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4/17/2012 Congruent triangles What does congruent mean? Congruent Triangles What are the short cuts for proving triangles congruent? What are the short cuts for proving right triangles congruent? Proofs In a proof what are some things that could be given to us to tell us that segments are congruent? In a proof what are some things that could be given to us to tell us that angles are congruent? What can a figure tell us? If segment SR and NP bisect each other are the triangles congruent? S P O N R If the triangles are congruent what is your reason? If Segment SN is parallel to PR and O is the midpoint of segment SR, are the triangles congruent? S If <P = <S and S0 = PO are the triangles congruent? S P O P O N N R If the triangles are congruent, what is the reason? R If the triangles are congruent, what is the reason? 1 4/17/2012 X is the midpoint of AC, And BX is perpendicular to AC, are the triangles congruent? If triangle ABC is Isosceles with AB = BC and x is the midpoint are the triangles congruent? B B C X A A C X If the triangles are congruent, what is the reason? IF the triangles are congruent, what is the reason? If <A = <C and segment BX bisects < ABC , are the triangles congruent? If AD = CB, and AB = CD are the triangles congruent? A B B D X A If the triangles are congruent, what is the reason? If ABCD is a rectangle are the two triangles congruent? A C C If they are congruent what is the reason? If AB is parallel to CD are the two triangles congruent? A B B D D C C If the triangles are congruent, what is the reason? If the triangles are congruent what is the reason? 2 4/17/2012 If AC and BD bisect each other what triangles are congruent? A If ABCD is a rectangle, is triangle ACD congruent to triangle BDC? B A B E E D D C C If the triangles are congruent what is the reason? What are the reasons for the triangles to be congruent Proving Properties If ABCD is a square, are the four small triangles congruent? A B E D C • Last unit, we learned that if a quadrilateral has diagonals that bisect each other, it’s a parallelogram. • Let’s prove it: • Given: Quadrilateral ABCD with diagonals that intersect at E; AC = BD; E is the midpoint of AC and BD If they are congruent what is the reason? Draw a picture and mark the given information A B Proofs Given Isosceles triangle ABC and BX bisects angle ABC is triangle ABX congruent to CBX? E B D C A X C Sides/Angles Congruent short cut ______ How do you know ? 3 4/17/2012 Given the following figure with SN parallel to PR SR bisects PN. Show: ΔSNO ≅ ΔRPO. S Given Isosceles trapezoid ABCD is triangle ABC congruent to triangle DCB? P short cut _________ short cut _____ O N R Sides/Angle congruent How do you know ? Sides/angles congruent How do you know? HW Given parallelogram ABCD with diagonal AC prove that triangle ABC is congruent to triangle CDA ( draw figure) short cut ________ Sides/angles congruent How do you know 1. Given a square ABCD with diagonals AC and BD which intersect at point E. Prove triangle ABE is congruent to triangle CBE. Fold paper in half twice: You should have 4 sections. Copy the problem, Draw a figure , and prove using the same process as the notes. 2. Given Isosceles trapezoid ABCD whose diagonals AC and BD intersect at point E. Prove that triangle AED is congruent to triangle BEC. 4 4/17/2012 3. Given kite ABCD with diagonals AC and BD intersecting at E. AC is the shorter diagonal. Prove that triangle ABE is congruent to triangle CBE. 4. Given Quadrilateral ABCD with diagonals AC and BD intersecting at E. If triangle AED is congruent to triangle CEB, prove that triangle ABC is congruent to triangle CDA. 5