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Congruence Same size AND same shape. Congruent figures can be mapped onto each other such that there intersection is the same A as the figures themselves. A Corresponding Parts are match up in a one-to-oneA correspondence. A!D A AB ↔ DE C C ↔ EF BC C!F AC ↔ DF C B up. B The vertices andBsides must match B B C A B!E B We can therefore write C ABC!DEF A E C point A matches up to point D point B matches up to point E point C matches up to point F The order of the letters is important D F Angles are congruent if they have the same measure. Sides are congruent if they have the same length. Congruence is therefore an equation between geometric figures, while equality is an equation between numbers. CONGRUENCE AND EQUALITY ARE NOT INTERCHANGEABLE!!!!!!! !ABC ! !DBF means the angles have the same measure. !ABC= !DBF means the angles are the same angle. This is true for triangles as well. You can only say two triangles are equal if they are the same triangle. Two triangles are congruent if there is a correspondence between all pairs of sides and all pairs of angles. ∆ABC! ∆DEF tells us 12 things at once. AB ≅ DE, AB = DE, AC ≅ DF, AC = DF, BC ≅ EF, BC = EF ∠A ≅ ∠D, m∠A = m∠D, ∠B ≅ ∠E, m∠B = m∠E, ∠C ≅ ∠F, m∠C = m∠F B A C A side of the triangle is said to be included by the angles whose vertices are the endpoints of the segment. Side AC is included by !A and !C. An angle of the triangle is said to be included by the sides of the triangle which lie on the sides of the angle. !C is included by sides AC and BC. Congruence for segments is an equivalence relation. Congruence for triangles is an equivalence relation. A B C Create ∆ABC using the following specifications. 1. AB = 3 inches, BC = 4 inches, m!B = 45!SAS 2. AB = 3 inches, AC = 4 inches, m!B = 45!SSA 3. m!A = 75!, m!B = 45!, AB = 3 inches ASA 4. m!B = 45!, m!C = 60!, AB = 3 inches AAS 5. AB = 3, BC = 4, AC = 6 SSS 6. m!A = 75!, m!B = 45!, m !C = 60! AAA SSS: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, then the two triangles are congruent. B SAS: If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts B of the other triangle, then the two triangles are congruent. ASA: If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, then the two triangles are congruent. F A C F A C D E F A B D E C E D The order of the letters is important. SAS: Every SAS correspondence is a congruence. F A ∆ABC ! ∆FED B Corresponding vertices C E D E B Given: ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF Pr ove: ΔABC ≅ ΔFED A D C F What congruence should be used? Given: A pair of congruent angles. (!A ! !F) Therefore ASA and SAS are the only possibilities. ( Given: A piece of a side, can you get the entire side? AD ≅ CF Given: Another Angle. (!EDA ! !BCF) Therefore ASA and is the only possibility. Find the third angle of the triangle. Brainstorm your statements. Write the proof. ) Side Angle E B Given: ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF Pr ove: ΔABC ≅ ΔFED Angle Statements 1. ∠A ≅ ∠F, AD ≅ CF, ∠EDA ≅ ∠BCF 2. AC ≅ DF 3. ∠ADC and ∠DCF are straight ∠s 4. ∠EDA and ∠EDF are sup plementary ∠BCF and ∠BCA are sup plementary 5. ∠BCA ≅ ∠EDF 6. ΔABC ≅ ΔFED A D C F Reasons 1. Given 2. segment addition property 3. assumed from diagram 4. If the sum of 2 ∠s is a straight ∠, then they are sup plementary 5. Supp. of ≅ ∠s are ≅ 6. ASA (1,2,5) Order is important!!