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Congruency is used often in proving statements in geometry. These geometric proofs can vary in complexity from simple statements such as an angle being congruent to itself to more advanced concepts. The basic idea behind doing a proof in geometry is to start with a given statement and using various rules of geometry in a logical sequence to prove a concluding statement. Each rule is a link in the chain of logic that connects the given statement to the conclusion. When doing a twocolumn proof, the statement is placed in the left hand column while the rule justifying the statement is placed in the right hand column. Many proofs using congruency are used to show that geometric figures such as triangles, quadrilaterals and polygons are congruent. Proof Involving Three AnglesThe following proof shows that two angles are congruent when each is adjacent and supplementary to one of two given congruent angles the other of which is included as a part. Given that CAB DAE. Prove that DAB CAE.
Proof Involving A Transversal With Three Parallel LinesWhen two parallel lines are cut by a transversal, the corresponding angles as well as the alternate interior and exterior angles are congruent. This can be extended to three parallel lines. Given: j  k and k  l (j, k and l parallel to each other), transversal m Prove: 1 4
Since congruency is transitive, it is a possible to combine steps 6 and 7 into one substitution leaving the final statement as 1 4. Geometry Proofs For Congruent TrianglesWhen doing congruency proofs for triangles the three main congruency postulates are used, namely, the SSS, SAS and ASA postulates. AAS or Angle  Angle  Side can be used indirectly using the sum of angles in a triangle property. SSS PostulateShowing that three sides of one triangle are congruent to another proves that both triangles are congruent. Given: AB CD and E is a midpoint for AD and BC. Prove: ABE CED
Since ABE CED are vertical angles and therefore congruent, it is also possible to use the SAS postulate. SAS PostulateWhen using SAS it is important to remember that the angle is included between two sides of the triangle. For this reason the letters SAS and ASA are written in exact order to indicate the corresponding parts being compared. It is also important to distinguish from SSA which does not prove triangles to be congruent. Given: BC CD and AC BD Prove: ABE ACD
ASA PostulateIf two angles with an included side is congruent to the corresponding two angles and included side of another triangle then the two triangles are congruent. Given: A F, AC DF and EDA BCF, line m. Prove: ABC ACD
AAS TheoremGiven: A F, B D, BC DE Prove: ABC DEF
Midsegment TheoremA midsegment of a triangle is parallel to the side not including the any of the endpoints and is half the length of that side. Given: ABC, midsegment DE Prove: DE  BC, DE = 1/2BC
Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite these sides are congruent. This proof involves a correspondence of ABC ↔ ACB as if there were two separate isosceles triangles. Given: ABC , AB AC Prove: B C

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