triangle congruency criteria · 2019. 8. 15. · congruent triangles proofs 1. mark the given and...

Modules 5 & 6 Proofs Part 2

Honors: 5.2: 7-‐9 5.3: 8-‐9 5.4: 10, 13, 22, 26 6.2: 13-‐14, 25 6.3 6, 8-‐9, 20

Regular: 5.2: 7-‐8 5.3: 8-‐9 5.4: 10, 13, 22 6.2: 13-‐14 6.3: 6, 8-‐9

Triangle Congruency Criteria

Homework

Congruent Triangles Proofs

1. Mark the Given and what it implies.2. Mark … Reflexive Sides / Vertical Angles3. 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?

146 146

Choose a Problem.

147

Problem #1

Problem #2

Problem #3

End Slide Show

D

A B

C

E

C

D

AB

Z

W Y

X

SSS

SAS

ASA

147

Choose a Problem.

148

Problem #4

Problem #5

End Slide Show

AAS

HL

E

C

D

AB

CB D

A

D

A B

C

Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA

Problem #1

149 149

Step 1: Mark the Given

150

D

A B

C


150

• Reflexive Sides• Vertical Angles

D

A B

C


Step 2: Mark . . .

… if they exist.151

Step 3: Choose a Method

SSSSASASAAASHL

D

A B

C


152

Step 4: List the Parts

D

A B

C


STATEMENTS REASONS

1. AB ≅ CD2. BC ≅ DA3. AC ≅ AC

… in the order of the Method

SSS

153

Step 5: Fill in the Reasons

(Why did you mark those parts?)

D

A B

C


STATEMENTS REASONS


1. Given2. Given3. Reflexive Prop.

SSS

154

SSS

Step 6: Is there more?

155

D

A B

C


STATEMENTS REASONS


1. Given2. Given3. Reflexive Prop.

4. ABC ≅ CDA 4. SSS (pos.)

The “Prove” Statementis always last !

155


156

E

C

D

AB

Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD

156

Problem #2


… if they exist.

Step 2: Mark . . .

157

E

C

D

AB


157


158

SSSSASASAAASHL

E

C

D

AB


158


159 … in the order of the Method

STATEMENTS REASONS

E

C

D

AB


1. AB ≅ CB2. ∠ABE ≅ ∠CBD

3. EB ≅ DB

SAS

159


160 (Why did you mark those parts?)

STATEMENTS REASONS

E

C

D

AB



3. EB ≅ DB

1. Given2. Vertical ∠s (thm.)

3. Given

SAS

160


161

STATEMENTS REASONS

E

C

D

AB


4. ABE ≅ CBD


3. EB ≅ DB

1. Given2. Vertical ∠s (thm.)

3. Given4. SAS (pos.)


SAS

161

Problem #3

ZW Y

XGiven: ∠XWY ≅ ∠ZWY ∠XYW ≅ ∠ZYWProve: WXY ≅ WZY

162 162


163

ZW Y



… if they exist.

Step 2: Mark . . .

164

ZW Y



165

SSSSASASAAASHL

ZW Y


165


166 … in the order of the Method

STATEMENTS REASONSZ

W YXGiven: ∠XWY ≅ ∠ZWY

∠XYW ≅ ∠ZYWProve: WXY ≅ WZY

1. ∠XWY ≅ ∠ZWY

2. WY ≅ WY3. ∠XYW ≅ ∠ZYW

ASA



STATEMENTS REASONSZ





1. Given2. Reflexive (pos.)3. Given

ASA


168

STATEMENTS REASONSZ





4. WXY ≅ WZY

1. Given2. Reflexive (pos.)3. Given4. ASA (pos.)


ASA

Problem #4

Statements Reasons

169

AAS

Given

Given

Vertical Angles Thm

AAS Postulate

Given: ∠A ≅ ∠C BE ≅ BDProve: ABE ≅ CBD

E

C

D

AB

4. ABE ≅ CBD

Problem #5

3. AC AC≅

Statements Reasons

170

CB D

AHL

Given

Given

Reflexive Property

HL Postulate 4. ABC ≅ ADC

1. ΔABC, ΔADC right Δs

Given ΔABC, ΔADC right Δs, Prove:

AB AD≅

ABC ADCΔ ≅ Δ

2. 𝐴𝐵  ≅ 𝐴𝐷 

Congruence Proofs1. Mark the Given.2. Mark …

Reflexive Sides or Angles / Vertical AnglesAlso: mark info implied by given info.3. Choose a Method. (SSS , SAS, ASA, AAS, HL)4. List the Parts …



171 171

Step 1: Mark or draw with the Given Step 2: Decided the kind of proof to use:____

172

ASA Extra

Prac7ce


173

SAS Extra Prac7ce


174

SSS Extra Prac7ce

176

AAS Extra

Prac7ce

177

AAS Extra

Prac7ce

ANSWER

178

HL Extra Prac7ce

179 10/11/19

Given implies Congruent Parts

184

midpoint

parallel

segment bisector

angle bisector

perpendicular

segments≅

angles≅

segments≅

angles≅

angles≅184

Example Problem

CB D

AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC

185 185


186

… and what it implies

CB D



Step 2: Mark . . .

187

… if they exist.

CB D


187


188

SSSSASASAAASHL

CB D


188


189

STATEMENTS REASONS

… in the order of the Method

CB D


∠BAC ≅ ∠DAC

AB ≅ AD

AC ≅ AC

S

AS

189



STATEMENTS REASONS

CB D


∠BAC ≅ ∠DAC

AB ≅ AD

AC ≅ AC

Given

Def. of Bisector Reflexive (prop.)

S

AS

Angle

190

S

AS


191

STATEMENTS REASONS

CB D


∠BAC ≅ ∠DAC

AB ≅ AD

AC ≅ AC

Given AC bisects ∠BAD Given

Def. of Bisector

Reflexive (prop.)ABC ≅ ADC SAS (pos.)

1.2.3.4.5.

1.2.3.4.5.

Angle

191

Midpoint implies ≅ segments.

192

Back

CB D

A

STATEMENTS REASONS

1. C is the midpoint of BD

S

Given: C is the midpoint of BD AB ≅ AD Prove: ABC ≅ ADC

2. BC ≅ CD 2. Def. of Midpoint

1. Given

… AB ≅ AD3. 3. Given

192

Parallel implies ≅ alternaHng angles.

193

Back

STATEMENTS REASONSD

A B

C

Given: AB DC AD BC Prove: ABC ≅ CDA

2. ∠BAC ≅ ∠DCA 2. Alt. Int. ∠s (thm.)1. AB DC

3. AD BC 3. Given 4. ∠DAC ≅ ∠BCA 4. Alt. Int. ∠s (thm.)

A

A

1. Given

193

Seg. bisector implies ≅ segments.

194

Back

STATEMENTS REASONS

CBA

DE

2. EB ≅ BC

4. AB ≅ BD

1. AD bisects EC 1. Given 2. Def. of bisect

3. EC bisects AD 3. Given 4. Def. of bisect

S

S

Given: AD bisects EC EC bisects ADProve: ABE ≅ DBC

… 194

Angle bisector implies ≅ angles.

195

Back

STATEMENTS REASONS

CB D


2. ∠BAC ≅ ∠DAC1. AC bisects ∠BAD 1. Given

A

…

2. Def. of bisect

195

⊥ implies ≅ right angles.

196

Back

STATEMENTS REASONS

CB D

AGiven: AC ⊥ BD BC ≅ DCProve: ABC ≅ ADC

2. ∠ACB and ∠ACD are right ∠s3. ∠ACB ≅ ∠ACD

1. AC ⊥ BD 1. Given 2. ⊥ lines form 4 rt. ∠s (thm)3. All rt. ∠s are ≅ (thm)

A

… S 2. BC ≅ CD 4. Given 4. 196

Congruent Triangles Proofs

1. Mark the Given and what it implies.2. Mark … Reflexive Sides / Vertical Angles3. Choose a Method. (SSS , SAS, ASA, AAS, HL)4. List the Parts …



197 197

Using CPCTC in Proofs

•  According to the defini7on 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.

198 198

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 ver7cal angles. •  So triangle ADM is congruent to triangle BCM by ASA.

• This means sides AD and BC are congruent by CPCTC.

199 199


• A two column proof that sides AD and BC are congruent in the figure at right is shown below:

Statement Reason

MA≅MB Given

∠A ≅∠B Given

∠1≅ ∠2 Vertical angles

ΔADM≅ ΔBCM ASA

AD ≅ BC CPCTC


•  Some7mes it is necessary to add an auxiliary line in order to complete a proof

•  For example, to prove ∠𝑅≅∠𝑂 in this picture

Statement Reason

𝐹𝑅 ≅ 𝐹𝑅  Given

𝑅𝑈 ≅ 𝑂𝑈  Given

𝑈𝐹 ≅ 𝑈𝐹  Same segment

ΔFRU≅ ΔFOU SSS

∠R ≅∠O CPCTC

5.2/5.3/5.4/6.2/6.3 Classwork PART 2 Proofs

• GO ONLINE and complete HW Due in Two Days?

•  In Class: Honors: 5.2: 7, 8, 9 5.3 8, 9, 5.4 10, 13, 22, 26, 6.2 13, 14, 25, 6.3 6, 8, 9, 20 Regular: 5.2: 7, 8 5.3 8, 9, 5.4 10, 13, 22, 6.2 13, 14, 6.3 6, 8, 9,

Reminders: • Guess what?! A quiz is nearing.

202

triangle congruency criteria · 2019. 8. 15. · congruent triangles proofs 1. mark the given and...

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