G 6 Proving Triangles Congruent 1 Visit www

  • Slides: 47
Download presentation
G. 6 Proving Triangles Congruent 1 Visit www. worldofteaching. com For 100’s of free

G. 6 Proving Triangles Congruent 1 Visit www. worldofteaching. com For 100’s of free powerpoints.

2 The Idea of Congruence Two geometric figures with exactly the same size and

2 The Idea of Congruence Two geometric figures with exactly the same size and shape. F B A C E D

3 How much do you need to know. . . about two triangles to

3 How much do you need to know. . . about two triangles to prove that they are congruent?

4 Corresponding Parts Previously we learned that if all six pairs of corresponding parts

4 Corresponding Parts Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B 1. AB DE 2. BC EF 3. AC DF 4. A D 5. B E 6. C F A C ABC DEF E F D

5 Do you need all six ? NO ! SSS SAS ASA AAS HL

5 Do you need all six ? NO ! SSS SAS ASA AAS HL

6 Side-Side (SSS) If the sides of one triangle are congruent to the sides

6 Side-Side (SSS) If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. B E Side A C 1. AB DE 2. BC EF 3. AC DF Side F D Side ABC DEF The triangles are congruent by SSS.

7 Included Angle The angle between two sides HGI G GIH I GHI H

7 Included Angle The angle between two sides HGI G GIH I GHI H This combo is called side-angle-side, or just SAS.

8 Included Angle Name the included angle: E Y S YE and ES YES

8 Included Angle Name the included angle: E Y S YE and ES YES or E ES and YS YSE or S YS and YE EYS or Y The other two angles are the NON-INCLUDED angles.

Side-Angle-Side (SAS) 9 If two sides and the included angle of one triangle are

Side-Angle-Side (SAS) 9 If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. included angle B E Side F A C 1. AB DE 2. A D 3. AC DF D Side Angle ABC DEF The triangles are congruent by SAS.

10 Included Side The side between two angles GI HI GH This combo is

10 Included Side The side between two angles GI HI GH This combo is called angle-side-angle, or just ASA.

11 Included Side Name the included side: E Y S Y and E YE

11 Included Side Name the included side: E Y S Y and E YE E and S ES S and Y SY The other two sides are the NON -INCLUDED sides.

Angle-Side-Angle (ASA) 12 If two angles and the included side of one triangle are

Angle-Side-Angle (ASA) 12 If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. included side B E Side A C 1. A D 2. AB DE 3. B E Angle F D Angle ABC DEF The triangles are congruent by ASA.

Angle-Side (AAS) 13 If two angles and a non-included side of one triangle are

Angle-Side (AAS) 13 If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. Non-included side Angle B E F A C 1. A D 2. B E 3. BC EF Side D Angle ABC DEF The triangles are congruent by AAS.

14 Warning: No SSA Postulate There is no such thing as an SSA postulate!

14 Warning: No SSA Postulate There is no such thing as an SSA postulate! Side Angle Side The triangles are NOTcongruent!

15 Warning: No SSA Postulate There is no such thing as an SSA postulate!

15 Warning: No SSA Postulate There is no such thing as an SSA postulate! NOT CONGRUENT!

BUT: SSA DOES work in one situation! 16 If we know that the two

BUT: SSA DOES work in one situation! 16 If we know that the two triangles are right triangles! Side Angle

17 We call this HL, for “Hypotenuse – Leg” Hypotenuse Leg RIGHT Triangles! These

17 We call this HL, for “Hypotenuse – Leg” Hypotenuse Leg RIGHT Triangles! These triangles ARE CONGRUENT by HL! Remember! The triangles must be RIGHT!

Hypotenuse-Leg (HL) 18 If the hypotenuse and a leg of a right triangle are

Hypotenuse-Leg (HL) 18 If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Right Triangle Hy pot 1. AB HL 2. CB GL 3. C and G are rt. ‘s enu se Leg ABC DEF The triangles are congruent by HL.

19 Warning: No AAA Postulate There is no such thing as an AAA postulate!

19 Warning: No AAA Postulate There is no such thing as an AAA postulate! Same Shapes! E B A C D NOT CONGRUENT! Different Sizes! F

Congruence Postulates and Theorems 20 • SSS • SAS • ASA • AAS •

Congruence Postulates and Theorems 20 • SSS • SAS • ASA • AAS • AAA? • SSA? • HL

21 Name That Postulate (when possible) SAS SSA Not enough info! ASA AAS

21 Name That Postulate (when possible) SAS SSA Not enough info! ASA AAS

22 Name That Postulate (when possible) AAA SSA Not enough info! SSS SSA HL

22 Name That Postulate (when possible) AAA SSA Not enough info! SSS SSA HL

23 Name That Postulate (when possible) Not enough info! SSA HL Not enough info!

23 Name That Postulate (when possible) Not enough info! SSA HL Not enough info! AAA

Vertical Angles, Reflexive Sides and Angles 24 When two triangles touch, there may be

Vertical Angles, Reflexive Sides and Angles 24 When two triangles touch, there may be additional congruent parts. Vertical Angles Reflexive Side shared by two triangles

25 Name That Postulate (when possible) Reflexive Property SAS Vertical Angles AAS Vertical Angles

25 Name That Postulate (when possible) Reflexive Property SAS Vertical Angles AAS Vertical Angles SAS Reflexive Property SSA Not enough info!

26 Reflexive Sides and Angles When two triangles overlap, there may be additional congruent

26 Reflexive Sides and Angles When two triangles overlap, there may be additional congruent parts. Reflexive Side shared by two triangles Reflexive Angle angle shared by two triangles

Let’s Practice 27 Indicate the additional information needed to enable us to apply the

Let’s Practice 27 Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F

28 What’s Next Try Some Proofs End Slide Show

28 What’s Next Try Some Proofs End Slide Show

29 Choose a Problem #1 SSS Problem #2 SAS Problem #3 ASA End Slide

29 Choose a Problem #1 SSS Problem #2 SAS Problem #3 ASA End Slide Show

AAS Problem #4 Statements Reasons Given Vertical Angles Thm Given AAS Postulate 55

AAS Problem #4 Statements Reasons Given Vertical Angles Thm Given AAS Postulate 55

HL Problem #5 Given ABC, ADC right s, Prove: Statements 1. ABC, ADC right

HL Problem #5 Given ABC, ADC right s, Prove: Statements 1. ABC, ADC right s Reasons Given Reflexive Property HL Postulate 57

58 Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles

58 Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles / Vertical Angles Also: mark info implied by given info. 3. Choose a Method. (SSS , SAS, ASA, AAS , HL) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

59 Given implies Congruent Parts midpoint parallel segment bisector segments angles segments angle bisector

59 Given implies Congruent Parts midpoint parallel segment bisector segments angles segments angle bisector angles perpendicular angles

60 Example Problem

60 Example Problem

61 Step 1: Mark the Given … and what it implies

61 Step 1: Mark the Given … and what it implies

62 Step 2: Mark. . . • Reflexive Sides • Vertical Angles … if

62 Step 2: Mark. . . • Reflexive Sides • Vertical Angles … if they exist.

63 Step 3: Choose a Method SSS SAS ASA AAS HL

63 Step 3: Choose a Method SSS SAS ASA AAS HL

64 Step 4: List the Parts STATEMENTS REASONS S A S … in the

64 Step 4: List the Parts STATEMENTS REASONS S A S … in the order of the Method

65 Step 5: Fill in the Reasons STATEMENTS REASONS S A S (Why did

65 Step 5: Fill in the Reasons STATEMENTS REASONS S A S (Why did you mark those parts? )

66 Step 6: Is there more? STATEMENTS S 1. 2. A 3. S 4.

66 Step 6: Is there more? STATEMENTS S 1. 2. A 3. S 4. 5. REASONS 1. 2. 3. 4. 5.

72 Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark

72 Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

73 Using CPCTC in Proofs According to the definition of congruence, if two triangles

73 Using CPCTC in Proofs According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

74 Corresponding Parts of Congruent Triangles For example, can you prove that sides AD

74 Corresponding Parts of Congruent Triangles For example, can you prove that sides AD and BC are congruent in the figure at right? The sides will be congruent if triangle ADM is congruent to triangle BCM. Angles A and B are congruent because they are marked. Sides MA and MB are congruent because they are marked. Angles 1 and 2 are congruent because they are vertical angles. So triangle ADM is congruent to triangle BCM by ASA. This means sides AD and BC are congruent by CPCTC.

Corresponding Parts of Congruent Triangles 75 A two column proof that sides AD and

Corresponding Parts of Congruent Triangles 75 A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Given Ð 1 @ Ð 2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles 76 A two column proof that sides AD and

Corresponding Parts of Congruent Triangles 76 A two column proof that sides AD and BC are congruent in the figure at right is shown below: Statement Reason MA @ MB Given ÐA @ ÐB Given Ð 1 @ Ð 2 Vertical angles DADM @ DBCM ASA AD @ BC CPCTC

Corresponding Parts of Congruent Triangles 77 Sometimes it is necessary to add an auxiliary

Corresponding Parts of Congruent Triangles 77 Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU Given UF @ UF reflexive prop. DFRU @ DFOU SSS ÐR @ ÐO CPCTC

Corresponding Parts of Congruent Triangles 78 Sometimes it is necessary to add an auxiliary

Corresponding Parts of Congruent Triangles 78 Sometimes it is necessary to add an auxiliary line in order to complete a proof For example, to prove ÐR @ ÐO in this picture Statement Reason FR @ FO Given RU @ OU Given UF @ UF Same segment DFRU @ DFOU SSS ÐR @ ÐO CPCTC