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
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
150
• Reflexive Sides• Vertical Angles
D
A B
C
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
Step 2: Mark . . .
… if they exist.151
Step 3: Choose a Method
SSSSASASAAASHL
D
A B
C
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
152
Step 4: List the Parts
D
A B
C
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
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
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
STATEMENTS REASONS
1. AB ≅ CD2. BC ≅ DA3. AC ≅ AC
1. Given2. Given3. Reflexive Prop.
SSS
154
SSS
Step 6: Is there more?
155
D
A B
C
Given: AB ≅ CD BC ≅ DAProve: ABC ≅ CDA
STATEMENTS REASONS
1. AB ≅ CD2. BC ≅ DA3. AC ≅ AC
1. Given2. Given3. Reflexive Prop.
4. ABC ≅ CDA 4. SSS (pos.)
The “Prove” Statementis always last !
155
Step 1: Mark the Given
156
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
156
Problem #2
• Reflexive Sides• Vertical Angles
… if they exist.
Step 2: Mark . . .
157
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
157
Step 3: Choose a Method
158
SSSSASASAAASHL
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
158
Step 4: List the Parts
159 … in the order of the Method
STATEMENTS REASONS
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
1. AB ≅ CB2. ∠ABE ≅ ∠CBD
3. EB ≅ DB
SAS
159
Step 5: Fill in the Reasons
160 (Why did you mark those parts?)
STATEMENTS REASONS
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
1. AB ≅ CB2. ∠ABE ≅ ∠CBD
3. EB ≅ DB
1. Given2. Vertical ∠s (thm.)
3. Given
SAS
160
Step 6: Is there more?
161
STATEMENTS REASONS
E
C
D
AB
Given: AB ≅ CB EB ≅ DBProve: ABE ≅ CBD
4. ABE ≅ CBD
1. AB ≅ CB2. ∠ABE ≅ ∠CBD
3. EB ≅ DB
1. Given2. Vertical ∠s (thm.)
3. Given4. SAS (pos.)
The “Prove” Statementis always last !
SAS
161
Problem #3
ZW Y
XGiven: ∠XWY ≅ ∠ZWY ∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
162 162
Step 1: Mark the Given
163
ZW Y
XGiven: ∠XWY ≅ ∠ZWY ∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
• Reflexive Sides• Vertical Angles
… if they exist.
Step 2: Mark . . .
164
ZW Y
XGiven: ∠XWY ≅ ∠ZWY ∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
Step 3: Choose a Method
165
SSSSASASAAASHL
ZW Y
XGiven: ∠XWY ≅ ∠ZWY ∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
165
Step 4: List the Parts
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
Step 5: Fill in the Reasons
167 (Why did you mark those parts?)
STATEMENTS REASONSZ
W YXGiven: ∠XWY ≅ ∠ZWY
∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
1. ∠XWY ≅ ∠ZWY
2. WY ≅ WY3. ∠XYW ≅ ∠ZYW
1. Given2. Reflexive (pos.)3. Given
ASA
Step 6: Is there more?
168
STATEMENTS REASONSZ
W YXGiven: ∠XWY ≅ ∠ZWY
∠XYW ≅ ∠ZYWProve: WXY ≅ WZY
1. ∠XWY ≅ ∠ZWY
2. WY ≅ WY3. ∠XYW ≅ ∠ZYW
4. WXY ≅ WZY
1. Given2. Reflexive (pos.)3. Given4. ASA (pos.)
The “Prove” Statementis always last !
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 …
in the order of the method.5. Fill in the Reasons …
why you marked the parts.6. Is there more?
171 171
Step 1: Mark or draw with the Given Step 2: Decided the kind of proof to use:____
172
ASA Extra
Prac7ce
Step 1: Mark or draw with the Given Step 2: Decided the kind of proof to use:____
173
SAS Extra Prac7ce
Step 1: Mark or draw with the Given Step 2: Decided the kind of proof to use:____
174
SSS Extra Prac7ce
175
176
AAS Extra
Prac7ce
177
AAS Extra
Prac7ce
ANSWER
178
HL Extra Prac7ce
179 10/11/19
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181
182
183
Given implies Congruent Parts
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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
Step 1: Mark the Given
186
… and what it implies
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
• Reflexive Sides• Vertical Angles
Step 2: Mark . . .
187
… if they exist.
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
187
Step 3: Choose a Method
188
SSSSASASAAASHL
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
188
Step 4: List the Parts
189
STATEMENTS REASONS
… in the order of the Method
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
∠BAC ≅ ∠DAC
AB ≅ AD
AC ≅ AC
S
AS
189
Step 5: Fill in the Reasons
190 (Why did you mark those parts?)
STATEMENTS REASONS
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
∠BAC ≅ ∠DAC
AB ≅ AD
AC ≅ AC
Given
Def. of Bisector Reflexive (prop.)
S
AS
Angle
190
S
AS
Step 6: Is there more?
191
STATEMENTS REASONS
CB D
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
∠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
AGiven: AC bisects ∠BAD AB ≅ ADProve: ABC ≅ ADC
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 …
in the order of the method.5. Fill in the Reasons …
why you marked the parts.6. Is there more?
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
Corresponding Parts of Congruent Triangles
• 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
Corresponding Parts of Congruent Triangles
• 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.
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