Aim: SAS – Triangle Congruence Course: Applied Geometry Do Now: Aim: Are there any shortcuts to prove triangles are congruent? In triangle ABC, the measure of angle B is twice the measure of angle A and an exterior angle at vertex C measures 120 o. Find the measure of angle A.
Aim: SAS – Triangle Congruence Course: Applied Geometry Is ABCDE the exact same size and shape as STUVW ? Congruence C B A E D W S T U V How would you prove that it is? Measure to compare. Measure what? 5 sides 5 angles If the 5 side pairs and 5 angle pairs measure the same, then the two polygons are exactly the same.
Aim: SAS – Triangle Congruence Course: Applied Geometry Corresponding Parts CORRESPONDING PARTS IF AB BC CD DE EA ST TU UV VW WS THEN THE POLYGONS ARE CONGRUENT ARE CONGUENT ABCDEABCDE STUVWSTUVW Corresponding Parts – pairs of segments or angles that are in similar positions in two or more polygons. C B A E D W S T U V
Aim: SAS – Triangle Congruence Course: Applied Geometry 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. True for all polygons, triangles our focus. CPCPC CPCTC Corresponding Parts of Congruent Triangles are Congruent.
Aim: SAS – Triangle Congruence Course: Applied Geometry Model Problem Hexagon ABCDEF hexagon STUVWX. Find the value of the variables? AB and ST are corresponding sides x = 10 x = F & X are corresponding ’s ED and WV are corresponding sides 2y = 8y = 4
Aim: SAS – Triangle Congruence Course: Applied Geometry Corresponding Parts. Is ABC the exact same size and shape as GHI? How would you prove that it is? Measure corresponding sides and angles. What are the corresponding sides? angles?
Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side I. 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 angle of the other triangle. S represents a side of the triangle and A represents an angle. A BB’CC’ A’ If CA = C'A', A = A', BA = B'A', then ABC = A'B'C' If SAS SAS, then the triangles are congruent
Aim: SAS – Triangle Congruence Course: Applied Geometry Model Problem Each pair of triangles has a pair of congruent angles. What pairs of sides must be congruent to satisfy the SAS postulate?
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.
Aim: SAS – Triangle Congruence Course: Applied Geometry Do Now: Aim: Are there any shortcuts to prove triangles are congruent? Is the given information sufficient to prove congruent triangles? 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 angle of the other triangle.
Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side Is the given information sufficient to prove congruent triangles? AB C D
Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side Given that C is the midpoint of AD and AD bisects BE, prove that ABC CDA. A B D E C C is the midpoint of AD means that CA CD. BCA DCE because vertical angles are congruent. AD bisects BE means that BE is cut in to congruent segments resulting in BC CE. The two triangles are congruent because of SAS SAS (S S) (A A) (S S)
Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side In ABC, AC BC and CD bisects ACB. Explain how ACD BCD A D B C
Aim: SAS – Triangle Congruence Course: Applied Geometry Side-Angle-Side In ABC is isosceles. CD is a median. Explain why ADC BDC. A D B C
Aim: SAS – Triangle Congruence Course: Applied Geometry Sketch 12 – Shortcut #1 SAS SAS B A C Copied 2 sides and included angle: AB A’B’, BC B’C’, B B’ Copied 2 sides and included angle: AB A’B’, BC B’C’, B B’ B’ A’ C’B’ A’ C’ Shortcut for proving congruence in triangles: Measurements showed: ABC A’B’C’
Aim: SAS – Triangle Congruence Course: Applied Geometry The Product Rule