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# Congruent Triangles

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• 1. Image Source: http://resources3.news.com.au
• 2. Identical Twins have the exact same size and shape, and we instantly recognise that the two people are are exactly the same. When two items have the exact same size and shape, we say that they are "Congruent". This Presentation is all about "Congruent Triangles", eg. Groups of Triangles which have the exact same size and shape.
• 3. In the Real World Congruent Triangles are used in Construction when we need to reinforce structures so that they are strong and stable, and do not bend or buckle in strong winds or when under load.
• 4. 7cm 9cm Identical Triangles have all three Sides, and all three Angles exactly the same sizes. 5cm 60o 85o 35o 5cm 9cm 60o 7cm 85o 35o If we gave several people three sticks: 5cm, 7cm and 9cm long, they would each only be able to make the exact same Triangle.
• 5. 7cm 9cm Congruent Triangles do not have to be in the same orientation or position. They only have to be identical in size and shape. 5cm 60o 85o 35o 5cm9cm 60o 7cm 85o35o The Green Triangle on the right, is an upside down version of the Pink Triangle. They are both the same size and shape.
• 6. Identical Twins have over 1000 body parts which are exactly the same. HOWEVER, we only have to check a few key items to tell that they are "Congruent". For "Congruent Triangles”, it turns out that we do not actually have to check that all three Angles and all three sides to prove they are Identical. There are “Shortcut” Rules which we can use.
• 7. 7cm 9cm 5cm 5cm 9cm 7cm Any Triangle with sides of 5cm, 7cm and 9cm long, can only be one specific shape. In that shape the angles will be specific values. Two triangles are congruent if: All three Sides of the triangles are equal in length.
• 8. Triangles can have markings on their sides to indicate that pairs of sides are the exact same length. The “SSS” rule can be applied. Two triangles are congruent if: All three Sides of the triangles are equal in length.
• 9. Triangles are labelled with Letters and special symbols used. A B C D E F ABC DEF (by SSS Rule) AB = DE and BC = EF and AC = DF
• 10. We need to be careful with labelling when orientation is different. A B C D E F ABC = DEF (by SSS Rule) AB = DE and BC = EF and AC = DF ~
• 11. 7cm 5cm 5cm 7cm If we label the Sides as “S”, and the Angle as “A”; then the pattern as we trace around the Triangle is “Side Angle Side” or “SAS”. Two triangles are congruent if: Two matching sides have equal lengths, and the angle in between these two sides is the same. 65o 65o
• 12. 5cm 5cm If we label the Side as “S”, and the Angles as “A”; then the pattern as we trace around the Triangle is “Angle Angle Side” or “AAS”. Two triangles are congruent if: Two matching angles are equal, and one matching side is the same length in both triangles. 65o 65o40o 40o
• 13. 7cm 7cm There was an old rule called “ASA” Angle Side Angle, but this is now part of the “AAS” Rule. (Two Angles and One Side). Two triangles are congruent if: The order of the AAS Rule is not important, as long as two angles are equal & one side the same. 65o 65o40o 40o
• 14. 8mm 8mm This Rule works because of Pythagoras for 90 Degree Triangles means that the missing side lengths are equal, so really “SSS”. Two Right Angled Triangles are congruent if: The Hypotenuse and one matching side are equal in length. They also have a 90 degree angle equal. 10mm 10mm
• 15. Two sides and the included angle equal. (SAS) Two angles and any matching side equal . (AAS) Right angle, Hypotenuse and a Side (RHS) Three sides equal. (SSS)
• 16. A B C 10cm 3 cm9 cm X Z Y 10cm 3 cm9 cm AB = XZ (Corresponding Equal Sides) BC = ZY (Corresponding Equal Sides) AC = XY (Corresponding Equal Sides) ABC XYZ (by SSS Rule) By identifying matching items, prove the Triangles are Congruent
• 17. A B C 9 AB = PQ (Corresponding Equal Sides) B = Q (Corresponding Equal Angles) BC = QR (Corresponding Equal Sides) ABC PQR (by SAS Rule) By identifying matching items, prove the Triangles are Congruent 5 85o P Q R 9 5 85o
• 18. A B C 12mm 65o 20o P Q R 12mm 65o 20o A = P (Corresponding Equal Angles) C = R (Corresponding Equal Angles) AC = PR (Corresponding Equal Sides) ABC PQR (by AAS Rule) By identifying matching items, prove the Triangles are Congruent
• 19. A B CD M AB is parallel to DC and M is the midpoint of DB. Show that the segment AM = CM by proving Congruent Triangles ABM = CDM (Alternate Interior angles) BAM = DCM (Alternate Interior angles) Triangles ABM CDM are congruent ( by AAS) AM = CM The order of the lettering is important when naming congruent triangles. It would be wrong in this example to say that triangles ABM and DCM are congruent.
• 20. P Q RS In the diagram, PQRS is a Parallelogram with opposite sides parallel. Prove that PQ = RS and that PS = RQ PRS = RPQ (Alternate Interior angles) PRQ = RPS (Alternate Interior angles) Triangles PQR and RSP are congruent (AAS) PQ = RS and PS = RQ Triangles PQR and RSP both contain side PR