Theorem 54 also known as “AAS”

1. Theorem 54also known as “AAS” By Chenxia Liu

2. Proof of AAS theorem Statements Reasons ~ • G K • H M • JH OM • J O • GHJ KMO 1.Given 2. No-Choice Theorem 3. ASA = ~ If there exists a correspondence between the vertices of two triangles such that two angles and a non-included side of one are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS) = ~ = ~ = ~ =

3. History Euclid, known as the “Father of Geometry”, is known for his book, The Elements. He outlined the Angle-Angle-Side theorem in Book I of The Elements.

4. Textbook Proof #1 Statements Reasons • O • SOV TOW • WSO VTO • 2. WO VO • 3. SOW TOV • WOS VOT • 5. SO TO 1. Given 2. All radii of a circle are 3. Subtraction 4. AAS 5. CPCTC ~ = ~ = ~ = ~ = ~ = ~ = ~ =

5. Textbook Proof #2 Statements Reasons 1. PD and PC lie in plane m BP m C D 2. BP PD and PC 3. BPC and BPD are rt s 4. BPC BPD 5. BP BP 6. CBP DBP 7. PBC PBD 1. Given ~ = 2. a segment to a plane is to all lines on that plane 3. segments form rt s ~ ~ 4. rt s are 5. Reflexive 6. AAS 7. CPCTC = = ~ = ~ = ~ =

6. Black Book Proof Statements Reasons 1. Given ~ 1. A B CD AB 2. CD CD 3. CDA is a rt CDB is a rt 4. CDA CDB 5. CDA CDB = 2. Reflexive 3. segments form rt s ~ = ~ ~ 4. rt s are 5. AAS = = ~ =

7. Statements Reasons 1. CA CB EG CA DF CB DF EG 2. EGA is a rt DFB is a rt 3. EGA DFB 4. GAB FBD ~ = 1. Given ~ = 2. segments form rt s ~ ~ 3. rt s are 4. If two sides of a triangle are , then the angles opposite the sides are = = ~ = ~ = ~ = ~ 5. GAE FBD 6. FDA GEB = 5. AAS 6. CPCTC ~ =