Proving Triangle Congruence: SSS and SAS

SSS CONGRUENT TRIANGLE PROOFS If all corresponding sides of two triangles are congruent, then the triangles are congruent. However, you can prove triangles are congruent using fewer parts.   SSS (Side-Side-Side) CONGRUENCE THEOREM If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.   AB ≅ DE (Side) BC ≅ EF (Side) AC ≅ DF (Side) Then, △ABC ≅ △DEF 1. Given: PQ ≅ ST, QR ≅ TR, R is the midpoint of PS Prove: △PQR ≅ △STR Statements Reasons 1. PQ ≅ ST 1. Given 2. QR ≅ TR 2. Given 3. R is the midpoint of PS 3. Given 4. PR ≅ SR 4. Definition of midpoint 5. △PQR ≅ △STR 5. SSS 2. Given: JL ≅ NM, K is the midpoint of JN and LM Prove: △JKL ≅ △NKM Statements Reasons 1. JL ≅ NM 1. Given 2. K is the midpoint of JN and LM 2. Given 3. JK ≅ NK 3. Definition of midpoint 4. LK ≅ MK 4. Definition of midpoint 5. △JKL ≅ △NKM 5. SSS 3. Given: AC ≅ AD, CB ≅ DB Prove: ∆ABC ≅ ∆ABD Statements Reasons 1. AC ≅ AD 1. given 2. CB ≅ DB 2. given 3. AB ≅ AB 3. reflexive 4. ∆ABC ≅ ∆ABD 4. SSS SAS CONGRUENT TRIANGLE PROOFS SAS (Side-Angle-Side) CONGRUENCE THEOREM If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.   AB ≅ DE (Side) ∠B ≅ ∠E (Angle) BC ≅ EF (Side) Then, ∆ABC ≅ ∆DEF "INCLUDED" MEANS THE ANGLE BETWEEN THE SIDES! 4. Given: JM ≅ LN, ∠JMN ≅ ∠LNM Prove: ∆JMN ≅ ∆LNM Statements Reasons 1. JM ≅ LN 1. given 2. ∠JMN ≅ ∠LNM 2. given 3. MN ≅ MN 3. Reflexive 4. ∆JMN ≅ ∆LNM 4. SAS 5. Given: AB ≅ CB, BD bisects ∠ABC Prove: ∆ABD ≅ ∆CBD Statements Reasons 1. AB ≅ CB 1. given 2. BD bisects ∠ABC 2. given 3. ∠ABD ≅ ∠CBD 3. def. of angle bisector 4. BD ≅ BD 4. reflexive 5. ∆ABD ≅ ∆CBD 5. SAS 6. Given: V is the midpoint of WZ and XY Prove: ∆WVX ≅ ∆ZYV Statements Reasons 1. V is the midpoint of WZ and XY 1. given 2. WV ≅ ZV 2. def. of midpoint 3. ∠WVX ≅ ∠ZYV 3. vertical angles 4. XV ≅ YV 4. def. of midpoint 5. ∆WVX ≅ ∆ZYV 5. SAS 7. Given: HG ≅ HJ, HG || GJ Prove: ∆GHJ ≅ ∆IHJ Statements Reasons 1. HG ≅ HJ 1. given 2. HG GJ 3. ∠GHJ ≅ ∠IHJ 3. alternate interior angles 4. HJ ≅ HJ 4. reflexive 5. ∆GHJ ≅ ∆IHJ 5. SAS TRIANGLE CONGRUENCY: SSS vs. SAS Directions: State whether the triangles can be proven congruent, if possible, by SSS or SAS. Include a congruency statement for all congruent triangles.   SAS: ∆OYT ≅ ∆CKP SAS: ∆ACD ≅ ∆CAB SAS: ∆JMK ≅ ∆LKM Not congruent SSS: ∆ECF ≅ ∆EDF Not congruent SSS: ∆PQR ≅ ∆STU SAS: ∆JDA ≅ ∆MEA Directions: Complete each proof using the most appropriate method (SSS or SAS). 1. Given: AB ≅ CD, AB || CD, D is the midpoint of BE Prove: ∆ABD ≅ ∆CDE Statements Reasons 1. AB ≅ CD 1. given 2. AB CD 3. D is midpoint of BE 3. given 4. BD ≅ ED 4. def. of midpoint 5. ∠ABD ≅ ∠CDE 5. corresponding angles 6. ∆ABD ≅ ∆CDE 6. SAS 10. Given: KM ≅ NM, LK ≅ LN Prove: ∆KLM ≅ ∆NLN | Statements | Reasons | |---|---| | 1. KM ≅ NM | 1. given | | 2. LK ≅ LN | 2. given | | 3. LM ≅ LM | 3. reflexive | | 4. ∆KLM ≅ ∆NLM | 4. SSS 11. Given: AD ≅ AB, AC bisects ∠DAB Prove: ∆ADC ≅ ∆ABC | Statements | Reasons | |---|---| | 1. AD ≅ AB | 1. given | | 2. AC bisects ∠DAB | 2. given | | 3. ∠DAC ≅ ∠BAC | 3. def. of angle bisector | | 4. AC ≅ AC | 4. Reflexive | | 5. ∆ADC ≅ ∆ABC | 5. SAS 12. Given: JM ≅ NM, L is the midpoint of JN Prove: ∆JLM ≅ ∆NLM | Statements | Reasons | |---|---| | 1. JM ≅ NM | 1. given | | 2. L is the midpoint of JN | 2. given | | 3. JL ≅ NL | 3. def. of midpoint | | 4. LM ≅ LM | 4. Reflexive | | 5. ∆JLM ≅ ∆NLM | 5. SSS