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Reasons

# Reasons - Euclidean Geometry(Theorems in abbreviation for...

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Euclidean Geometry (Theorems in abbreviation for reference, Middle 1 to Middle 3) Theorem 1. x y A O B D If AOB is a straight line, then x + y = 180 ° (adj. s on st. line) [ by p ] Theorem 2, 3 (converse of Theorem 1) x y A O B D Corollary x y z w If x + y = 180 ° , then AOB is a straight line (adj. s supp.) [ [ ] u [ ] ( s at a pt.) [ ] Theorem 4. x y B O C D A If two straight lines AOB, COD meet at O then x = y (vert. opp. s) [ ] Theorem 5, 6, 7 c a b d C A B D If AB // CD then (1) a = b (corr. s, AB//CD) [ , AB//CD ] (2) c = b (alt. s, AB//CD) [ , AB//CD ] (3) c + d = 180 ° (int. s, AB//CD) [ , AB//CD ]

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Theorem 8, 9, 10 (converse of Theorem 5, 6, 7) a b c d B C D A (1) If a = b then AB//CD (corr. s equal) [xë ] (2) If c = b then AB//CD (alt. s equal) [x ] (3) If c + d = 180 ° then AB//CD (int. s supp.) [xë p ] Theorem 11. A B C D F E If AB//CD and AB//EF then CD//EF (// to the same st. line) [ˆë p ] Theo rem 12, 13 a c d A B C D b In ABC (1) a + b + c = 180 ° ( sum of ) [ ] (2) d = a + b (ext. of ) [ ] Theorem 14,15,16,17, 18. (Test for Congruent s) A B C P Q R a b c p q r In ABC, PQR (1) If AB = PQ, b = q , BC = QR then ABC 2245 PQR (S.A.S) (2) If b = c , BC = QR, c = r then ABC 2245 PQR (A.S.A.) (3) If a = p , b = q , BC =QR then ABC 2245 PQR (A.A.S.) (4) If AB = PQ, BC = QR, CA = RP then ABC 2245∆ PQR (S.S.S.) (5) If B = Q = 90 ° , AC = PR, BC = QR then ABC 2245 PQR
A B C P Q R (R.H.S.) Theorem 19, 20,21 (Tests for Similar Triangles) a p b c q r A B C P Q R In ABC, PQR (1) If a = p and b = q and c = r then ABC PQR (A.A.A) (2) If a = p and AB PQ AC PR = then ABC PQR (ratio of 2 sides, inc. ) [ ] (3) If AB PQ AC PR BC QR = = then ABC PQR (3 sides proportional) p ] Theorem 22, 23. (1) The sum of the interior angles of a convex polygon with n sides is (n-2)x180 ° ( sum of polygon) p ] (2) If the sides of a convex polygon are produced in order, the sum of the exterior angles so formed is 360 ° (sum of ext. s of polygon) [ p ] Theorem 24 A B C ABC is isosceles such that AB = AC then B = C (base s, isos. ) [ ] Theorem 25 (converse of Theorem 24) A B C If B = C then AC = AB (sides opp. Equal s) ]

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Theorem 26 A B C If AB = BC = CA then A = B = C = 60 ° (Property of equilateral ) [ ] Theorem 27, 28,29, 30.
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