Geo 4-6 Teacher Powerpoint
Download
Report
Transcript Geo 4-6 Teacher Powerpoint
4-6
TriangleCongruence:
Congruence: CPCTC
CPCTC
4-6 Triangle
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
4-6 Triangle Congruence: CPCTC
Do Now
1. If ∆ABC ∆DEF, then A
2. If 1 2, why is a||b?
Holt Geometry
?
and BC
?
.
4-6 Triangle Congruence: CPCTC
Objective
Use CPCTC to prove parts of triangles
are congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Vocabulary
CPCTC
Holt Geometry
4-6 Triangle Congruence: CPCTC
CPCTC is an abbreviation for the phrase
“Corresponding Parts of Congruent
Triangles are Congruent.” It can be used
as a justification in a proof after you have
proven two triangles congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Remember!
SSS, SAS, ASA, AAS, and HL use
corresponding parts to prove triangles
congruent. CPCTC uses congruent
triangles to prove corresponding parts
congruent.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 1: Engineering Application
A and B are on the edges
of a ravine. What is AB?
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 2
A landscape architect sets
up the triangles shown in
the figure to find the
distance JK across a pond.
What is JK?
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 3: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY YZ.
Prove: XYW ZYW
Statements
Reasons
1. 𝑌𝑊𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑋𝑍
1. Given
2. 𝑋𝑌 𝑌𝑍
2. Given
3. 𝑋𝑊 𝑍𝑊
3. Def. segment bisector
4. 𝑌𝑊 𝑌𝑊
4. Reflexive POC
5. 𝑋𝑌𝑊 ZYW
5. SSS
6. 𝑋𝑌𝑊ZYW
6. CPCTC
Holt Geometry
Z
4-6 Triangle Congruence: CPCTC
Example 4
Given: PR bisects QPS and QRS.
Prove: PQ PS
Statements
Reasons
1. 𝑃𝑅 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆 𝑎𝑛𝑑 QRS
1. Given
2. 𝑄𝑃𝑅SPR
2. Def. bisector
3. 𝑄𝑅𝑃SRP
3. Def. bisector
4. 𝑃𝑅 𝑃𝑅
4. Reflexive POC
5. 𝑄𝑃𝑅 SPR
5. ASA
6. 𝑃𝑄 𝑃𝑆
6. CPCTC
Holt Geometry
4-6 Triangle Congruence: CPCTC
Helpful Hint
Work backward when planning a proof. To
show that ED || GF, look for a pair of angles
that are congruent.
Then look for triangles that contain these
angles.
Holt Geometry
4-6 Triangle Congruence: CPCTC
Example 5: Using CPCTC in a Proof
Given: NO || MP, N P
Prove: MN || OP
Statements
1
4
2
Reasons
1. 𝑁𝑂 𝑀𝑃
1. Given
2. 𝑁O
2. Given
3. 12
3. Alt. int. th.
4. 𝑀𝑂 𝑀𝑂
4. Reflexive POC
5. 𝑁𝑂𝑀 PMO
5. AAS
6. 34
6. CPCTC
7. 𝑀𝑁 𝑂𝑃
7. Converse of alt. int. th.
Holt Geometry
3
4-6 Triangle Congruence: CPCTC
Example 6
Given: J is the midpoint of KM and NL.
Prove: KL || MN
3
1
Statements
Reasons
2
4
1. 𝐽 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐾𝑀 𝑎𝑛𝑑𝑁𝐿
1. Given
2. 𝐾𝐽 𝑀𝐽
2. Def. midpoint
3. 𝑁𝐽 𝐿𝐽
3. Def. midpoint
4. 12
4. Vertical th.
5. 𝐾𝐽𝐿 MJN
5. SAS
6. 34
6. CPCTC
7. 𝐾𝐿 𝑀𝑁
7. Converse of alt. int. th.
Holt Geometry
4-6 Triangle Congruence: CPCTC
You Try It!
Given: X is the midpoint of AC .
1 2
Prove: X is the midpoint of BD.
Statements
1. 𝑋 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶
2. 12
Reasons
1. Given
2. Given
3. 𝐴𝑋 𝐶𝑋
3. Def. midpoint
4. 34
4. Vertical th.
5. 𝐴𝑋𝐷 CXB
5. ASA
6. 𝐷𝑋 𝐵𝑋
6. CPCTC
7. 𝑋 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷
7. Def. midpoint
Holt Geometry
4
3