Creating Definitions and Angle Relationships Homework: Lesson 1.3/1-24 Assignment from my web page: View PPT “Conjectures: aka Theorems and Postulates”. Answer the ‘Time to Ponder’ questions/1-16 Due: Wednesday, October 2

Defining Terms This statement defines a protractor: “A protractor is a geometry tool used to measure angles.” First, you classify what it is (a geometry tool), then you say how it differs from other geometry tools (it is the one you use to measure angles).

Define a square “A square is a figure with four equal sides.” Is this a good definition? Test the definition a. Can you sketch a counterexample? counterexample – a figure that satisfies the definition but is NOT a square b. If you can produce a counter example, then you need to write a better definition for a square.

Counterexamples

A Good Definition “A square is a quadrilateral that has all sides congruent and all angles measuring 90 degrees.” (a 4-sided polygon with equal sides and equal angles)

Steps to Creating a Good Definition Classify your term. What is it? What class/group does it fit into? Differentiate your term. How does it differ from others in that class/group? Test your definition by looking for counterexamples.

Picture markings for parallel lines h g Notation: h // g Definitions: Parallel lines are coplanar lines that do not intersect. Skew lines are non-coplanar lines that do not intersect Picture markings for parallel lines Line a is skew to line b

Perpendicular Lines l k Definition:   k Definition: Perpendicular lines are intersecting lines that form 90˚ angles.

7 Angle relationships to define Right angle Acute angle Obtuse angle Complementary angles Supplementary angles Linear pair of angles Vertical angles

Textbook Investigation: Lesson 1.3 http://acr.kendallhunt.com/KeyPressPortalV5.0/Viewer/HomeFrame.htm

Types of Angles An Acute angle is an angle that measures between 0° and 90° A Right angle is an angle that has a measure of 90° An Obtuse angle is an angle that measures between 90° and 180°

Complementary Angles Complementary angles are two angles whose sum of their degree measurements equals 90 degrees. Example: These two angles are complementary. Their sum is 90˚.                                                         58° + 32° = 90°

Adjacent Complementary Angles. Those same two angles can be "pasted" together to form a right angle! These 2 angles are adjacent – they share a vertex and a side.

Supplementary Angles Supplementary angles are two angles whose sum of their degree measurements equals 180 degrees. Example: These two angles are supplementary. The sum of their measures is 180˚                                                                        139° +41° = 180 °

Adjacent Supplementary Angles Linear Pair of Angles Adjacent Supplementary Angles A linear pair of angles are two angles that are adjacent and supplementary angles.

Linear Pair of angles If Then A linear pair of angles are two angles that are adjacent and supplementary angles. Conjecture: If two angles form a linear pair, then they are supplementary. If Then

Vertical Angles Vertical angles are opposite angles formed by intersecting lines.      For any two lines that meet, as in the diagram below, <AEB and <DEC are called vertical angles. <BEC and <AED are also vertical angles.         110 70 70 110

Vertical Pair of Angles Vertical angles are two angles that are created by two intersecting lines and are opposite each other. < 1 & <3 are a vertical pair of angles < 2 & <4 are a vertical pair of angles Conjecture: If two angles are vertical angles, then they are congruent. If Then

Angles Around a Point Angles around a point will always add up to 360 degrees. The angles above all add to 360° 53° + 80° + 140° + 87° = 360°

Sum of known angles = 110° + 75° + 50° + 63° = 298° We can find an unknown angle using this sum. Example: What is angle “C"? To find the measure of < C, find the sum of the known angles and subtract that from 360° . C Sum of known angles = 110° + 75° + 50°  + 63° = 298° m< C = 360° − 298° m < C = 62°

Review State whether the following are acute, right, or obtuse. 1. 3. 5. 2. 4. ? ?

Complementary and Supplementary Find the missing angle. Two angles are complementary. One measures 65 degrees. Two angles are supplementary. One measures 140 degrees.

Complementary and Supplementary Find the missing angle. You do not need a protractor. Use the clues in the pictures. 9. 8. x x 165° 55°

Vertical Angles & Linear Pair 10. Find the missing angles. You do not need a protractor. Use the clues in the pictures. y x 58°

More drawings 11. Find the measure of all the angles with vertex C C E Box in the corner indicates a right angle. D C G J H

Final Drawing 12. Find the measure of all the angles with vertex G. B C G A D F E