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Chapter 4 Notes Congruent Polygons: Symbol: β Definition: -Simplified: shapes that have the same angles or sides -Text Book: two polygons are congruent if they are the same size and shape - that is, if their corresponding angles and sides are equal Steps to Naming: 1. Congruent Polygons: B C F G A D E H Name ALL of the CONGRUENT angles and sides: Congruent Polygons: B C F G A D E H Give the congruence Statement for the two figures: Congruent Polygons: If LAMPS β BRITE, What are the congruent corresponding parts? L A M S E T P I R B Congruent Polygons: If FEAR β RUNS, 1. Draw the figure. 2. List the congruent corresponding parts. Congruent Triangles The congruent statement βπ΄π΅πΆ β βπ ππ --tell us that triangles ABC and RST are congruent. Order matters! A corresponds to R. (Aβ R) B corresponds to S. (Bβ S) C corresponds to T. (Cβ T) Congruent Triangles If DNA β YOU, then list the congruent corresponding parts. Congruent Right Triangles β’ Same as before except: the RIGHT ANGLE has to be in the MIDDLE of the triangleβs name β’ SAM β ROB Congruent Triangles Congruent Triangles Words you need to know to do PROOFS: β’Congruent β’Vertical angles β’Addition Property β’Subtraction Property β’Multiplication Property Words you need to know to do PROOFS: β’Division Property β’Substitution Property β’Symmetric Property β’Reflexive Property β’Third Angle Theorem β’Midpoint Words you need to know to do PROOFS: β’Parallel β’Alternate Interior β’Alternate Exterior β’Interior β’Exterior β’Perpendicular Bisector Introduction to PROOFS: β’ 2 Column Proofs: STATEMENT REASONS 1. 2. 3. 1. 2. 3. Introduction to PROOFS: Write a proof for the following: STATEMENT 5x + 3 = 18 REASONS 1. 1. 2. 2. 3. 3. Introduction to PROOFS: Write a proof for the following: 2x β 6 = 10 STATEMENT 3 REASONS 1. 1. 2. 2. 3. 3. Introduction to PROOFS: Write a proof for the following: 3x β 8 = 11 STATEMENT 2 REASONS 1. 1. 2. 2. 3. 3. Proving Triangles are Congruent β’ Steps: 1. If not given a picture, draw one. 2. Mark what you know on the picture (not what you are trying to prove). 3. Decide how you know the triangles are congruent (SSS, SAS, ASA, AAS). 4. State the given and the prove. 5. Complete the proof with the information. Triangle Congruence Principles Side-Side-Side (SSS) If the three sides of one triangle have the same measures as the three sides of a second triangle, then those triangles are congruent. If: π΄π΅ β π π π΄πΆ β π π πΆπ΅ β ππ Then: βπ΄πΆπ΅ β βπ ππ Side-Side-Side (SSS) Given: ππ β ππΏ πΏπ β ππ Prove: βπππΏ β βππΏπ STATEMENT REASONS 1. 1. Given 2. 2. 3. 3. Side-Side-Side (SSS) E B F Given: πΈπ΅ β πΉπΆ πΈπΆ β πΉπ΅ Prove:βπΉπ΅πΈ β βπΈπΆπΉ C STATEMENT REASONS 1. πΈπ΅ β πΉπΆ πΈπΆ β πΉπ΅ 2. 1. Given 2. 3. 3. Side-Side-Side (SSS) Triangle Congruence Principles Side-Angle-Side (SAS) If two sides and the included angle of one triangle have the same measure as two sides and the included angle of a second triangle, then those triangles are congruent. ππ β π΅π΄ π π β π΄πΆ β ππ π β β π΅π΄πΆ Then: βππ π β βπ΅π΄πΆ If: Side-Angle-Side (SAS) STATEMENT REASONS 1. π΅π΄ β π΅πΆ 1. Given β π΄π΅π β β πΆπ΅π 2. 2. Prove that ΞABM β ΞCBM. 3. 3. Side-Angle-Side (SAS) Given: π»πΌ β πΊπΌ I is the midpoint of π½πΈ Prove: π»π½ β πΊπΈ STATEMENT 1. π»πΌ β πΊπΌ I is the midpoint of π½πΈ 2. REASONS 1. Given 3. β π½πΌπ» β β πΈπΌπΊ 2. Definition of Midpoint 3. 4. βπΌπ½π» β βπΌπΈπΊ 4. 5. 5. Side-Angle-Side (SAS) T C STATEMENT REASONS 1. 1. Given G A 2. β πΆπ΄π β β πΊπ΄π΅ 2. B Prove: πΆπ β π΅πΊ 3. 3. 4. πΆπ β πΊπ΅ 4. Triangle Congruence Principles Angle-Side-Angle (ASA) If two angles and an included side of one triangle have the same measures as two angles and an included side of a second triangle, then those triangles are congruent. π΄π΅ β π π β π΅π΄πΆ β β ππ π β πΆπ΅π΄ β β πππ Then: βπ΄πΆπ΅ β βπ ππ If: Angle-Side-Angle (ASA) R o y STATEMENT 1.β πππ β β πππ½ β π ππ β β π½ππ 2. 2. 3. J Prove: βπ ππ β βπ½ππ REASONS 3. 1. Given Angle-Side-Angle (ASA) A C N T O Prove: βπΆπ΄π β β πππ STATEMENT REASONS 1. 1. Given 2. 2. 3. 3. Angle-Side-Angle (ASA) U B S Prove: π΅π β π π STATEMENT REASONS 1. 1. Given 2. 2. 3. 3. R Triangle Congruence Principles Angle-Angle-Side (AAS) If two angles and a non-included side of one triangle have the same measures as two angles and a nonincluded side of a second triangle, then those triangles are congruent. π΄π΅ β π·πΈ β π΅π΄πΆ β β πΈπ·πΉ β π΄πΆπ΅ β β π·πΉπΈ Then: βπ΄π΅πΆ β βπ·πΈπΉ If: Angle-Angle-Side (AAS) STATEMENT J R A 1. π½π β π·π 1. Given β π½π΄π β β π·ππ 2. 2. 3. O D Prove: βπ ππ· β βπ½π΄π REASONS 3. Angle-Angle-Side (AAS) X Z Y W T Prove: βπππ β βπππ STATEMENT REASONS 1. 1. Given 2. 2. 3. 3. THERE IS NO ASS CONGRUENCIES IN Triangle Congruence Principles Hypotenuse-Leg (HL) If the hypotenuse and a leg of one RIGHT triangle have the same measures as the hypotenuse and a leg of a second RIGHT triangle, then those triangles are congruent. π΄π΅ β π·πΈ π΅πΆ β πΈπΉ β π΅πΆπ΄ β β πΈπΉπ· Then: βπ΄π΅πΆ β βπ·πΈπΉ If: Hypotenuse-Leg (HL) O B Prove: βπ΅ππ β βπππ S X STATEMENT REASONS 1. 1. Given 2. 2. 3. 3. Hypotenuse-Leg (HL) STATEMENT REASONS 1. 1. Given 2. 2. A 3. E H Given: ππ» is a perpendicular bisector to πΈπ΄ ππΈ β ππ΄ Prove: βππ»πΈ β βππ΄π» 3. T Hypotenuse-Leg (HL) L G J K Prove: πΏπΎ β πΊπΌ I H STATEMENT REASONS 1. 1. Given 2. 2. 3. 3.