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Transcript
Aim: Are there any shortcuts to prove triangles are congruent? Do Now: In triangle ABC, the measure of angle B is twice the measure of angle A and an exterior angle at vertex C measures 120o. Find the measure of angle A. Aim: SAS – Triangle Congruence Course: Applied Geometry Congruence Is ABCDE the exact same size and shape as STUVW? S A T B E D V W 5 sides 5 angles C U How would you prove that it is? Measure to compare. Measure what? If the 5 sideAim:pairs and 5 angle pairs measure the SAS – Triangle Congruence Course: Applied Geometry same, then the two polygons are exactly the same. Corresponding Parts Corresponding Parts – pairs of segments or angles that are in similar positions in two or more polygons. S A B E AB BC CD DE EA IF CORRESPONDING PARTS W V D ARE CONGUENT T THEN THE C U A S ST POLYGONS ARE TU B T CONGRUENT UV C U V VW Aim: SAS – Triangle Congruence D Geometry Course: Applied WS E W Congruence Definitions & Postulates Two polygons are congruent if and only if 1. corresponding angles are . 2. corresponding sides are . Corresponding parts of congruent polygons are congruent. CPCPC True for all polygons, triangles our focus. Corresponding Parts of Congruent Triangles are Congruent. CPCTC Aim: SAS – Triangle Congruence Course: Applied Geometry Model Problem B Hexagon ABCDEF hexagon STUVWX. A 10 the value of the variables? Find B A AB and ST are corresponding sides x = 10 C 10 120 F C 120 F T 8 D E X T 8 E S D F & X are corresponding ’s x = 1200 S U X X U 2y X W V 2y ED and WV are corresponding sides V 2y = 8 y =W4 Aim: SAS – Triangle Congruence Course: Applied Geometry Corresponding Parts. Is ABC the exact same size and shape as GHI? G C A B I H How would you prove that it is? Measure corresponding sides and angles. What are the corresponding sides? angles? A I AC GI B H AB IH C G BC GH Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side SAS = SAS I. Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included angle of the other triangle. S represents a side of the triangle and A represents an angle. A’ A B C B’ C’ If CA = C'A', A =A', BA = B'A', then ABC = A'B'C' If SAS SAS , Aim: SAS – Triangle Congruence Course: Applied Geometry then the triangles are congruent Model Problem Each pair of triangles has a pair of congruent angles. What pairs of sides must be congruent to satisfy the SAS postulate? C D E A CE and EB; AE and ED C B G H B F A GH and BC ; FG and AB Aim: SAS – Triangle Congruence Course: Applied Geometry Model Problem Each pair of triangles is congruent by SAS. List the given congruent angles and sides for each pair of triangles. E F B D AB DE; BC EF , B E C A E D DE DG; DF DF , F EDF GDF G Aim: SAS – Triangle Congruence Course: Applied Geometry Aim: Are there any shortcuts to prove triangles are congruent? Do Now: Is the given information sufficient to prove congruent triangles? F C A B D E SAS = SAS Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included of Congruence the other triangle. Aim:angle SAS – Triangle Course: Applied Geometry Side-Angle-Side Is the given information sufficient to prove congruent triangles? F C A A B F B C D E E D B D C D A B A Aim: SAS – Triangle Congruence C Course: Applied Geometry Side-Angle-Side Given that C is the midpoint of AD and AD bisects BE, prove that ABC CDA. B D C A E • C is the midpoint of AD means that (S S) CA CD. • BCA DCE because vertical angles are congruent. (A A) • AD bisects BE means that BE is cut in (S S) to congruent segments resulting in BC CE. The two triangles are congruent because of SAS SAS Course: Applied Geometry Aim: SAS – Triangle Congruence Side-Angle-Side In ABC, AC BC and CD bisects ACB. Explain how ACD BCD C A D B Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side In ABC is isosceles. CD is a median. Explain why ADC BDC. C A D B Aim: SAS – Triangle Congruence Course: Applied Geometry Sketch 12 – Shortcut #1 A A’ B B’ C C’ Copied 2 sides and included angle: AB A’B’, BC B’C’, B B’ A’ B’ C’ Measurements ABC showed: Shortcut for proving congruence in Aim: SAS – Triangle Congruence triangles: A’B’C’ SAS SAS Course: Applied Geometry The Product Rule Aim: SAS – Triangle Congruence Course: Applied Geometry