ANGLE

The union of two rays with a endpoint, called the vertex.

http://www.mathsisfun.com/angles.html

Table of Contents

Angle

Bisector

Midpoint

Congruence

Constructions

Parallel/Perpendicular

Dilation

Vector

Similarity

Transformations

Circle

Quadrilaterals

Special Right Triangles

Trigonometry

Proofs

Real Life Ex.

Type of Angle Description

Acute Angle: an angle that is less than 90°

Right Angle: an angle that is 90° exactly

Obtuse Angle: an angle that is greater than 90° but less than 180°

Straight Angle: an angle that is 180° exactly

Reflex Angle: an angle that is greater than 180°

**Bisector**

To divide into two congruent parts.

Midpoint of

Segment

B

A line, ray or segment which cuts

another line

segment into two equal parts

Bisector (segment)

Midpoint

The midpoint splits it into two equal sides

M

A

12x+3

10x+5

12x+3=10x+5

2x=2

x=1

http://www.core-learning.com/downloads/resources/math/geo_apdx1.pdf

The bisector of an angln is the line

or line segment that divides the

angle into two equal parts.

A

Ex: If FH bisects <EFG & m <EFG=120o, what is m<EFH?

Bisector (angles)

D

C

B

BD is an angle bisector <ABC

E

H

G

F

120

2

=

60

m<EFH = 60

Real Life EX.

**Midpoint**

A point on a line segment that divides the segment into two congruent segments.

**c**

A

B

Point C bisects

of the

segments of AB

C=[3+(-5)] /2

-5 -4 -3 -2 -1 0 1 2 3 4 5

C=(-2) /2

C=-1

**Formula**

**Real Life**

the lower part of his torso is

the midpoint of his elevation

**Congruence**

**Figures or angles that have**

the same size and shape.

the same size and shape.

**Real Life**

Congruence Mapping

(Isometrics)

It means a mapping which

preserves distances

Corresponding parts

Two triangles are congruent when the three sides and

the three angles of one triangle

have the same measurements as three sides

and three angles of another triangle.

The parts of the two

triangles that have the

same measurements

A theorem: For any points A and B there exists a translation mapping A to B.

A translation is an isometry.The three points A, B and X can be completed in a unique way to a parallelogram or any points X, Y the quadrilateral XY T(Y )T(X) is a parallelogram,since XT(X)||AB||Y T(Y )

Therefore, XY =T(X)T(Y ), so T is isometricry

Proof:

Congruent Triangles

(AAS Theorem): If two angles and a side not between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent

(ASA Postulate): If two angles and the side between

them in one triangle are congruent to the

corresponding parts in another triangle,

then the triangles are congruent

(SSS Postulate): If each side of

one triangle is congruent to the

corresponding side of another

triangle, then the triangles

are congruent

(HL Postulate): If the hypotenuse and

leg of one right triangle are congruent

to the corresponding parts

of another right triangle,

then the triangles are congruent

(LA Theorem): If one leg and an acute angle

of one right triangle are

congruent to the corresponding parts

of another right triangle,

then the triangles are congruent

http://planetmath.org/Congruence.html

**Formulas**

Congruence formulas obtained by counting irreducible.

Concept of role and competencies

Competency management framework

Competency identification

Competency assessment

Competency development

Expectation of significant others and self

Linking concept

individual

team

organization

Different from position

Skill:

Ability accomplish

Talent:

Inherent ability

Competency:

**Constructions**

**To draw shapes, angles,or lines accurately.**

In constructions you use

compass, straightedge

(i.e.ruler) and a pencil.

A compass is a metal

V-shaped tool that

helps draw arcs or circles.

straight edge which can be

used to draw a line segment.

This is another term for the Stitched Fine Edge.In math helps us draw, write, and solve problems

Real Life:

http://www.mathsisfun.com/geometry/constructions.html

Affinities

Circulations

Site Planning

Environment

land use

Zoning

Cost

Labor productivity

Materials

Structure

MEP

Assembly

Systems

Scale

Proportion

Texture & Color

Form

Symbolism

Rhythm

Landscaping

Functions

Technology

Esthetics

What Geometric Construction is in the real word

**Parallel/Perpendicular Lines**

a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end...

**AB**

**CD**

Intersecting - Lines cross or meets

http://www.regentsprep.org/Regents/math/ALGEBRA/AC3/Lparallel.htm

To better explain

Example

**Formulas:**

The slope of L1 is and 3/5 . Find the slope of L2 .

Since the lines are perpendicular, the slopes are negative reciprocals.

The slope of L2 is -5/3 . ANSWER

Parallel- describes lines in the same plane that never cross.

The slope of L1 is 3/5 and L1||L2 . Find the slope of L2 .Since the lines are parallel, the slopes are the same.The slope of L2 is also 3/5 . ANSWER

Example

3/5||3/5

**Real Life:**

**Dilation**

**A dilation is a type of transformation**

that changes the size of the image

that changes the size of the image

**Real Life**

http://www.mathwarehouse.com/transformations/dilations/dilations-in-math.php

In this image the scale factor is 2.

Therefor This dilation means

image A' is twice the size of A

Scale Factor

describe how much the figure

is made bigger or reduced.

Ex. A dilation with scale factor of X, you can find the image of a point by multiplying each coordinate by X: (a ,b) (ka , kB)

http://physics.about.com/od/mathematics/a/VectorMath.htm

**Vectors**

Its a number physic and geometric applications, which values from its ability to represent magnitude and direction simultaneously.

Real Life:

Wind, for example, had both a speed and a direction and, hence, is conveniently

expressed as a vector. The same can be said of moving objects and forces. The location of a points on a Cartesian coordinate plane is usually expressed as an ordered pair (x, y),

which is a specific example of a vector. Being a vector, (x, y) has a a certain distance

(magnitude) from and angle (direction) relative to the origin (0, 0).

Vectors are quite useful in simplifying problems from three-dimensional geometry.

**notation:**

In these diagrams magnitude (length) and direction are shown.

The second step in molecular cloning is to join the passenger DNA to the DNA of a suitable cloning vehicle.

These vehicles (or vectors) have the property that they replicate themselves and any attached passenger

DNA so that the passenger is amplified and can be eventually isolated. A number

of different vectors have been developed for genetic engineering. Each has special distinguishing properties..

SuperV

Vectors can also be cloning vectors

V

**Formula:**

**Similarity**

**Real life:**

Two shapes are Similar if the

only difference is size

SSS

Side-Side-Side Similarity - If all pairs of corresponding sides of two triangles are proportional, then the triangles are similar.

SAS

Side-Angle-Side Similarity - If one angle

of a triangle is congruent to one

angle of another triangle and the sides

that include those angles are

proportional, then the two

triangles are similar.

AA

http://hotmath.com/hotmath_help/topics/angle-angle-similarity.html

If two angles of a triangle are

congruent to two angles

of another triangle, then the two

triangles are similar.

Scale Factor

Corresponding sides change by the same scale factor.

all the sides of the small figure are multiplied by the

same number to obtain the lengths of the

corresponding sides of the large figure.

in the picture below

The scale factor of figure A to B is: 3 (3 * 3 = 9; 5 * 3=15)

The scale factor of figure B to A is: 1/3 (9 * 1/3 = 3; 15 * 1/3 = 5)

A transformation consisting of rotations and translations which leaves a given arrangement unchanged.

**Transformations**

**Real Life:**

Resizing

dilation, contraction, compression,

enlargement or even expansion

http://www.harcourtschool.com/glossary/math_advantage/images/slide7.gif

**Circle**

**Draw a curve that is "radius" away from a central point.**

A movement of a

geometric figure to a new

position without

turning or flipping it

Translation Slide

description: 7 units to the left and 3 units down.

mapping:

notation:

vectors:

v

http://mrsdell.org/geometry/motion.html

Reflection flip

flipping a geometric figure over

a line of reflection to obtain a

mirror image

Rotation turn

A turning of a figure

about a fixed point

Parts of a Circle

http://www.science.co.il/formula.asp

Real Life:

Formula:

for the distance of the circle

A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.

is to find the surface of the circle

Pythagorean Theorem

**Pythagorean Theorem**

This proof is by the distinguished Dutch mathematician E. W. Dijkstra (1930 - 2002). The proof itself is, like Proof #18, a generalization of Proof #6 and is based on the same diagram. Both proofs reduce to a variant of Euclid VI.31 for right triangles (with the right angle at C). The proof aside, Dijkstra also found a remarkably fresh viewpoint on the essence of the theorem itself:

If, in a triangle, angles α, β, γ lie opposite the sides of length a, b, c, then sign( + - ) = sign(a² + b² - c²),

where sign(t) is the signum function. BCN = CAB and ACL = CBA so that ACB = ALC = BNC. The details and a dynamic illustration are found in a separate page.

**Quadrilaterals**

A four-sided figure.

Real Life:

http://www.mathsisfun.com/quadrilaterals.html

Types of quadrilaterals:

Sides: a, b, c, d

Angles: A, B, C, D

Around the quadrilateral are a, A, b, B, c, C, d, D, and back to a, in that order

Altitudes: ha , etc.

Diagonals: p = BD, q = AC, intersect at O

Angle between diagonals: theta

Perimeter: P

Semiperimeter: s

Area: K

Radius of circumscribed circle: R

Radius of inscribed circle: r

Formulas:

**Special Right triangles**

A special right triangle is a right triangle

with some regular feature that makes

calculations on the triangle easier.

**Real life:**

https://www.khanacademy.org/math/geometry/right_triangles_topic/special_right_triangles/v/30-60-90-triangle-side-ratios-proof

The 30° and 60° angles give this one away.

x = 6

2x =12

z = x3√=63√

**Trigonometry**

45 45 90

If we represent the legs of an

isosceles right triangle by 1, we can use the Pythagorean Theorem to establish pattern relationships between the lengths of the legs and the hypotenuse. These relationships will be stated as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means.

There are two pattern formulas that apply ONLY to the 45º-45º-90º triangle.

30 60 90

If you draw an altitude in an equilateral triangle, you will form two congruent 30º- 60º- 90º triangles. Starting with the sides of the equilateral triangle to be 2, the Pythagorean Theorem will allow us to establish pattern relationships between the sides of a 30º- 60º- 90º triangle. These relationships will be stated here as "short cut formulas" that will allow us to quickly arrive at answers regarding side lengths without applying trigonometric functions, or other means.

There are three pattern relationships that we can establish that apply ONLY to a 30º-60º-90º triangle.

The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

**Real Life:**

**Formulas:**

They've given me an angle measure and the length of the side "opposite" this angle, and have asked me for the length of the hypotenuse. The sine ratio is "opposite over hypotenuse", so I can turn what they've given me into an equation:

sin(20°) = 65/x

x = 65/sin(20°) Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

I have to plug this into my calculator to get the value of x: x = 190.047286...

x = 190.047

A theorem attributed to Pythagoras that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other

http://mathworld.wolfram.com/PythagoreanTheorem.html

**Formula:**

**Proofs**

A geometric proof involves writing reasoned,

logical explanations that

use definitions,

axioms, postulates,

and previously proved theorems to

arrive at a conclusion about a

geometric statement

http://www.basic-mathematics.com/geometry-proofs.html

Since angle C is bisected, angle (x) = angle (y)Segment AC = segment BC ( This one was given) Segment CF = segment CF (Common side is the same for both triangle ACF and triangle BCF)Triangles ACF and triangle BCF are then congruent by SAS or side-angle-side In other words, by

AC-angle(x)-CF and BC-angle(y)-CF Since triange ACF and triangle BCF are congruent, angle A = angle B

By Aalaysia Lindsey