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Ready to begin! DIRECTIONS: • Review the material by clicking pictures and buttons • Do not click on the next object until you have fully read what you needed to, because when you click on an image, you can only read what is presented one time!!! • If you accidentally click the new image, you will have to start over Ready to begin! TO REVIEW, SELECT FROM THE FOLLOWING OPTIONS: SAS Congruent figures AAS HL DONE! Try some examples! WHICH FIGURES ARE CONGRUENT? Yes! Since we have parallel lines, <F = <J and <G = <K by alternate interior angles. Then <FHG = <KHJ because they’re vertical angles. So now we know all pairs of corresponding parts are congruent!!! Yes! <LNM = < PNQ because they’re vertical angles. Then <M = <P by the third angles theorem. So now we know Yes! Because of the third all pairs theorem of corresponding parts angles <JHG = <JHI. congruent!!! Also, JHare = JH by reflexive. So now we know all pairs of corresponding parts are congruent!!! Go back to main menu • • • • We know <Q = <T because it was given We know <QSR = <TSV because they’re vertical angles We know <R = <V by the third angle theorem But, we have no information about the sides of the triangles, and there is no theorem or postulate proving triangles are congruent without knowing something about the sides. Go back Example: Try some examples! WHICH TRIANGLES ARE CONGRUENT BY SSS? YES! All three pairs of corresponding sides are congruent! YES! All three pairs of corresponding sides are congruent because KO = KO! YES! All three pairs of corresponding sides are congruent because YQ = YQ! Go back to main menu • • • • We know that ZW = ZX because it was given We know that <W = <X because it was given We know ZY = ZY by the reflexive property But, we know nothing about the third sides (WY and XY), so we can’t use SSS here Go back Example: Try some examples! *remember that EF = EF WHICH TRIANGLES ARE CONGRUENT BY SAS? Yes! <WXZ = <XZY because they’re alternate interior angles and XZ = XZ by reflexive. Yes! We have two pairs of congruent corresponding sides, and the included angle is marked right. All Yes! <PTF = <STG because right angles are congruent they’re vertical by Theorem 2.4 angles. Go back to main menu • We know that <ONR = <NRE because it was given • We know NR = NR by the reflexive property • But, we know nothing about any of the other sides, and we need to know that NO = RE to use SAS because the sides need to include the angle Go back Example: Try some examples! WHICH TRIANGLES ARE CONGRUENT BY ASA? Yes! IJ = IJ by reflexive. Yes!Yes! <JLK <MLPtwo pairs We=have because ofthey’re congruent vertical angles. angles, corresponding and their included sides are also marked congruent. Go back to main menu Go back • We know that <JML is a right angle • Since our book allows us to assume that all things that appear to be lines, are lines, we also know <JMK is right since <JMK and <JML are supplementary (they form a linear pair) • We know JM = JM by the reflexive property • But, we know nothing about any of the other angles, and we need to know that <MJL = <JMK to use ASA because the angles need to include the congruent side Example: Try some examples! WHICH TRIANGLES ARE CONGRUENT BY AAS? Yes! GF = GF by reflexive. Yes! <LMO = <NMP because they’re vertical angles. Yes! <HED = <GEF because they’re vertical angles. Go back to main menu • We know that <ABC = <CDE because all right angles are congruent (Theorem 2.4) • We know <BCA = <ECD because they are vertical angles • But, we know nothing about any of the sides, and we need to know that either AC = CE or that AB = DE to use AAS Go back Example: Try some examples! WHICH TRIANGLES ARE CONGRUENT BY HL? Yes! PR = PR by reflexive. Yes! BD = BD by reflexive. Yes! We have two right triangles, GJ = JI, and GH = IH, which is all we need of HL. Go back to main menu • In the left triangle, we have a leg marked as 7 units, and a hypotenuse marked as 25 units • In the right triangle, the sides marked 7 and 25 units are both legs • We can’t use HL unless there are two hypotenuses marked congruent, and a pairs of corresponding legs marked congruent Go back Go back to main menu • First of all, it spells a bad word, and your parents would be disappointed in your teachers • Second, if you had two triangles with a pair of congruent angles not included by two pairs of congruent sides, this would not be enough information to prove the triangles are congruent (believe me, many intelligent mathematicians have tried!) • It has actually been proven that ASS will never work on any triangle