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To Prove that AG is the Angle Bisector of CAB To Prove that CD bisects AB at M.

Congruent Triangles Remember: Two shapes are congruent if all sides and angles in one, are equal to the sides and angles in the other. Congruent triangles are of particular importance in mathematics because they enable us to determine/prove many geometrical properties/theorems. Euclid of Alexandria In his book “The Elements” Euclid proved four theorems concerning the conditions under which triangles are guaranteed to be congruent. He used some of these theorems to help establish proofs of other important theorems such as the Theorem of Pythagoras and the bisection of a chord. O The Windmill

Conditions for Congruency of Triangles Two sides and the included angle equal. (SAS) Two angles and a corresponding side equal. (AAS) Right angle, hypotenuse and side (RHS) Three sides equal. (SSS)

35 o 120 o 8 cm 10 cm 4 cm 8 cm 10 cm 8 cm 120 o 4 cm 25 o 35 o 120 o 25 o 4 cm 35 o 120 o 10 cm 8 cm 4 cm 10 cm 120 o 8 cm 35 o Not to Scale! Decide which of the triangles are congruent to the yellow triangle, giving reasons. SSS SAS AAS RHS    SAS  SSS  AAS  SAS

5 cm 12 cm 13 cm 20 o 13 cm 20 o 70 o 13 cm 12 cm 13 cm 12 cm 5 cm 13 cm 5 cm 13 cm 70 o 13 cm 70 o 20 o Decide which of the triangles are congruent to the yellow triangle, giving reasons. SSS SAS AAS RHS Not to Scale!  RHS  SSS   RHS  AAS  SAS

Proving Relationships A B C D M In the diagram AB is parallel to DC and M is the midpoint of DB. Prove that AM = MC Angle ABM = angle CDM (Alternate angles) Angle BAM = angle DCM (Alternate angles) Triangles ABM and CDM are congruent (AAS)  AM = MC The order of the lettering is important when naming congruent triangles. Corresponding sides are identified by ordered letter pairs. AB  CD AM  CM BM  DM It would be wrong in this example to say that triangles ABM and DCM are congruent.

Proving Relationships P Q R S In the diagram, PQRS is a quadrilateral with opposite sides parallel. Prove that PQ = SR and that PS = QR Angle PRS = angle RPQ (Alternate angles) Angle PRQ = angle RPS (Alternate angles) Triangles PQR and RSP are congruent (AAS)  PQ = SR and PS = QR

Proving Relationships O S T U In the diagram, TP is a line perpendicular to the chord SU that passes through the centre of the circle at O. Prove that the chord SU is bisected by line TP. P In triangles OST and OUT, OS = OU (radii of the same circle) Also, OT is common to both triangles Angle OTS = angle OTU (angles on a straight line) Triangles OST and OUT are congruent (RHS)  ST = TU

Perp Bisect Draw the locus of the point that remains equidistant from points A and B. A B 1. Join both points with a straight line. 2. Place compass at A, set over halfway and draw two arcs. 3. Place compass at B, with same distance set and draw two arcs to intersect first two. 4. Draw the perpendicular bisector through the points of intersection. Congruent triangles can be used to prove results from some of our earlier work on loci. An example of this would be proving the construction of a line bisector.

Perp Bisec Proof To prove that CD bisects AB at M. Arcs lie on the circumference of circles of equal radii. AC = AD = BC = BD (radii of the same circle). Triangles ACD and BCD are congruent with CD common to both (SSS). A B C D M So Angle ACD = BCD Triangles CAM and CBM are congruent (SAS) Therefore AM = BM QED

Angle Bisect A B C Draw the locus of the point that remains equidistant from lines AC and AB. 1. Place compass at A and draw an arc crossing both arms. 2. Place compass on each intersection and set at a fixed distance. Then draw two arcs that intersect. 3. Draw straight line from A through point of intersection for angle bisector. Congruent triangles can be used to prove results from some of our earlier work on loci. Another example of this would be proving the construction of an angular bisector.

Ang Bisect Proof To prove that AG is the Angle Bisector of CAB A B C D E F G AD = AE (radii of the same circle) DG = EG (both equal to radius of circle DE) Triangle ADG is congruent to AEG (AG common to both) SSS So angle EAG = DAG Therefore AG is the angle bisector of CAB QED

Worksheets 35 o 120 o 8 cm 10 cm 4 cm 8 cm 10 cm 8 cm 120 o 4 cm 25 o 35 o 120 o 25 o 4 cm 35 o 120 o 10 cm 8 cm 4 cm 10 cm 120 o 8 cm 35 o Not to Scale! Decide which of the triangles are congruent to the yellow triangle, giving reasons. SSS SAS AAS RHS

5 cm 12 cm 13 cm 20 o 13 cm 20 o 70 o 13 cm 12 cm 13 cm 12 cm 5 cm 13 cm 5 cm 13 cm 70 o 13 cm 70 o 20 o Decide which of the triangles are congruent to the yellow triangle, giving reasons. SSS SAS AAS RHS Not to Scale!

Proving Relationships

To Prove that CD bisects AB at M. A B C D M

To Prove that AG is the Angle Bisector of CAB A B C D E F G