must have the same weight. Therefore, the thing holding it up must be placed directly in the middle so each side has an equal weight. Congruence Definition: To be equal in size and shape. 4 Isometry: A transformation that preserves distance/ length. These could be rotations, reflections, vectors and translations. Mapping: This is the correspondence of points. Triangle Congruence: If all sides and interior angles are congruent, you have triangle congruence. This can be proven by side-angle-side congruence, side- side-side congruence, angle-angle-side congruence, and side-angle-side congruence. congruent by AAS. - The clocks are the same size and shape therefore they are congruent. It doesn't matter what time they show. web link: http://www.mathopenref.com/congruent.html Real Life Examples: - One example of congruence would be your eyes. Both eyes are the same size and shape.

- Another example would be a key and a lock. The key and lock have to have the same shape and size or you couldn't open the lock Constructions Definition: To accuatly draw a line, angle, or shape using a compass, protractor, and/ or ruler. - This can be used to copy segments, copy angles, find the midpoint, bisect an angle, and much more. Real Life Examples: - Architects use constructions everyday. To make a blue print of a building, they must use constructions to create it.

- Furniture designers need constructions to complete their designs correctly. http://www.mathsisfun.com/definitions/construction-geometry-.html Parallel/Perpendicular Lines Parallel Lines: Lines in the same plane that never intersect and have the same slope.

Perpendicular Lines: Lines that intersect and create right angles. Their slope is -1. Real Life Examples: - One example of parallel lines are lines on a roadway. These divide roadways for better driving.

- An example of parallel lines would be writing a "t" or a cross. web link: http://www.northstarmath.com/sitemap/ParallelandPerpendicularLines.html Dilation 5 6 7 Definition: When a figure is enlarged or shrunk by a scale factor. Scale Factor:

Two similar figures with the ratio of any two comparable lengths.

Notation: Math symbols used to represent different things. Real Life Examples: - One example would be cropping a picture. When you crop a picture you make a certain part of it larger by a certain scale factor. Dilation Notation:

D (0,0) A=A' - Another example would be when your pupil dilates. http://www.mathopenref.com/dilate.html Vectors Definition: A ray with a magnitude and direction. - A unit vector has a magnitude of one. Real Life Examples: - One example of a vector is a ramp. It goes up in direction and its magnitude could be any length.

- Another example is drawing. When you draw something, it could have any magnitude and could go in any direction. web link: http://www.mathsisfun.com/algebra/vectors.html Pythagorean Theorem 9 8 Definition: A theorem stating that a squared plus b squared will give you c squared in a right triangle. Proof: 1) You have four copies of the same right triangle.

2) Rotate them correctly and you will get 3) Each triangles area is ab/2.

4) The square hole inside this square has the side lengths of (a-b). So its area would be (a-b) squared and 2ab.

5) The area of the four triangles would be 4 times ab/2.

6) Therefore,

c^2= (a-b)^2+2ab

c^2= a^2-2ab+b^2+2ab

c^2= a^2+b^2 a b c Pythagorean Theorem and Distance: AC=(x1,y1)-(x1,y1) = (x2-x1)

AB=(x1,y2)-(x1,y1) = (y2-y1)

BC^2=AC^2+AB^2

BC^2=(x2,x1)^2+(y2+y1)^2

BC = (x2-x1)^2+(y2-y1)^2 Real Life Examples: - One example is finding the measurements of a T.V. You must diagonally measure the T.V. and find the length and width. You could find any of these lengths by using the Pythagorean Theorem.

- Another example is if you wanted to find the height of something. If you had the measurements of the hypotenuse and the length, you could do this. http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html A B C (X2-X1) (Y2-Y1) Similarity 10 Definition: If the only difference between figures is size, then they are considered similar. Transformations Definition: Moving a figure by a translation (slide), reflection (flip), or rotation (turn) without changing the size, area, angles, and line lengths. http://www.mathsisfun.com/definitions/transformation.html Circle Definition: A shape made by a loop with every point being the same distance away from the center of it. a b c d e f g h i Radius: eh

Diameter: df

Chord: di

Secant: line cg

Tangent: line ja or line jb

Central Angle: angle feh

Inscribed Angle: angle kgi j k web link: http://www.mathsisfun.com/definitions/circle.html Real Life Examples: One example of a circle would be a Ferris Wheel. Any point on the rim of the Ferris Wheel is equidistant from the center. There are also chords running through it to support the seats.

- Another example could be a clock. A clock has many rays lined on the inside of it and a radius, or the hand of the clock. Formulas: Quadrilaterals Definition: A 2-dimensional figure with four straight sides and angles that add to 360 degrees. web link: http://www.mathsisfun.com/quadrilaterals.html Special Right Triangles Definitions- 45-45-90 Triangle: A right triangle with two sides and two angles that are equal (45 degrees, 45 degrees, 90 degrees) and a ratio of 1:1: square root 2. web link: http://www.icoachmath.com/math_dictionary/Isosceles_Right_Triangle.html 30-60-90 Triangle: A triangle with degrees of 30, 60, and 90 with a ratio of 1:2: square root 3. Trigonometry Definition: The study of angles and angle relationships in planar and 3-dimensional figures. web link: http://mathworld.wolfram.com/Trigonometry.html Web Link: http://mathworld.wolfram.com/Trigonometry.html AA Similarity: When two angles correspond to two angles in another similar triangle.

SAS Similarity: When the ratio of two sides and a included angle correspond to another similar triangle.

SSS Similarity: When three sides of a triangle are proportional to another corresponding, similar triangle. AA sas sas Scale Factor: The the ratio of two corresponding lengths in two similar figures. Ratio: Shows how two or more figures relate in size. Real Life Examples: - One example of similarity would be comparing people. You may wonder if someone is taller or shorter than you and compare your heights.

- Another example of similarity would be different sizes in clothing. You have to know which size will fit you. They may both be the same pants, but different sizes, and only one will fit you. Math Glossary By: Haley Leavitt January 5, 2013

Period: 1 Table of Contents Subject Page Angles......................................................................................... 1

Bisectors.................................................................................. 2

Midpoint..................................................................................... 3

Congruence............................................................................. 4

Constructions....................................................................... 5

Parallel/Perpendicular Lines....................................... 6

Dilation....................................................................................... 7

Vectors....................................................................................... 8

Pythagorean Theorem........................................................ 9

Similarity.................................................................................. 10

Transformations.................................................................. 11

Circles........................................................................................ 12

Quadrilaterals....................................................................... 13

Special Right Triangles.................................................... 14

Trigonometry.......................................................................... 15

Proofs.......................................................................................... 16 Mapping: This is the correspondence of points. coordinate notation: (x,y) -----> x+n, y+m Reflection over the...

y-axis: (x, y) ---> (x, -y)

x-axis: (x, y) ---> (-x, y) Counterclockwise rotation:

90 degrees: (x, y) ---> (-y, x)

180 degrees: (x, y) ---> (-x, -y)

270 degrees: (x, y) ---> (y, -x) Glide Reflections: When a translation and a reflection are combined into one transformation. Real Life Examples: - One example of a transformation would be a mirror. This you can relate to reflecting a figure over the x or y-axis. It appears identical, but is reversed.

- Another example would be chess or checkers when you slide your piece (translate) to a different square. You have changed the coordinates of your image. Kites:

-Two pairs of consecutive sides.

- One pair of opposite angles are congruent.

- One diagonal bisects another.

- Diagonals are perpendicular.

-One diagonal bisects one pair of opposite angles.

Parallelograms:

- Both pairs of opposite sides are parallel.

- Both pairs of opposite sides are congruent.

- Diagonals bisect each other.

- Both pairs of opposite angles are congruent.

- Both pairs of consecutive angles are supplementary.

Trapezoid:

- Exactly one pair of parallel sides. Isosceles Trapezoid:

- One pair of congruent legs.

- Diagonals are congruent.

- Two pairs of base angles are supplementary.

Rhombus:

- All sides are congruent.

- Diagonals bisect the angles.

- Diagonals are always perpendicular.

Rectangle:

- All angles are right angles.

- The diagonals are congruent.

Squares:

- Every property above this is always true about a square.

- It's a parallelogram with equal sides and equal angles. Real Life Examples: - One example of a special triangle is used when building a house. Builders have to make the ceiling 90 degrees exactly for proper structure, and they use these triangles to do that.

- Another example would be young gymnasts learning to do a somersault. They use what they call a "cheese wedge" to roll down. This would be a 45-45-90 triangle. Law of Sines Real Life Examples: - Trigonometry is used all the time by engineers and architects when having to create designs and build things. If they don't have an angles correct, the building isn't going to be capable of building.

- Another example could be when planes land. If they don't land at the right degree, they won't have a very smooth landing. Proofs Definition: The explanation, through laws and definitions, of how a certain part of a figure or formula came to be. Real Life Examples: - I honestly have no idea why anyone will use a proof and I can't find ANY real life examples on it. So, I am just going to explain why we need to learn them. Math isn't all about learning to use things in real life; but, you can use it to develop skills such as logical reasoning. Learning about proofs will help develop strategic skills and help us work through things better step by step. This could be used in school or with real life drama.

Full transcriptZero Angle: An angle who's measure is zero degrees.

Acute Angle: An angle who's measure is less than ninety degrees.

Obtuse Angle: An angle who's measure is more than ninety degrees,

Right Angle: An angle who's measure is exactly ninety degrees.

Straight Angle: An angle who's measure is exactly one hundred and eighty degrees.

Reflex Angle: An angle who's measure is greater than one hundred and eighty degrees.

Complete Angle: A angle who's measure is exactly three hundred and sixty degrees. Real Life Examples: - One example of angle use is in a bike spoke. The angles on the bike spokes make it possible for them to evenly pull the rim toward the hub. The stiffness from the rim prevents it from collapsing. - Another example is a steel square. This is a tool carpenters usually use to create ninety degree angles. These can be used in creating stairs, roofs, or just for measurements. Web Link: http://www.mathopenref.com/angle.html Bisector Definition: A line that evenly divides a figure, angle or line segment into two equal parts and/or angles. 2 A B C - Angle C is bisected by line segment CD to create two equal angles. (angle C1 and angle C2). C1 c2 web link: http://www.wyzant.com/Help/Geometry/Triangles/Bise Real Life Examples: - One example of a bisector would be an intersection. This creates a perpendicular bisector with ninety degrees for every angle.

- Another example of a bisector is when you diagonally cut a sandwich in half. The cut down the center divided it into two equal parts and the angle that it was cut is now evenly divided into each part. 3 Midpoint Definition: A point on a line segment or arc that creates two equidistant parts when measured from both endpoints. web link: http://www.regentsprep.org/Regents/math/Geometry/GCG2/Lmidpoint.htm Real Life Examples: - One example of midpoint could be used when traveling.If you were planning on traveling a long distance, you might want to take a break in the middle somewhere. This break would make each driving distance equal.

- Another example would be a seesaw. In order for it to be balanced, each side must have the same weight. Therefore, the thing holding it up must be placed directly in the middle so each side has an equal weight. Congruence Definition: To be equal in size and shape. 4 Isometry: A transformation that preserves distance/ length. These could be rotations, reflections, vectors and translations. Mapping: This is the correspondence of points. Triangle Congruence: If all sides and interior angles are congruent, you have triangle congruence. This can be proven by side-angle-side congruence, side- side-side congruence, angle-angle-side congruence, and side-angle-side congruence. congruent by AAS. - The clocks are the same size and shape therefore they are congruent. It doesn't matter what time they show. web link: http://www.mathopenref.com/congruent.html Real Life Examples: - One example of congruence would be your eyes. Both eyes are the same size and shape.

- Another example would be a key and a lock. The key and lock have to have the same shape and size or you couldn't open the lock Constructions Definition: To accuatly draw a line, angle, or shape using a compass, protractor, and/ or ruler. - This can be used to copy segments, copy angles, find the midpoint, bisect an angle, and much more. Real Life Examples: - Architects use constructions everyday. To make a blue print of a building, they must use constructions to create it.

- Furniture designers need constructions to complete their designs correctly. http://www.mathsisfun.com/definitions/construction-geometry-.html Parallel/Perpendicular Lines Parallel Lines: Lines in the same plane that never intersect and have the same slope.

Perpendicular Lines: Lines that intersect and create right angles. Their slope is -1. Real Life Examples: - One example of parallel lines are lines on a roadway. These divide roadways for better driving.

- An example of parallel lines would be writing a "t" or a cross. web link: http://www.northstarmath.com/sitemap/ParallelandPerpendicularLines.html Dilation 5 6 7 Definition: When a figure is enlarged or shrunk by a scale factor. Scale Factor:

Two similar figures with the ratio of any two comparable lengths.

Notation: Math symbols used to represent different things. Real Life Examples: - One example would be cropping a picture. When you crop a picture you make a certain part of it larger by a certain scale factor. Dilation Notation:

D (0,0) A=A' - Another example would be when your pupil dilates. http://www.mathopenref.com/dilate.html Vectors Definition: A ray with a magnitude and direction. - A unit vector has a magnitude of one. Real Life Examples: - One example of a vector is a ramp. It goes up in direction and its magnitude could be any length.

- Another example is drawing. When you draw something, it could have any magnitude and could go in any direction. web link: http://www.mathsisfun.com/algebra/vectors.html Pythagorean Theorem 9 8 Definition: A theorem stating that a squared plus b squared will give you c squared in a right triangle. Proof: 1) You have four copies of the same right triangle.

2) Rotate them correctly and you will get 3) Each triangles area is ab/2.

4) The square hole inside this square has the side lengths of (a-b). So its area would be (a-b) squared and 2ab.

5) The area of the four triangles would be 4 times ab/2.

6) Therefore,

c^2= (a-b)^2+2ab

c^2= a^2-2ab+b^2+2ab

c^2= a^2+b^2 a b c Pythagorean Theorem and Distance: AC=(x1,y1)-(x1,y1) = (x2-x1)

AB=(x1,y2)-(x1,y1) = (y2-y1)

BC^2=AC^2+AB^2

BC^2=(x2,x1)^2+(y2+y1)^2

BC = (x2-x1)^2+(y2-y1)^2 Real Life Examples: - One example is finding the measurements of a T.V. You must diagonally measure the T.V. and find the length and width. You could find any of these lengths by using the Pythagorean Theorem.

- Another example is if you wanted to find the height of something. If you had the measurements of the hypotenuse and the length, you could do this. http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html A B C (X2-X1) (Y2-Y1) Similarity 10 Definition: If the only difference between figures is size, then they are considered similar. Transformations Definition: Moving a figure by a translation (slide), reflection (flip), or rotation (turn) without changing the size, area, angles, and line lengths. http://www.mathsisfun.com/definitions/transformation.html Circle Definition: A shape made by a loop with every point being the same distance away from the center of it. a b c d e f g h i Radius: eh

Diameter: df

Chord: di

Secant: line cg

Tangent: line ja or line jb

Central Angle: angle feh

Inscribed Angle: angle kgi j k web link: http://www.mathsisfun.com/definitions/circle.html Real Life Examples: One example of a circle would be a Ferris Wheel. Any point on the rim of the Ferris Wheel is equidistant from the center. There are also chords running through it to support the seats.

- Another example could be a clock. A clock has many rays lined on the inside of it and a radius, or the hand of the clock. Formulas: Quadrilaterals Definition: A 2-dimensional figure with four straight sides and angles that add to 360 degrees. web link: http://www.mathsisfun.com/quadrilaterals.html Special Right Triangles Definitions- 45-45-90 Triangle: A right triangle with two sides and two angles that are equal (45 degrees, 45 degrees, 90 degrees) and a ratio of 1:1: square root 2. web link: http://www.icoachmath.com/math_dictionary/Isosceles_Right_Triangle.html 30-60-90 Triangle: A triangle with degrees of 30, 60, and 90 with a ratio of 1:2: square root 3. Trigonometry Definition: The study of angles and angle relationships in planar and 3-dimensional figures. web link: http://mathworld.wolfram.com/Trigonometry.html Web Link: http://mathworld.wolfram.com/Trigonometry.html AA Similarity: When two angles correspond to two angles in another similar triangle.

SAS Similarity: When the ratio of two sides and a included angle correspond to another similar triangle.

SSS Similarity: When three sides of a triangle are proportional to another corresponding, similar triangle. AA sas sas Scale Factor: The the ratio of two corresponding lengths in two similar figures. Ratio: Shows how two or more figures relate in size. Real Life Examples: - One example of similarity would be comparing people. You may wonder if someone is taller or shorter than you and compare your heights.

- Another example of similarity would be different sizes in clothing. You have to know which size will fit you. They may both be the same pants, but different sizes, and only one will fit you. Math Glossary By: Haley Leavitt January 5, 2013

Period: 1 Table of Contents Subject Page Angles......................................................................................... 1

Bisectors.................................................................................. 2

Midpoint..................................................................................... 3

Congruence............................................................................. 4

Constructions....................................................................... 5

Parallel/Perpendicular Lines....................................... 6

Dilation....................................................................................... 7

Vectors....................................................................................... 8

Pythagorean Theorem........................................................ 9

Similarity.................................................................................. 10

Transformations.................................................................. 11

Circles........................................................................................ 12

Quadrilaterals....................................................................... 13

Special Right Triangles.................................................... 14

Trigonometry.......................................................................... 15

Proofs.......................................................................................... 16 Mapping: This is the correspondence of points. coordinate notation: (x,y) -----> x+n, y+m Reflection over the...

y-axis: (x, y) ---> (x, -y)

x-axis: (x, y) ---> (-x, y) Counterclockwise rotation:

90 degrees: (x, y) ---> (-y, x)

180 degrees: (x, y) ---> (-x, -y)

270 degrees: (x, y) ---> (y, -x) Glide Reflections: When a translation and a reflection are combined into one transformation. Real Life Examples: - One example of a transformation would be a mirror. This you can relate to reflecting a figure over the x or y-axis. It appears identical, but is reversed.

- Another example would be chess or checkers when you slide your piece (translate) to a different square. You have changed the coordinates of your image. Kites:

-Two pairs of consecutive sides.

- One pair of opposite angles are congruent.

- One diagonal bisects another.

- Diagonals are perpendicular.

-One diagonal bisects one pair of opposite angles.

Parallelograms:

- Both pairs of opposite sides are parallel.

- Both pairs of opposite sides are congruent.

- Diagonals bisect each other.

- Both pairs of opposite angles are congruent.

- Both pairs of consecutive angles are supplementary.

Trapezoid:

- Exactly one pair of parallel sides. Isosceles Trapezoid:

- One pair of congruent legs.

- Diagonals are congruent.

- Two pairs of base angles are supplementary.

Rhombus:

- All sides are congruent.

- Diagonals bisect the angles.

- Diagonals are always perpendicular.

Rectangle:

- All angles are right angles.

- The diagonals are congruent.

Squares:

- Every property above this is always true about a square.

- It's a parallelogram with equal sides and equal angles. Real Life Examples: - One example of a special triangle is used when building a house. Builders have to make the ceiling 90 degrees exactly for proper structure, and they use these triangles to do that.

- Another example would be young gymnasts learning to do a somersault. They use what they call a "cheese wedge" to roll down. This would be a 45-45-90 triangle. Law of Sines Real Life Examples: - Trigonometry is used all the time by engineers and architects when having to create designs and build things. If they don't have an angles correct, the building isn't going to be capable of building.

- Another example could be when planes land. If they don't land at the right degree, they won't have a very smooth landing. Proofs Definition: The explanation, through laws and definitions, of how a certain part of a figure or formula came to be. Real Life Examples: - I honestly have no idea why anyone will use a proof and I can't find ANY real life examples on it. So, I am just going to explain why we need to learn them. Math isn't all about learning to use things in real life; but, you can use it to develop skills such as logical reasoning. Learning about proofs will help develop strategic skills and help us work through things better step by step. This could be used in school or with real life drama.