Life is what you make it. Mr. H s dad

Transcription

1 Life is what you make it. Mr. H s dad

2 You can classify triangles by if their sides are congruent. Scalene Triangle This triangle has no congruent sides. Isosceles Triangle This triangle has at least 2 congruent sides Equilateral Triangle This triangle has all sides being congruent.

3 You can classify a triangle by the angles that form the triangle. Acute Triangle Right Triangle Obtuse Triangle Equiangular Triangle All angles of the triangle are acute. One angle of the triangle is a right angle. One angle of the triangle is an obtuse angle. All angles of the triangle are congruent.

4 A triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle. 1) A(2, 3), B(6, 3), C(2, 7) Isosceles right triangle 2) A(1,9), B(4, 8), C(2, 5) Scalene triangle

5 Triangle Sum Theorem: The sum of the measure of the interior angles of a triangle is 180. A Interior Angles C B m A+m B+m C=180

6 Find the measure of the interior angles of the given triangles. A 1) 2) 55 D (3x) (x-10) B C E (2x+10) F G 3) (3x) H (2x) (x) I

7 Exterior angletheorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. A Exterior Angles C m A+m C=m 1 B 1 No special name

8 Find the value of the numbered angles

9 Find the value of the numbered angles

10 Find the value of the numbered angles

11 Find the value of x. 40 (3x) (x+80)

12 Bad is never good until worse happens. Danish Proverb

13 Congruent-same size and shape. Figures-two dimensional shapes. Congruent Figures-Two shapes that are the same size. All parts of one figure are congruent to the corresponding parts of the other figure. E K A D F J ABCDE FGHJK B C G H This is a congruence statement!

14 E K A D F J ABCDE FGHJK B C G H Corresponding Angles A F B G C H D J E K Corresponding Sides AE FK ED KJ DC JH CB HG AB FG

15 A E B C F D ABC DEF Corresponding Angles A D B E C F Corresponding Sides AB DE BC EF AC DF

16 Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. MNP QRS Corresponding Sides MN QR NP RS MP QS Corresponding Angles M Q N R P S

17 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. A D B C E F If A D and B E, then C F.

18 Find the value of x. 1) 50 (3x) (x+30) 50 2) (2x) 50

19 Reflexive Property For any triangle ABC, ABC ABC. A triangle is congruent to itself! Symmetric Property If ABC DEF, then DEF ABC. Transitive Property If ABC DEF and DEF GHI, then ABC GHI. Two triangles congruent to the same triangle are congruent to each other.

20 Name the property illustrated. If ABC GHI, then GHI ABC. Symmetric Property If CBA HIF and HIF TOR, then CBA TOR. Transitive Property DEF DEF. Reflexive Property

21 Step 1: Write given. Make a two column proof. Step 2: Show all sides are congruent to corresponding sides. Use Reflexive property, definition of midpoint, etc. Step 3: Show all angles are congruent to corresponding angles. Use AIA, AEA, CA, CIA, Third angle theorem, etc. Step 4: Write congruence statement. Ex. ABC DEF

22 Given:AC CE, BC CD, AB DE, AB DE Prove: ABC EDC A B C D E

23 Things could be worse. Suppose your errors were counted and published every day, like those of a baseball player. Anon.

24 Flow Proof-uses a flow chart to show reasons and statements of proofs. Main reason to use this is because it shows cause and effect.

25 Flow Proof-uses a flow chart to show reasons and statements of proofs. Statement 1 These are the effect of the statement before. Reason 1 Statement 2 Reason 2 Statement 3 Reason 3 Statement 4 Reason 4 These are usually from the diagram/given of the problem.

26 Prove the Alternate Interior Angles Converse Theorem. Given: 1 2 Prove: mlln 1 m Statements Reasons Given Vertical Angles Congruence Theorem 2 3 n Transitive Property 4. mlln 4. Corresponding Angles Converse Postulate

27 Statements Reasons Given Vertical Angles Congruence Theorem Transitive Property mlln 4. Corresponding Angles Converse Postulate Given 3 2 Vertical Angles Congruence Theorem 3 1 Transitive Property m n Corresponding Angles Converse Postulate

28 If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Corresponding Sides MN QR NP RS MP QS This causes MNP QRS

29 Given:C is the midpoint of AE and BD, AB DE, Prove: ABC EDC A C D AB DE B E Given C is the midpoint of AE and BD Given AC CE Definition of Midpoint BC CD Definition of Midpoint ABC EDC SSS Congruence Postulate

30 Is ABC DEF? The coordinates of the vertices are A(1, 4), B(4, 4), C(1, -1), D(-2, -2), E(-5, -2), and F(-2, 3). Explain. Use distance formula to check to see if AB=DE, BC=EF, and AC=DF. If yes, then triangles are congruent. If no, then triangles are not congruent

31 Because there is only one possible triangle with three given sides, they are the most stable shape.

32 Makes it stable because now we have two triangles that don t change. Two shapes that use the same four lengths.

33 Are the shapes below stable? 1) 2) 3)

34 The mind of man(kind) is capable of anything because everything is in it, all the past as well as all the future. Joseph Conrad

35 Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. A D Corresponding Sides AB DE AC DF A D This causes B C E F BAC EDF Included angle because it is between two congruent sides.

36 Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate. 1) 2) 3)

37 What is the third piece of information to show the triangle is congruent using the SAS Congruence Postulate. A 1. AB DE, AC DF, 2. A D, AC DF, 3. F C, AC DF, B D C 4. AB DE, BC EF, E F

38 SSA and the triangles are not congruent. There is no SSA Congruence Postulate because the triangles don t have to be congruent. One exception (HL)

39 Sides adjacent to the right angle are legs Side opposite (across) from the right angle is the hypotenuse. Leg Hypotenuse Leg

40 What are the parts of the triangle called. a is the/a b is the/a c is the/a C is the/a a c C b

41 Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. A Corresponding Sides AB DE AC DF C and F are right angles This causes C D B BAC EDF F E

42 Decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem. 1) 2)

43 A Given: AB BD Prove: ABC DBC B C D AB BD Given A and D are right angles. Given (Diagram) ABC and DBC are right triangles Definition of Right Triangle BC BC Reflexive Property ABC DBC HL Congruence Theorem

44 That which is bitter to endure may be sweet to remember. Thomas Fuller

45 If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. A D Corresponding Parts AB DE B E A D This causes B C E F BAC EDF Included side because it is between two congruent angles.

46 If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. A D Corresponding Sides/Angle BC EF B E A D This causes B C E F BAC EDF

47 Explain how you can prove the triangles congruent. If it is not possible, state it. A D 1) 2) A D B C E ABC DEF F B C ABC DEF E F 3) A D 4) A D B C E F B C E F

48 Is ABC DEF? Explain how you know. 1. A D, AC DF, and B E A 2. C F, AC DF, and B E 3. A D, AC DF, and C F B D C 4. AB EF, AC DF, and B E 5. A D, C F, and B E E F

49 SSS Congruence Postulate SAS Congruence Postulate HL Congruence Theorem ASA Congruence Postulate AAS (SAA) Congruence Theorem Doesn t work to show triangles congruent: SSA AAA

50 A B C Given: FE DE, A C Proven: ADE CFE A F E D C D F FE DE Given A C Given E E E Reflexive Property E ADE CFE AAS Congruence Postulate

51 Memory is the thing you forget with. Alexander Chase

52 By definition, if congruent triangles have congruent corresponding parts. We can use this idea to find distances across rivers and other hard to measure distances.

53 A Explain how you use the given information to prove that the parts are congruent. Given: CAB CAD, ACB ACD Prove: BC CD The first step is to show the two triangles are congruent. ABC ADC are congruent because ASA congruence postulate because of the two angles given and AC AC. Now that I have the triangles being congruent, BC CD by Corresponding Parts of Congruent Triangles are Congruent (CPCTC). B C D

54 D C 1 2 E 3 4 A Use this information to write a plan for a proof. Given: 1 2, 3 4 Prove: BCE DCE In BCE and DCE we know 1 2 and CE CE. If we can show that BC CD, then BCE DCE. In BCA and DCA we know 1 2, 3 4, and CA CA. Therefore BCA DCA by ASA congruence postulate. Therefore BC CD by CPCTC and consequently BCE DCE by SAS congruence postulate. B

55 Prove the distance across the river is 1 mile. Where segment AB is across the river and B segment BE is along the riverside. You also know BC CE and DE=1 mile. A C D E

56 B A BC CE Given C E and B are right angles Given(Diagram) BCA ECD E B All right s are D ADE CFE AAS Congruence Postulate E BA DE CPCTC Vertical Angles are Congruent DE=1 mile Given AB = 1 mile Substitution Property BA=DE Definition of Congruence

57 Memory is the thing you forget with. Chinese proverb

58 Congruent sides are called legs. Non-congruent side is called the base. Angle opposite the base is called the vertex. Angles adjacent to the base are called the base angle(s). Leg Vertex Leg Base angles Base

59 What are the parts of the triangle. a is the/a b is the/a c is the/a A is the/a c B is the/a C is the/a B a A b C

60 Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent. Converse to the Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent.

61 Find the value of x in each problem. 1) 2) (5x) (65) (45) (5x) 3) 4) 6m x ft 15 ft (3x)

62 Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular. Corollary to the Converse of the Base Angles Theorem If a triangle is equiangular, then it is equilateral.

63 Find the value of x and y. 1) 2) 5 x (6x) y 3) y x

64 How is it that we remember the least triviality that happens to us, and yet not remember how often we have recounted it to the same person? La Rochefoucauld

65 Transformation is an operation that moves or changes a geometric figure in some way to produce a new figure called an image. Translation moves every point of a figure the same distance in the same direction. Reflection uses a line of reflection to create a mirror image of the original figure. Rotation turns a figure about a fixed point, called the center of rotation.

66

67 Line of reflection

68 Center of rotation

69 Name the type of transformation demonstrated in each picture. 1) 2) 3)

70 This notation can describe any translation. (x,y) (x+a, y+b) a is how far the image has moved horizontally (positive is to the right and negative is to the left) b is how far the image has moved vertically (positive is upward and negative is downward).

71 Use coordinate notation to describe the translation. 1) 4 units to the right and 3 units up. (x,y) (x+4,y+3) 2) 6 units to the left and 2 units down. (x,y) (x-6,y-2) 3) 5 units to the left and 7 units up. (x,y) (x-5,y+7)

72 Draw the figure after the given translation. (x,y) (x+11,y+8) D(6,7) A E(3,1) F(9,1) B C

73 Reflection over the x-axis. (x, y) (x, -y) Reflection over the y-axis. (x, y) (-x, y)

74 A point on the original figure and the transformation is given. Find the corresponding point on the image. 1) Point on original shape: (5,4); Transformation: (x,y) (x+2, y-1) Point on image: (5+2, 4-1) or (7, 3) 2) Point on original shape: (-3,2); Transformation: (x,y) (x,-y) Point on image: (-3, -2)

75 A point on the image and the transformation is given. Find the corresponding point on the original figure. 1) Point on image: (7,6); Transformation: (x,y) (x+2, y-1) Point on original shape: (7-2, 6+1) or (5, 7) 2) Point on image: (-2,5); Transformation: (x,y) (x,-y) Point on original shape: (-2, -5)