Isosceles Trap. Materials Needed:

Compass

Pencil

Straight Edge(Ruler) Interactive Glossary Angles: The space within two lines or planes beginning at a common plane. singular point, or within two planes diverging from a > 90° 90° < 90°

> 180° 180° < 180°

> 360° 360° Table of Contents 1 Angles

2 Bisectors

3 Midpoint

4 Congruence

5 Constructions

6 Parallel/Perpendicular

Lines

7 Dilations

8 Vectors 9 Pythagorean

Theorem

10 Similarity

11 Transformations

12 Circle

13 Quadrilaterals

14 Special Right

Triangles

15 Trigonometry

16 Proofs Parts of an Angle

The corner point is called a VERTEX.

Each straight line is an ARM.

The amount of turn between arms is called the ANGLE. Real-Life Examples One real-life example is a bicycle wheel. In this particular radial-laced wheel, the adjacent spokes create tiny acute angles, they look about 30°-40°. Another real-life example of angles a door hinge. You can find these throughout any building you go into. They can start at 0 degrees and swing the door open to a full 180 degrees. Weblink: http://www.mathopenref.com/angle.html Bisectors: A line/plane that splits an angle or segment perfectly in half. As you can see here, a bisector cuts an angle/line/etc. in two halves. Web link: http://www.mathopenref.com/bisectorline.html Real-Life Examples One real-life example of an angle bisector is an analog clock. This is quite crude, but here you can see the angle of the minute and hour and being bisected by the seconds hand. A real-life example of a segment bisector can be a file cabinet. The center line that is between the two drawers(a) can somewhat bisect the line on the side that is the lengths of both drawers(b). a b 3 2 1 Midpoint: A point directly in the middle of a line segment. The Midpoint Formula: This will allow you to find the midpoint of a line on a coordinate plane. The answer here would be (1,2). Here is a diagram of the midpoint of a segment. Using the formula; The coordinate from point B is 2. The coordinate from point A would be 1. Therefore, Real-Life Examples Weblink: http://www.purplemath.com/modules/midpoint.htm a A real-life example of midpoint could be the center of two windows (point a). When the windows are closed, the line in the center of the windows(b) have a midpoint at (a) shown by the handles. b Another example of a midpoint that can be found in real-life is a rope with a knot in the center. The knot is in the center of the rope, showing the rope's midpoint. Congruence: same in all aspects. Two objects that are the These two triangles are congruent. These two triangles are NOT congruent. Mappings Web Link: http://www.mathsisfun.com/geometry/congruent-angles.html Real-Life Examples A real-life example of congruency is this metal bridge. Engineers that draw out bridges need to keep the beams congruent, and you can see that in the triangular shapes here. Another example of congruency, is in a fan. Ceiling fan, desk fan, doesn't matter. The blades on a fan will always be congruent, in order to balance out the machine and keep the air flow even. Constructions: 4 The drawing of geometric shapes using a compass, straight edge and pencil. Take a look at this informational video on how to construct and equilateral triangle. Web Link: http://www.mathsisfun.com/geometry/constructions.html Real-Life Examples A real-life example of a construction is simply a construction site. An engineer or an architect will do constructions on paper, like we do in class, and then workers will translate that person's ideas into a real building. Another example in real-life of constructions is an architect's job. Like I stated above, they use compasses, straight edges and pencils to construct geometric shapes and sketch out buildings. 5 Mappings in geometry are just like functions in algebra. A

B

C D

E

F This is a mapping(function) because each point 'maps' to exactly one point. No point maps to two different points. A

C D

E

F This is NOT a mapping(function) because point A maps to both D and E, which leads it to not being a function. Parallel/Perpendicular Lines: next to each other extended in the same direction, or lines that intersect and create 90 degree angles. Lines that either lie right a b c d e f g h a & b; c & d; e & f; h & g: Supplementary

a & c; b & d; e & g; f & h: Vertical

a & e; b & f; d & h; c & g: Corresponding

c & e; d & f: Alternate Interior

a & g; b & h: Alternate Exterior Angles in a Transversal Don't be fooled, both sets of lines here are parallel. These lines here are perpendicular, because they create a 90° angle. Web Link: http://www.purplemath.com/modules/slope2.htm Slope With parallel lines, the slope of both lines are the same. Because the lines are next to eachother heading the same direction, they have the same rise and run. (Both lines have the same rise/run) With perpendicular lines, the lines are at right angles from each other. Because of this, their rise/run is flipped. Line A Line B Line A has a slope of 3/4. Line B has a slope of 4/3. Do you see what I mean when I say the rise/run is 'flipped'? Real-Life Examples An example of parallel lines in real life is a set of railroad tracks. The railroad tracks need to be parallel, or else a train wouldn't be able to ride on it. An example of perpendicular lines in real life is a tennis court. The lines on a court need to be perpendicular in order to show the separate zones of the court. Dilations: Shapes/Points/Lines that are changed to be larger or smaller than the original shape. Notation In a dilation;

angle measures (remain the same)

parallelism (parallel lines remain parallel)

colinear lines (points stay on the same lines)

midpoint (midpoints remain the same in each figure)

orientation (lettering order remains the same) When writing a dilation in words, there's a special notation you must use. Dilation Notation:

D(0,0),K D stands for "Dilate". (0,0) or sometimes O stands for the center of dilation. This is the point which the dilation happens about. (Usually written in subscript) Scale Factor K is the Scale factor. 6 is the ratio of any two corresponding lengths in two similar figures. 4 4 2 2 A B Square A has a side length of 4. Square B has a side length of 2. 2/4 is 2, therefore the scale factor of these two shapes is 2. SCALE FACTOR:

2 Real-Life Examples A real-life example of a Dilation is a human body. It may be weird to think of this as an example, but when we're born we're all very small. As we get older, our bones and bodies 'dilate' and get larger. Another example of a dilation in real-life is a pupil. When light hits your eye, your pupils shrink up. When it's darker out, your pupils dilate and grow to allow more light in. 7 Vectors: A line that moves in one direction and has a magnitude. A vector has both a magnitude and a direction. The magnitude is marked by the size of the vector. The direction is marked by the way the arrow is pointing. To add two vectors together, simply set them head to toe. When subtracting vectors, there is two steps. First, you must reverse the vector that you wish to subtract (in this case, it's b). Next you must add them as you normally would. Multiplying a vector by a scalar (regular/real number) is quite simple. For example, say our vector is (7,3). Multiplying it by a scalar would give us (7x3,3x3) which would equal (21,9). We tripled the vector. Real-Life Examples Web Link:

http://www.math.com/tables/oddsends/vectors.htm A car is a real-life example of a vector, in a way. They travel in a direction, and have a magnitude (length). Cars have the same components as a vector. A football is also a real-life example of a vector. It too travels in a direction from the quarterback's hands to the receiver, and has a magnitude. 8 Pythagorean Theorem: The theorem that the square of the hypotenuse of a right triangle equals the squares of the two legs. As you can see here, the square of side a and the square of side b equal the square of side c. Hence, a² + b² = c². 9 Ancient Greek philosopher Pythagoras discovered this phenomenon. The Pythagorean theorem as we know, allows you to find the length of the hypotenuse of a right triangle if you have the length of the legs. To apply it to the distance formula, think of the difference between the x coordinates on a coordinate plane as one leg of a right triangle and the difference between the y coordinates as the other leg of the triangle. If you draw a horizontal line(b) out from the lowest point on the plane and a vertical line(a) down from the highest point, they will intersect and form a right triangle. Relating the Distance Formula to The Pythagorean Theorem a b President Garfield's Proof Web Link: http://www.cut-the-knot.org/pythagoras/index.shtml The formula for the area of a trapezoid is: half the sum of the bases times the height.

So, the area of the trapezoid in the diagram can be shown as:

(a + b)/2 · (a + b)

The area can also be calculated as the sum of the three triangles:

ab/2 + ab/2 + c · c/2

Which leads to:

(a + b)/2 · (a + b) = ab/2 + ab/2 + c · c/2

That simplifies down:

(a + b)² = 2ab + c²

And then:

a² + b² + 2ab - 2ab = c²

Finally:

c² = a² + b² Real-Life Examples A ladder leaning against a wall can be a real-life example of the pythagorean theorem. If you know the distance from the wall to the end of the ladder(d) and the height of the ladder up the wall(h) you can calculate the length of the ladder(l). A baseball field is also a great example of the pythagorean theorem in real-life. If you know the distance from Home base to First base(a), and From First base to Second Base(b) you can calculate the distance from Home to Second(c). Home First Second a b c Similarity: Two shapes that share angle measure, aren't congruent but their sides are proportional. These two rectangles are similar. The reason being is because they both have 90° angles, but the sides aren't equal, but proportional. A B Say rectangle A has a length of 15. Rectangle B has a length of 5. This means that the scale factor of A to B is 3. SAS Similarity 3 9 4 2 47° Since triangle ABC has side lengths of 9 and 4, and triangle DEF has side lengths of 3 and 2, we can tell that those corresponding sides are proportional. There's our two sides(SS). Now, we also know that corresponding angles B and F are both 47°. That gives us our similar angle(A). So, we can prove these triangles to be similar by SAS similarity. A B C D F E 10 Web Link: http://www.mathsisfun.com/geometry/triangles-similar.html AA Similarity A B C D E F 52° 24° In this problem, we weren't given any side lengths to prove these triangles similar. So, we have to work with the angle measures. In triangle ABC, angle B has a measure of 52°. Luckily, we also know that angle E in triangle DEF is also 52°. Plus, We know angles A and D both have a measure of 24°. So, we know these triangles to be similar by AA similarity. Real-Life Examples A real-life example of similarity can be seen in trees of a forest. Not all of the trees are the same length or thickness, but some can be proportional. This just shows how nature and math can collide. Another real-life example of similarity can be seen in a pizza pie. All the slices are never cut exactly the same. Due to a pizza slice being a triangle, they will usually have similar sized sides and angles. 11 Transformations: A rigid transformation is one that only moves the location of the object, not the shape. Rigid Transformations There are three kinds of rigid transformations. "Rigid" just means that the size/shape of the object stays the same. (If you only do these three transformations, your image for your pre-image will be congruent!) Rotate

(Turn) Reflect

(Flip) Translate

(Slide) pre-image image pre-image image pre-image image Web Link: http://www.mathwarehouse.com/transformations/ Rotation 90° counterclockwise:

(x,y) = (-y,x)

Rotation 180° counterclockwise:

(x,y) = (-x,-y)

Rotation 270° counterclockwise:

(x,y) = (y,-x)

Reflection over x-axis:

(x, y) = (x, -y)

Reflection over y -axis:

(x, y) = (-x, y)

Reflection over line y = x:

(x, y) = (y, x)

ANY Translation will just be adding how many spaces the shape is moved:

(x,y) = (#+x,#+y) Transformation Rules Real-Life Examples A real-life example of a transformation can be our home, planet earth. Earth rotates about it's axis, which is obviously a rotation. It is still the same earth, but it is constantly turning. Another example of a transformation in real-life can be a person pushing a boat. If you're pushing a boat on water, it is 'gliding' on the water. That's showing a translation. Circle: A closed plane curve consisting of every 12 point from the center. Parts of a Circle Secant Line C Central Angle:

Angle AOB

Inscribed Angle:

Angle ACB Formulas: Area of a Circle:

A = πr² Circumference of a Circle:

C = 2πr Equation of a Circle:

r² = (x-h)² + (y-k)² r= radius

π= pi(3.14)

(h,k)= center Web Link: http://mathworld.wolfram.com/Circle.html Real-Life Examples A real-life example of a circle could be the equator, the tropics, and the arctic circle. These imaginary lines wrap the earth and give us a sense geography, and areas of the earth. Another real-life example of a circle is the famous Stonehenge. This ancient monument in England is shrouded in mystery as to why it is here. Whatever its purpose, it is a circle, even if it's somewhat crude. Quadrilaterals: A plane figure having four angles and sides. Properties Quadrilaterals Kites Parallelogram Trapezoid Rhombus Rectangle Square Polygon with 4 sides

Sum of interior angles is 360° Two pairs of consecutive sides congruent

One pair of opposite angles congruent

One diagonal bisects a pair of opposite angles

Diagonals are perpendicular

One diagonal bisects the other Both pairs of opposite sides parallel

Both pairs of opposite sides congruent

Both pairs of opposite angles are congruent

All pairs of consecutive angles are supplementary

Diagonals bisect each other Exactly one pair of parallel sides One pair of opposite sides congruent

Two pairs of base angles congruent

Diagonals are congruent Four congruent sides

Diagonals are perpendicular

Diagonals bisect the angles Four congruent angles

Diagonals are congruent Every property above the square is ALWAYS true!

A parallelogram with congruent sides and congruent angles 13 Real-Life Examples Web Link: http://www.mathsisfun.com/quadrilaterals.html A computer screen is a real-life example of a quadrilateral. As you can see even on your computer screen, It is a rectangle within a rectangle. Another example of a quadrilateral is a kite. Special Right Triangles: Triangles that have special angles that correspond to their sides with a special ratio. With Special right triangles comes two different kinds. There is the:

45-45-90 Triangle

and the

30-60-90 Triangle There is a ratio to the sides. If you have only one side, you can find the other two. The ratios are as follows:

45-45-90= x:x

30-60-90= x:x :2x x x x x 2x x . ÷ ÷ ÷ 2 . 2 . 14 Web Link: http://library.thinkquest.org/20991/geo/stri.html Real-Life Examples A real life example of special right triangles is the house-builder's 3-4-5 rule. They need to get the walls & roof at perfect right angles, so they measure off 9 & 16 foot sections of wood, and if the long side ends up being 25, they know the building is square and correct. Another real life example of a special right triangle is a soccer goal. If a goalie is standing in the goal and the angle from the top left corner of the goal to his feet is 60°, then you can use the height of the goal to find the other two sides: the length of the space between his feet and the goal post, and the hypotenuse. Trigonometry: 15 The branch of math that deals with triangles and the relations of it's angles and sides, plus the calculations involved. Basic Trig Sine Function:

sin(1) = Opposite / Hypotenuse

Cosine Function:

cos(1) = Adjacent / Hypotenuse

Tangent Function:

tan(1) = Opposite / Adjacent 1 Always remember!:

Sin(1)=cos(2)

Sin(2)=cos(1) 2 SOHCAHTOA:

Some Old Hobo Came Around Here Tripping On Acid Web Link: http://www.cliffsnotes.com/study_guide/Law-of-Cosines.topicArticleId-11658,articleId-11573.html Law of Sines sinA/a=sinB/b=sinC/c

-Relates the sine of an angle with it's opposite side.

If you have a triangle that isn't right, and you're only given two angles and a side, or vice versa. Real-Life Examples A real-life application for trigonometry is cave surveying. They find the angle of the corner to the top of the cave, and from the top to the bottom. once they have those, they can use the tangent to find the hypotenuse. Another example of trigonometry in real life is property specification. Every property is identified by its position to certain landmarks. Trigonometry is used to figure that out. Proofs: used to prove certain things in geometry, such as if two shapes are congruent or not. A B C D Formal Proof Given: AB = BD, BD = BC

Prove: B is the midpoint of AC 1. AB = BD

2. BD = BC

3. AB = BC

4. B is the midpt of AC 1. Given

2. Given

3. Transitive Property

4. Def. of a midpt 16 With proofs, you have many different properties, postulates, and definitions you can use to help prove the proof you're working on. Some of these include: Transitive Property, Segment Addition Postulate, Definition of a perpendicular bisector, and so on. Real-Life Examples Real life examples of proofs are quite scarce. I have one however. One example can be trying to prove that two identical houses had the same measurements. You can use somewhat the same rules. However, us humans prove many things everyday. Why we do some actions, why our car is the best, or why a certain website is better than the other. Proving something isn't just tied down in mathematics, it's all around you. Web Link: https://www.khanacademy.org/math/geometry/geometry-worked-examples/v/ca-geometry--more-proofs

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# Geometry Project - Interactive Glossary

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