Angles

 **__Marked 01/02/09__** __Angle__ In geometry, an angle is formed by 2 rays that share a common point called a **vertex.** (where the line segments originate) The following diagram shows the symbol for an angle. Angles are measured in degrees.

The following phrases all define an angle:
 * The figure formed by two lines diverging from a common point.
 * The figure formed by two planes diverging from a common line.
 * The rotation required to superimpose either of two such lines or planes on the other.
 * The space between such lines or surfaces.



The **initial side** of an angle is known as the starting position and the **terminal side** is known as the ending position of an angle. An angle is in standard position when it's initial sides lie on the positive x-axis. 

__Quick Definitions__ Ray - Has a beginning point but no end point. Think of sun's rays: they start at sun and go on forever... Vertex - The point at which the sides of an angle intersect. Adjacent Angles - Two angles that share a common side and a common vertex, but do not overlap.

__ Labeling Angles __When labeling angles, remember to give the names (letters) to the rays and the vertex. The point that names the vertex of the angle goes between the other two points.

Note: There can be more than 1 name for an angle. Refer to diagram.



For the following definitions, let 'x' represent the angle. Part of the definitions are expressed as inequalities. __Acute Angle__ An angle measuring between 0°and 89°. In other words, the angle is less than 90°. 0°<x<90°



__Right Angle__ An angle measuring 90°. Two lines or line segments that meet at a right angle are perpendicular. x=90°



__Obtuse Angle__ An angle whose measure in greater than 90°and less than 180°. 90°<x<180°

__Straight Angle__ An angle whose measure is exactly 180°. x=180°



__Reflex Angle__ An angle whose measure is greater than 180° and less than 360°. 180<x<360°



__A Perigon (Or A Revolution)__ An angle whose measure is exactly 360°. It is a full circle. x=360°



__Supplementary Angles__ Two angles are called supplementary angles if the sum of their degree measurements equal 180°. One of the supplementary angles is the supplement of the other. If the angles share a common vertex and a side, they form a straight, linear line. In other words, if the two supplementary angles are a djacent (i.e. have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a straight line.

If the two angles add to 180°, we say they **"Supplement"** each other. **Supplement** comes from Latin //supplere//, to complete or "supply" what is needed.

__Complementary Angles__ Two angles are called complementary angles if the sum of their degree measurements equal 90°. One of the complementary angles is the complement of the other angle. If the two complementary angles are adjacent (i.e. have a common vertex and share a side, but do not have any interior points in common) their non-shared sides form a right angle.


 * Complementary** comes from Latin //completum// meaning "completed" because the right angle is thought of as being a complete (full) angle.



Study Tip: How can you remember that Complementary is //90°// and Supplementary is //180°//? Luckily "C" comes before "S" in the alphabet and 90 comes before 180.

Note: Angles that are complementary or supplementary do not have to be together.

__Practice - (Answers are at after the question)__ Write the complement of the given angle. a) 9° b)45° c)57° d)63° e)89° Answers - a) 81°b)45°c)33° d)27° e)1° Write the supplement of the given angle. a)69° b)86° c)94° d)169° e)177° Answers - a)111°b)94°c)86° d)11° e)3°
 * Example 1.**
 * Example 2.**

Find the missing angle measure. a) b) c) Answers - a)90° b)37° c)39°
 * Example 3.**

i) Which of the following angles are acute?
 * Example 4.** (See Ex. 4 solutions below)

ii) Which of the following angles are obtuse?

iii) Which pairs of angles are complementary?

iv) Which pairs of angles are supplementary?

a = 21°

b = 90.1°

c = 90°

d = 134.2°

e = 69°

f = 45.8°

Solutions for Ex. 4 i) a, e and f

ii) b and d.

iii) a - e

iv) d - f

__Alternate Angles__ Pair of angles that lie on opposite sides and at opposite ends of a transversal (a line that cuts two or more lines in the same plane). The alternate angles formed by a transversal of two parallel lines are equal. Two angles formed on opposite sides of a line that crosses two other lines. The angles are both exterior or both interior, but not adjacent.

__Co-interior angles__ Co-interior angles are two angles on the same side of a 'C' like shape which can be slanted. The sum of two co-interior angles is always 180º

The sum of co-interior angles is 180° //a// and //b// are co-interior angles.

__Angles In Mathematics__ Angles are the basics and "roots" for geometry. Angles are frequently used in: __Shape and Space__ __- Geometric Properties of Circles and Polygons__
 * triangles/trigonometry (for example, the pythagorean theorem)
 * quadrilaterals
 * circles
 * parallel and perpendicular lines
 * The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
 * The inscribed angles subtended by the same arc are congruent
 * The angle inscribed in a semicircle is a right angle
 * The opposite angles of a cyclic quadrilateral are supplementary

All in all, every shape has angles.

__Angles In Science Optics__ Light bends when it moves at an angle from one transparent substance, such as air, to another substance, such as water. This bending of light is called refraction. We call the substances that light can move through mediums. Water, glass, and air are mediums. Light refracts at different angles depending on the density of the medium. Light refracts more when moving through glass than when moving through water. This is true because glass is denser than water.

__Application Of Angles In The Real World__ >
 * Corresponding, alternate, co-interior, and opposite angles are used when making divisions inside buildings, or when making cubicles, to specify how much each space should be worth and to ensure is equal
 * Reflex angles are used in trigonometry which is used in fields like astronomy and engineering
 * Acute angles are used to build objects like harpoons, arrow blades, bullets, blades and bolts or anything that needs to travel fast with minimum resistance
 * Angles were used in Greek and Roman architecture to create beautiful statues, buildings, and coliseums. Making sharp angles gave these buildings more character and brought them to life.
 * The construction of buildings use right angles (architecture)
 * The construction of bridges
 * Engineers and architects use angles for designs
 * Professional pool players must be familiar with angles in order to shoot billiard balls into the pockets of the pool table (Figure 1.)
 * In airplanes, pilots must be familiar with angles in order to fly
 * In war, armies must be familiar with angles. For example, when a missile is shot over from an enemy, armies must shoot another weapon towards the missile at an accurate angle so that the missile is destroyed
 * In war planes, the pilots must know how to control the speed and angle their plane/jet is flying at to reduce the chance of getting hit
 * Engineers and architects use angles for designs, roads, buildings and sporting facilities
 * Athletes use angles to enhance their performance
 * Artists use their knowledge of angles to sketch portraits and paintings
 * Carpenters use angles to make chairs, tables, sofas etc.

__Angles and Their Relationship With Trianges__ A triangle is made up of 3 angles. Refer to the Triangles Wikipage for more information.

Works Cited

__Very Important Note__: The following passages that we have cited do NOT appear to have a hanging indent. For some reason, the program terminated the hanging indent but there is supposed to be one so please pretend there is.

Beijing resumes construction of 'bird's nest'. (December 29, 2004). [On-line photograph]. Retrieved December 19, 2008, from http://www.chinadaily.com.cn/english/doc/2004-12/29/xin_261201291128961310987.jpg

= =  Complementary angles. (2008). [On-line photograph]. Retrieved November 30, 2008, from http://www.bartleby.com/images/A4images/A4comang.jpg Conrad, C. (2006, August). Angles and angle terms. //Welcome to the Math League.// Retrieved November 21, 2008, from http://www.mathleague.com/help/geometry/angles.htm Figure 1.: An angle defined as the rotation of a single ray. (2008). [On-line photograph]. Retrieved December 6, 2008, from http://www.sparknotes.com/math/trigonometry/angles/section1.html Man playing pool in Beijing, China. (August 16, 2008). [On-line photograph]. Retrieved December 10, 2008, from http://en.wikipedia.org/wiki/File:8ballpool.jpg Knill, G., et al. (1999). //Math Power 9 Ontario Edition//. Toronto: McGraw-Hill Ryerson.

Oak Framed Buildings. (2005). [On-line photograph]. Retrieved December 18, 2008, from http://www.wealdenoak.co.uk/mainimages/pic26_400.jpg Pierce, R. (2008, September 24). Angles – Acute, Obtuse, Straight and Right. //Math is Fun – Math Resources.// Retrieved November 21, 2008, from http://www.mathsisfun.com/angles.html Sellers, J. (n.d.). What is an angle exactly? //Welcome to the Krell Institute.// Retrieved December 10, 2008, from http://www.krellinst.org/uces/archive/resources/trig/node9.html Supplement. (2000). [On-line photograph]. Retrieved November 30, 2008, from http://www.bartleby.com/images/A4images/A4suplmt.jpg Terr, D. (2007). Uses of Angles. //Math Is Amazingly Powerful.// Retrieved December 10, 2008, from http://www.mathamazement.com/Math_&_Application/Uses-of-Angles.html The angle symbol. (July 20, 2006). [On-line photograph]. Retrieved November 21, 2008, from http://en.wikipedia.org/wiki/File:Angle_Symbol.svg Wikipedia. (2008, December 9). Angle. //Welcome to Wikipedia.// Retrieved November 30, 2008, from http://en.wikipedia.org/wiki/Angle Zidane in today's action. (n.d.). [On-line photograph]. Retrieved December 18, 2008, from http://johnbollwitt.com/uploads/2006/07/070906-zidane.jpg

This Wikipage was composed by J. Shen, D. Dickson, N. Burrows and K. Garbar

 