1. Warm-Up 2. 1. 3. 4. Which congruence shortcut, if any,
Supplies Needed:
Giant Whiteboard, Marker, Eraser
Warm-Up
Which congruence shortcut, if any,
can be used to prove the triangles congruent? 1. 2.
None
SSS 3. 4. HL
none

2. Bellringer
Complete Warm up!

3. Check Homework

4. A B C D E
Prove: ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶
Which one do you want to see first?
Paragraph proof
Two-column proof
Flow-chart proof

5. Paragraph Proof
Just write, using complete sentences, a logical argument that proves what you want to prove. For everything you state, you must say how you know it.

6. Paragraph Proof E B Prove: ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶 CD E
Prove: ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶
We know 𝑨𝑩 ≅ 𝑬𝑫 because it is given. We also know that ∠𝑨≅∠𝑬 because it is given. In addition, ∠𝑩𝑪𝑨≅∠𝑫𝑪𝑬 because they are vertical angles. Thus, ∆𝑨𝑩𝑪≅∆𝑬𝑫𝑪 by AAS.

7. Two-Column Proof
Organizes your proof into columns. One column is for your statements, and the other one is for your reasons. The last statement will always be the one you are trying to prove.

8. Two-Column Proof A B C D E Prove: ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶 Statement 1) ___________
2) ___________
3) ___________
4) ___________
Reason A S
∠𝑨≅∠𝑬
Given
∠𝑩𝑪𝑨≅∠𝑫𝑪𝑬
Vertical Angles Thm.
𝑨𝑩 ≅ 𝑬𝑫
Given
∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶
AAS

9. Flow Chart Proof
A visual depiction of your proof. Each “bubble” will have a statement and a reason in it. You draw arrows to show which statements lead to which other statements.

10. Flow-Chart Proof AAS E B Prove: ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶 C D A ∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶 Given: ∠𝑨≅∠𝑬
Vertical Angles Thm:
∠𝑩𝑪𝑨≅∠𝑫𝑪𝑬
AAS
∆𝐴𝐵𝐶≅∆𝐸𝐷𝐶
Given:
𝑨𝑩 ≅ 𝑬𝑫

11. Given: K is the midpoint of 𝐽𝐿 . Prove: ∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀 J K L M

12. Two-Column Proof M Given: K is the midpoint of 𝐽𝐿 . Prove: ∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀 LJ K L M
Two-Column Proof
Given: K is the midpoint of 𝐽𝐿 . Prove: ∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀
Reason
1) ___________
2) ___________
3) ___________
4) ___________
5) ___________
Statement
𝑴𝑲 ≅ 𝑴𝑲
Reflexive Property
Given
∠𝑱𝑲𝑴≅∠𝑳𝑲𝑴
K is the midpoint of 𝐽𝐿
Given
𝑱𝑲 ≅ 𝑳𝑲
Definition of midpoint
∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀
SAS

13. Flow-Chart Proof SAS M Given: K is the midpoint of 𝐽𝐿 .J K L M
Given: K is the midpoint of 𝐽𝐿 .
Prove: ∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀
Reflexive Prop.
𝑲𝑴 ≅ 𝑲𝑴
SAS
∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀
Given:
∠𝑱𝑲𝑴≅∠𝑳𝑲𝑴
Given:
K is the
midpoint of 𝐽𝐿
Def. of midpoint:
𝑱𝑲 ≅ 𝑳𝑲

14. P
Given: 𝑸𝑹 bisects ∠𝑷𝑸𝑺. Prove: ∆𝑷𝑸𝑹≅∆𝑺𝑸𝑹 Q R S

15. Prove: ∆𝑾𝑿𝒀≅∆𝒀𝒁𝑾 X W Y Z

16. B E
Given: 𝐴𝐵 ∥ 𝐷𝐸 Prove: ∆𝑨𝑩𝑪≅∆𝑬𝑫𝑪 C A D

17. Write a flowchart proof!
Given: JL bisects KLM
Prove: JKL  JML

19. Write a Two-Column Proof
Given: C is the midpoint of BD and AE.
Prove: ∆ABC  ∆EDC

20. 4. SAS 4. ABC  EDC 3. Vert. s Thm. 3. ACB  ECD 2. Def. of mdpt.
2. AC  EC; BC  DC
1. Given
1. C is mdpt. of BD and AE
Reasons
Statements

21. Write a Two-Column Proof!
Given: BC || AD, BC  AD
Prove: ∆ABD  ∆CDB
Statements
Reasons
1. Given
1. BC  AD
2. BC || AD
2. Given
3. CBD  ABD
3. Alt. Int. s Thm.
4. BD  BD
4. Reflex. Prop. of 
5. ∆ABD  ∆ CDB
5. SAS

22. Write a Two-Column Proof
Given: QP bisects RQS. QR  QS
Prove: ∆RQP  ∆SQP
Statements
Reasons
1. QR  QS
1. Given
2. QP bisects RQS
2. Given
3. RQP  SQP
3. Def. of angle bisector
4. QP  QP
4. Reflex. Prop. of 
5. ∆RQP  ∆SQP
5. SAS

23. 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.

24. SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.
Remember!

25. Write a Flowchart Proof!
Given: YW bisects XZ, XY  ZY.
Prove: XYW  ZYW Z

26. WY ZW Z

27. Write a Two Column Proof!
Prove: NMO  POM
Given: NO || MP, N  P

28. Statements
Reasons
1. N  P
1. Given
2. Given
2. NO || MP
3. NOM  PMO
3. Alt. Int. s Thm.
4. Reflex. Prop. of 
4. MO  MO
5. AAS
5. ∆MNO  ∆OPM
6. NMO  POM
6. CPCTC

29. Write a Two Column Proof!
Prove:
Given: J is the midpoint of KM and NL.
LKJ  NMJ

30. Statements
Reasons
1. Given
1. J is the midpoint of KM and NL.
2. KJ  MJ, NJ  LJ
2. Def. of mdpt.
3. KJL  MJN
3. Vert. s Thm.
4. ∆KJL  ∆MJN
4. SAS
5. LKJ  NMJ
5. CPCTC

31. Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)
Using CPCTC In the Coordinate Plane
Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1), H(0, 5), and I(1, 3)
Prove: DEF  GHI

32. Use the Distance Formula to find the lengths of the sides of each triangle.
Determine that ∆DEF  ∆GHI by SSS. DEF  GHI by CPCTC.

33. Given: J(–1, –2), K(2, –1), L(–2, 0), R(2, 3), S(5, 2), T(1, 1)
Using CPCTC In the Coordinate Plane
Given: J(–1, –2), K(2, –1), L(–2, 0), R(2, 3), S(5, 2), T(1, 1)
Prove: JKL  RST

34. Use the Distance Formula to find the lengths of the sides of each triangle.
Determine that ∆JKL  ∆RST by SSS. JKL  RST by CPCTC.

35. Proofs Worksheet