Angle Addition Postulate

1. Angle Addition Postulate -GOALS- To state what the angle addition postulate is Show examples of how to solve a problem with the angle addition postulate

2. Rules- A Better Understanding • The angle addition postulate basically states that if point M is in the interior of ∠JKL, then, ∠JKM + ∠LKM = ∠JKL. • The rule is that if an angle is split by a bisector, the two mini angles now are equivalent to the whole angle .

3. Practice Problems • If m∠ABC=72˚ and m∠CBD=38˚, what is m∠ABD? *ANSWER: ∠ABD is 110˚. • If m∠ABC=180˚ and x=12, what is m∠GBH? *ANSWER: ∠GBH is 30˚. • If m∠KML=55˚ and m∠JML=140˚, what is m∠JMK?? *ANSWER: ∠JMK is 85˚.

4. Links to Help You Study • These links are all about the angle addition postulate. • http://www.icoachmath.com/SiteMap/AngleAdditionPostulate.html • http://www.rio.k12.wi.us/MATH/geo1.html • http://www.schooltube.com/video/3322d1fcecd1487d8e40/Angle-Addition-Postulate By Jess Van Saders

5. Biconditional Statements Stephanie Friedkin

6. What is a Biconditional Statement? • The statement is usually represented by • If a conditional statement & its converse are both true, then it can be written as a biconditional statement, using “if and only if”.

7. Examples 1. Conditional:If the sum of two angle measures is 180 degrees, then the angles are supplementary. √ Converse:If the angles are supplementary, then the sum of two angle measures is 180 degrees. √ The sum of two angle measures is 180 degrees if and only if the angles are supplementary. 2.Conditional:If a triangle is isosceles, then the triangle has two congruent sides.√ Converse:If a triangle has 2 congruent sides, then the triangle is isosceles.√ A triangle is isosceles if and only if the triangle has two congruent sides. 3.Conditional:If a candidate becomes president, they have won the election. √ Converse: If a candidate wins the election, they become president. √ A candidate becomes president if and only if they win the election.

8. Helpful Links • Tutor Vista:http://www.tutorvista.com/content/math/boolean-algebra/boolean-algebra/conditional-biconditional-statements.php • Regents Prep: http://www.regentsprep.org/Regents/math/geometry/GP1/bicon.htm • Math Cuer: http://mcuer.blogspot.com/2007/07/geometry-chapter-22-definitions-and.html

9. Inductive Reasoningby Amanda Mezzina • The basics of geometry began when people started to recognize patterns. In fact, looking for patterns and making conjectures is part of the inductive reasoning process. • The goal of inductive reasoning is to be able to find and describe patterns, and to make real-life conjectures using this reasoning.

10. Using Inductive ReasoningGoal 2 Finding and Describing Patterns Goal 1 • There are three simple steps of inductive reasoning. • 1. Look for a pattern. Look at several examples. Use diagrams and table to help discover a pattern. • 2. Make a Conjecture. Use example to make a conjecture. A conjecture is an unproven statement based on observations. Discuss the conjecture with other, and if necessary modify it. • 3. Verify the Conjecture. Use logical reasoning to verify that the conjecture is true in all cases. 1. Sketch the next figure in the pattern. Each figure in the pattern looks like the previous figure with another row of squares added to the bottom. Each figure looks like a stairway.

11. Extra Examples Each row is adding another square. 52 32 42 1. Each number is four times the previous number. The next number is 256. --> 2. 1,4,16,64 --> 3. -5, -2, 4, 13 You add 3 to get the second number, then 6 to the third number, and you continue to add the next multiple of three.

12. Need more help? • http://www.basic-mathematics.com/examples-of-inductive-reasoning.html • http://www.tutorvista.com/math/inductive-reasoning-in-geometry • http://www.tutorvista.com/math/inductive-reasoning-in-geometry

13. Casey Spillane Period 2 1/20/11 Finding the midpoint of a segment on a coordinate plane The formula for finding the midpoint of a segment on a coordinate plane is relatively easy. It involves taking two points on a coordinate plane and finding the distance between them.

14. Formula • You take the two endpoint of the segment (x1, y1) (x2, y2) • And add the Xs together and divide the sum by 2, then do the same for the Ys. That should give you the coordinates of a point directly in the middle of the two

15. Practice Problems • Find the distance between the two points: 1) (3,-4) (6,8) Answer: (4.5, 2) • (10,-7) (5,4) Answer: (7.5, 1.5) • (3,9) (14,7) Answer: (8.5, 8) Ex:

16. Web Links • http://www.mathopenref.com/coordmidpoint.html • http://www.onlinemathlearning.com/coordinate-geometry.html • http://hotmath.com/help/gt/genericalg1/section_12_3.html

17. The Law of Detachment By Kyle Sorreta

18. The Law of Detachment The law of detachment is used for determining if a conditional statement is true or not. In this power-point I will be explaining the law, showing some examples, and giving some helpful websites.

19. Rules, Properties, and Formulas Rule: The law states that if P=Q then the value of P is also the value of Q

20. Examples • If two angles equal 90º, then they are complementary. Angle 1 and angle 2 add up to 90º. So angle 1 and angle 2 are complementary. • Kelly knows that if she misses practice before a game, then she will not start in the game. Kelly misses practice, so she concludes that she will not start. • Jim only eats sandwiches on Tuesdays. It is Tuesday , so Jim will eat a sandwich.

21. Helpful Sites • http://tutor-usa.com/free/geometry/worksheet/deductive-reasoning-law-detachment-law-syllogism • http://snippets.com/what-is-the-law-of-detachment-in-geometry.htm • http://scienceray.com/mathematics/geometry/geometry-help-deductive-reasoning/

22. Slope Intercept Form • It is probably one of the most frequently used ways to express the equation of a line. • One goal is to be able to find the slope of a line when you are given two coordinates. • We also are trying to find the y- intercept.

23. Important Rules • Formula for y intercept: y=mx+b • M is the slope of the line and b is the y- intercept. • When given two points of a line, the first thing you should find is the slope. You would find it using the slope formula. y2 - y1 x2 - x1 • Then once you’ve found m, all you have to do is plug it into the y intercept equation using an x and a y from one of the coordinates that they give you.

24. Example Problems 1. (-2, 8) and (6, 12) 12 -8 4 m=1 6 - 2 4 12= 1(6)+b b=6 Y= 1x+6 2. (0, 16) and (-8, 22) 22-16 6 m= 3/-4 -8 - 0 -8 16= 3/-4(0)+b b=16 Y= 3/-4x+16 3. (2, 4) and (1, -2) -2-4 m=6 1-2 4= 6(2)+b b= -8 Y= -8x+b

25. Helpful Sites • http://www.purplemath.com/modules/slopgrph.htm • http://www.purplemath.com/modules/strtlneq.htm • http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/study_guide/pdfs/alg1_pssg_G041.pdf

26. Summary • When you graph lines on a coordinate plane you’re given equations to use. • The goal of this presentation is to teach you or remind you of how to graph the lines with these equations.

27. Equations of lines • An equation of a line is usually in slope intercept form (y=mx+b) • y= point on y-axis, x= point on x-axis, m= slope, b= y-intercept • So an equation would look like y=2x+1 • To graph this line you would start with the y-intercept (+1) and put a point on the y-axis. Then you would follow the slope (2) which equals to 2/1. So you would move to spaces up and then one space to the right and make a point there. Continue to make points one after another. You can also go backwards if there isn’t enough room, 2/1 = down two and to the left one. Once you have your points you draw your line. In the picture, A is the y-intercept and the other points follow the slope.

28. Examples • y=2/3x-3 • Answer: • y=5 • Answer: • 2x-3y=6 • Answer:

29. Extra Help • http://www.purplemath.com/modules/slopgrph.htm • http://www.math.com/school/subject2/lessons/S2U4L2GL.html • http://www.algebralab.org/lessons/lesson.aspx?file=algebra_lineareqintercepts.xml

30. Vertical Angle Theorem By Chrissy Loganchuk

31. Summary The goal of this is to explain what the vertical angle theorem states and examples of when you would use the vertical angle theorem.

32. What is the Vertical Angle Theorem? If two angles are vertical angles their sides form two pairs of opposite rays. The theorem states that vertical angles are congruent.

33. Problems What is the measure of ? =77.61 What is the measure of ? =102.39 What is the measure of ? =47.27

34. Links http://library.thinkquest.org/2647/geometry/angle/proof2.htm http://hotmath.com/hotmath_help/topics/vertical-angles-theorem.html http://www.mathwarehouse.com/geometry/angle/vertical-angles.php

35. CONVERSE ALTERNATE INTERIOR ANGLE THEOREMMarissa SilvermanSummary & Goals • The alternate interior angle theorem states “if alternate interior angles are congruent then the two lines are parallel.” • The goal of this is to find out if the two lines are parallel

36. Rules & Properties(More Like Hints) • If you see a “z” and the angles inside of it are congruent the lines are parallel • That means that you are using the converse alternate interior angle theorem

37. Examples Are these lines parallel? Yes they are. There are arc on the angles show that the angles are congruent. Are these lines parallel? Yes they are. They angles are congruent. Are these lines parallel? Yes because the alternate interior angles are congruent.

38. Web Links • http://library.thinkquest.org/20991/geo/parallel.html

39. Congruent Triangle Proof By: Charlie Marshall

40. Summary Goal: To prove 2 triangles are congruent, using a proof table and multiple thereoms, postulates, and laws. The main idea of congruent triangle proofs is to use a proof table and show that two triangles are congruent using congruence shortcuts such as: ASA, AAS, SAS, SSS, HL

41. Rules Properties and Formulas • There are 5 postulates used in finding congruence. The one shown is ASA, or Angle Side Angle. When a triangle fits the descriptions of one of the congruence postulates, you can mathematically state it is congruent. • List of Postulates: • ASA- states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then these two triangles are congruent. • SAS- states that if two sides and the included angle of one triangle are ccongruent to two sides and the included angle of another triangle, then these two trianges are congruent. • SSS-states that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent. • AAS-states that if two angles and the non-included side one triangle are congruent to two angles and the non-included angle of another triangle, then these two triangles are congruent. • HL- states If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent

42. Examples AB and XY are congruent hypotenuses JK = MN; Triangles are isosceles

43. WEBSITES • http://www.mathwarehouse.com/geometry/congruent_triangles/ • http://www.mathwarehouse.com/geometry/congruent_triangles/isosceles-triangle-theorems-proofs.php