Triangle Proofs (SSS, SAS, ASA, AAS) - Geometry SMART Packet - Unit 4 Triangles (Part 1)

Sep 24, 2026

Uploaded by Asher

Geometry SMART Packet Unit 4 focuses on triangle congruence and writing proofs using common postulates. It covers SSS, SAS, ASA, and AAS, and emphasizes that AAA or SSA cannot establish triangle congruence. The packet guides students through setting up proofs with statements and reasons, listing key triangle properties and definitions such as reflexive property, vertical angles, midpoints, angle bisectors, perpendicular bisectors, parallel-line angle relationships, and the third angle theorem. Practice problems strengthen these proof methods.

LanguageEnglish

CategoryExam

FormatPDF

Pages13

Reading Time33 min

AccessFree to download

Unit 4: Triangles (Part 1)

Geometry SMART Packet

Triangle Congruence (SSS, SAS, ASA, AAS)

Standards (G.G.27, G.G.28)

Key Properties, Definitions & Theorems for Triangles

Practice: Identify congruence postulate

Practice: Fill in missing reasons (6-10)

Practice: Write a 2-column proof (11-13)

Regents Practice (14-17)

Unit 4: Triangles (Part 1) Geometry SMART Packet Triangle Proofs (SSS, SAS, ASA, AAS) Student: Date: Period: Standards G.G.27 Write a proof arguing from a given hypothesis to a given conclusion. G.G.28 Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles. SSS (Side, Side, Side) SAS (Side, Angle, Side) ASA (Angle, Side, Angle) AAS (Angle, Angle, Side) Note: We can NOT prove triangles with AAA or SSA!! How to set up a proof: Statement Intro: List the givens 9 Most Common Properties, Definitions & Theorems for Triangles | 1. Reflexive Property: AB = BA When the triangles have an angle or side in common
| 6. Definition of a Midpoint Results in two segments being congruent
| | --- | --- | | 2. Vertical Angles are Congruent When two lines are intersecting
| 7. Definition of an angle bisector Results in two angles being congruent
| | 3. Right Angles are Congruent When you are given right trianglesand/or a square/ rectangle
| 8. Definition of a perpendicular bisector
Results in 2 congruent segments and right angles.
| | 4. Alternate Interior Angles of Parallel Lines are congruent When the givens inform you that two lines are parallel
| 9. 3rd angle theorem
If 2 angles of a triangle are 􀀣 to 2 angles of another triangle, then the 3rd angles are 􀀣
| | 5. Definition of a segment bisector Results in 2 segments being congruent
| Note: DO NOT ASSUME ANYTHING IF IT IS NOT IN THE GIVEN | Directions: Check which congruence postulate you would use to prove that the two triangles are congruent. | 1.
| | | --- | --- | | 2.
| | | 3.
| | | 4.
| | | 5.
| | Practice. Fill in the missing reasons 6. Given: YLF 􀀣 FRY, RFY 􀀣 LFY Prove: 􀁕 FRY 􀀣 􀁕 FLY | Statement | Reason | | --- | --- | | 1. YLF 􀀣 FRY | | | 2. RFY 􀀣 LFY | | | 3. FY 􀀣 FY | | | 4. 􀁕 FRY 􀀣􀁕 FLY | | 7. Given: LT 􀀣 TR , ILT 􀀣 ETR, IT || ER Prove: 􀁕 LIT 􀀣 􀁕 TER | Statement | Reason | | --- | --- | | 1. LT 􀀣 TR | | | 2. ILT 􀀣 ETR | | | 3. IT || ER | | | 4. LTI 􀀣 ERT | | | 5. 􀁕 LIT 􀀣 􀁕 TER | | 8. Given: C is midpoint of BD AB 􀁁 BD BD 􀁁 DE Prove: 􀁕ABC 􀀣 􀁕 EDC | Statement | | | Reason | | --- | --- | --- | --- | |
1. C is midpoint of BD | | | | | 2. | AB 􀁁 BD and BD 􀁁 DE | | | | 3. | BC 􀀣 CD | | | | 4. | 􀂑BCA 􀀣 􀂑ECD | | | | 5. | 􀂑ABC and 􀂑EDCare right angles | | | | 6. | 􀂑ABC 􀀣 􀂑EDC | | | | 7. 􀀧ABC 􀀣 􀀧EDC
9. Given: BA 􀀣 ED
C is the midpoint of BEand AD
Prove: 􀁕ABC 􀀣 􀁕 DEC
| | | | | Statement | | | Reason | | 1. BA 􀀣 ED | | | | |
2. C is the midpoint of BE and | | AD | | | 3. BC 􀀣 EC | | | | | 4. AC 􀀣 DC | | | | | 5. 􀀧ABC 􀀣 􀀧DEC | | | | 10. Given: BC 􀀣 DA AC bisects 􀂑BCD Prove: 􀁕ABC 􀀣 􀁕 CDA | Statement | Reason | | --- | --- | | 1. BC 􀀣 DA | | | 2. AC bisects 􀂑BCD | | | 3. 􀂑BCA 􀀣 􀂑DCA | | | 4. AC 􀀣 AC | | | 5. 􀀧ABC 􀀣 􀀧CDA | | Practice. Write a 2-column proof for the following problems. 11. 12. Given: C is the midpoint of BD and AE Prove: 13. Given: AB 􀀣 CB, BD is a median of AC Prove: 􀀧ABD 􀀣 􀀧CBD Regents Practice 14. Which condition does not prove that two triangles are congruent? (1) (2) (3) (4) 15. In the diagram of and below, , , and . Which method can be used to prove ? (1) SSS (2) SAS (3) ASA (4) HL 16. In the accompanying diagram of triangles BAT and FLU, and . Which statement is needed to prove ? (1) (2) (3) (4) 17. In the accompanying diagram, bisects and . What is the most direct method of proof that could be used to prove ? (1) (2) (3) (4)

FAQs

Which congruence postulates are used in this packet to prove two triangles are congruent?

The packet uses SSS, SAS, ASA, and AAS.

Why can’t AAA or SSA be used to prove triangles congruent?

The packet states that triangles cannot be proved congruent using AAA or SSA.

How do students set up a triangle proof?

List the givens as the Intro, then write the proof using a statement-and-reason structure, using only information from the given.

Sep 24, 2026

Uploaded by Asher

Triangle Proofs (SSS, SAS, ASA, AAS) - Geometry SMART Packet - Unit 4 Triangles (Part 1)_Page 1

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