Page 1 Triangles Page 2 Triangles 30 0 30 0 65 0 65 0 30 0 30 0 65 0 65 0 6cm 8cm 30 0 30 0 65 0 65 0 6cm 6cm 6cm 4.5cm 3cm 8cm 6cm 4cm 8cm 6cm 4cm 8cm 6cm 4cm 6cm 4.5cm 30 0 30 0 8cm 6cm 8cm 6cm 30 0 30 0 8cm 6cm Angle - Angle (AA) Only two pairs of corresponding angles of two triangles is enough to prove similarity, but the same is not enough in the case of congruency. Angle - Side - Angle (ASA)/(AAS/SAA) Along with the 2 equal angles, a single side gives us the surety for congruency (equal side) of triangles & same is true for similarity (equal ratio) as well. Not applicable Side - Side - Side (SSS) When all the 3 corresponding sides of two triangles are in equal ratio, the triangles are similar and when all the sides are equal (i.e.ratio = 1), triangles are congruent. Side - Angle - Side (SAS) Two triangles can also be similar when any two corresponding sides are in equal ratio & an angle between them is equal whereas they will be congruent when two corresponding sides as well as an angle between them is equal. SIMILARITY Likeness CRITERIA CONGRUENCY Equal X TRIANGLES 15 Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles. Criteria for Similarity & Congruency of Triangles Page 3 Triangles 30 0 30 0 65 0 65 0 30 0 30 0 65 0 65 0 6cm 8cm 30 0 30 0 65 0 65 0 6cm 6cm 6cm 4.5cm 3cm 8cm 6cm 4cm 8cm 6cm 4cm 8cm 6cm 4cm 6cm 4.5cm 30 0 30 0 8cm 6cm 8cm 6cm 30 0 30 0 8cm 6cm Angle - Angle (AA) Only two pairs of corresponding angles of two triangles is enough to prove similarity, but the same is not enough in the case of congruency. Angle - Side - Angle (ASA)/(AAS/SAA) Along with the 2 equal angles, a single side gives us the surety for congruency (equal side) of triangles & same is true for similarity (equal ratio) as well. Not applicable Side - Side - Side (SSS) When all the 3 corresponding sides of two triangles are in equal ratio, the triangles are similar and when all the sides are equal (i.e.ratio = 1), triangles are congruent. Side - Angle - Side (SAS) Two triangles can also be similar when any two corresponding sides are in equal ratio & an angle between them is equal whereas they will be congruent when two corresponding sides as well as an angle between them is equal. SIMILARITY Likeness CRITERIA CONGRUENCY Equal X TRIANGLES 15 Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles. Criteria for Similarity & Congruency of Triangles Equilateral Triangle Isosceles Triangle Isosceles Right Angled Triangle Scalene Right Angled Triangle Scalene Triangle Right Angled Triangle A B C 60° 60° 60° A B C x x A B C A B C A B C • Three equal sides • Three equal angles • A triangle with one angle of 90° • Two equal sides • Two equal angles • Right triangle with two equal sides • Right triangle with no equal sides • All sides with di?erent lengths • All the angles with di?erent measurements A B C a b c A B C h A B C Here are some of the ways to calculate area of a triangle Area of a triangle (A) S = (Heron’s Formula) A = Base x Height x 1 2 a + b + c Sum of the angles of a triangle is 180° ?A + ?B + ?C = 180° 2 A = s (s - a) (s - b) (s - c) ? ? Triangles A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on varying side lengths and angle measurement. TRIANGLES 14 Page 4 Triangles 30 0 30 0 65 0 65 0 30 0 30 0 65 0 65 0 6cm 8cm 30 0 30 0 65 0 65 0 6cm 6cm 6cm 4.5cm 3cm 8cm 6cm 4cm 8cm 6cm 4cm 8cm 6cm 4cm 6cm 4.5cm 30 0 30 0 8cm 6cm 8cm 6cm 30 0 30 0 8cm 6cm Angle - Angle (AA) Only two pairs of corresponding angles of two triangles is enough to prove similarity, but the same is not enough in the case of congruency. Angle - Side - Angle (ASA)/(AAS/SAA) Along with the 2 equal angles, a single side gives us the surety for congruency (equal side) of triangles & same is true for similarity (equal ratio) as well. Not applicable Side - Side - Side (SSS) When all the 3 corresponding sides of two triangles are in equal ratio, the triangles are similar and when all the sides are equal (i.e.ratio = 1), triangles are congruent. Side - Angle - Side (SAS) Two triangles can also be similar when any two corresponding sides are in equal ratio & an angle between them is equal whereas they will be congruent when two corresponding sides as well as an angle between them is equal. SIMILARITY Likeness CRITERIA CONGRUENCY Equal X TRIANGLES 15 Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles. Criteria for Similarity & Congruency of Triangles Equilateral Triangle Isosceles Triangle Isosceles Right Angled Triangle Scalene Right Angled Triangle Scalene Triangle Right Angled Triangle A B C 60° 60° 60° A B C x x A B C A B C A B C • Three equal sides • Three equal angles • A triangle with one angle of 90° • Two equal sides • Two equal angles • Right triangle with two equal sides • Right triangle with no equal sides • All sides with di?erent lengths • All the angles with di?erent measurements A B C a b c A B C h A B C Here are some of the ways to calculate area of a triangle Area of a triangle (A) S = (Heron’s Formula) A = Base x Height x 1 2 a + b + c Sum of the angles of a triangle is 180° ?A + ?B + ?C = 180° 2 A = s (s - a) (s - b) (s - c) ? ? Triangles A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on varying side lengths and angle measurement. TRIANGLES 14 Theorem1 : A line parallel to one side of a triangle divides the other two sides in equal proportion Given: DE || BC A B C D < E > AD DB = AE EC AD DB = AE EC AD BD ........ (1) To Prove: Property Proof: 1 st 1 st Property 2 nd Step 2 nd Step 3 rd Step = = ? ( ADE) A ? ( BDE) A ? ( BDE) A [ Triangles with same height ] [ Triangles with same height ] [ Triangles with equal base & height ] AE EC ........ (2) ........ (3) { From........(1),(2) & (3)} = ? ( ADE) A ? ( CDE) A ? ( CDE) A -------------- ------------------- Triangles drawn between two parallel lines with same base have equal areas h b1 b1 b1 b2 b2 b2 Ratio of areas of triangles with equal height is equal to the ratio of their bases Base Height = = Basic Proportionality Theorem TRIANGLES 16 Learn these properties to prove BPT B = D = 90 o Given: ?ABC is a right angled triangle To prove: (Hyp) 2 = (Side1) 2 + (Side2) 2 (AC) 2 = (AB) 2 + (BC) 2 Theorem 2 : For any right angled triangle, square of hypotenuse is equal to the sum of squares of the other two sides. B = D = 90 o Draw a perpendicular line from the right angle to its opposite side. Proof: We’ll justify the original triangle is similar to the 2 newly formed right angled triangles. AB AD AC AB = = BC DB A is common In ?ABC & ?ADB ?ABC ?ADB AB 2 = AD x AC .......(1) AB BD BC DC = = AC BC C is common BC 2 = DC x AC ........(2) Construction: A B C A B C D A B C A B D A B C B C D AB 2 + BC 2 AB 2 + BC 2 = AC 2 On adding equation (1) & (2) we get = AD x AC + DC x AC = AC (AD+DC) = AC x AC = AC 2 Pythagoras Theorem In ?ABC & ?BDC . . . ?ABC ?BDC . . . Page 5 Triangles 30 0 30 0 65 0 65 0 30 0 30 0 65 0 65 0 6cm 8cm 30 0 30 0 65 0 65 0 6cm 6cm 6cm 4.5cm 3cm 8cm 6cm 4cm 8cm 6cm 4cm 8cm 6cm 4cm 6cm 4.5cm 30 0 30 0 8cm 6cm 8cm 6cm 30 0 30 0 8cm 6cm Angle - Angle (AA) Only two pairs of corresponding angles of two triangles is enough to prove similarity, but the same is not enough in the case of congruency. Angle - Side - Angle (ASA)/(AAS/SAA) Along with the 2 equal angles, a single side gives us the surety for congruency (equal side) of triangles & same is true for similarity (equal ratio) as well. Not applicable Side - Side - Side (SSS) When all the 3 corresponding sides of two triangles are in equal ratio, the triangles are similar and when all the sides are equal (i.e.ratio = 1), triangles are congruent. Side - Angle - Side (SAS) Two triangles can also be similar when any two corresponding sides are in equal ratio & an angle between them is equal whereas they will be congruent when two corresponding sides as well as an angle between them is equal. SIMILARITY Likeness CRITERIA CONGRUENCY Equal X TRIANGLES 15 Theorem 3: This chart will help you learn the criteria to determine Similarity and Congruency of two given triangles. Criteria for Similarity & Congruency of Triangles Equilateral Triangle Isosceles Triangle Isosceles Right Angled Triangle Scalene Right Angled Triangle Scalene Triangle Right Angled Triangle A B C 60° 60° 60° A B C x x A B C A B C A B C • Three equal sides • Three equal angles • A triangle with one angle of 90° • Two equal sides • Two equal angles • Right triangle with two equal sides • Right triangle with no equal sides • All sides with di?erent lengths • All the angles with di?erent measurements A B C a b c A B C h A B C Here are some of the ways to calculate area of a triangle Area of a triangle (A) S = (Heron’s Formula) A = Base x Height x 1 2 a + b + c Sum of the angles of a triangle is 180° ?A + ?B + ?C = 180° 2 A = s (s - a) (s - b) (s - c) ? ? Triangles A 2-dimensional ?gure with 3-sides and 3-angles is called a Triangle. This chart will help you learn the di?erent types of triangles based on varying side lengths and angle measurement. TRIANGLES 14 Theorem1 : A line parallel to one side of a triangle divides the other two sides in equal proportion Given: DE || BC A B C D < E > AD DB = AE EC AD DB = AE EC AD BD ........ (1) To Prove: Property Proof: 1 st 1 st Property 2 nd Step 2 nd Step 3 rd Step = = ? ( ADE) A ? ( BDE) A ? ( BDE) A [ Triangles with same height ] [ Triangles with same height ] [ Triangles with equal base & height ] AE EC ........ (2) ........ (3) { From........(1),(2) & (3)} = ? ( ADE) A ? ( CDE) A ? ( CDE) A -------------- ------------------- Triangles drawn between two parallel lines with same base have equal areas h b1 b1 b1 b2 b2 b2 Ratio of areas of triangles with equal height is equal to the ratio of their bases Base Height = = Basic Proportionality Theorem TRIANGLES 16 Learn these properties to prove BPT B = D = 90 o Given: ?ABC is a right angled triangle To prove: (Hyp) 2 = (Side1) 2 + (Side2) 2 (AC) 2 = (AB) 2 + (BC) 2 Theorem 2 : For any right angled triangle, square of hypotenuse is equal to the sum of squares of the other two sides. B = D = 90 o Draw a perpendicular line from the right angle to its opposite side. Proof: We’ll justify the original triangle is similar to the 2 newly formed right angled triangles. AB AD AC AB = = BC DB A is common In ?ABC & ?ADB ?ABC ?ADB AB 2 = AD x AC .......(1) AB BD BC DC = = AC BC C is common BC 2 = DC x AC ........(2) Construction: A B C A B C D A B C A B D A B C B C D AB 2 + BC 2 AB 2 + BC 2 = AC 2 On adding equation (1) & (2) we get = AD x AC + DC x AC = AC (AD+DC) = AC x AC = AC 2 Pythagoras Theorem In ?ABC & ?BDC . . . ?ABC ?BDC . . . TRIANGLES 17 Converse of Basic Proportionality Theorem Converse of Pythagoras Theorem Theorem: If a line divides any two sides of a triangle in a same ratio, then the lines is parallel to the third side. < > A D E B C < > A D E F B C Given :- AD DB AE EC = AD DB AF FC = AD DB AE EC = AF FC AE EC = AC FC AC EC = AC FC AC EC _ = 1 _ 1 To Prove :- DE || BC Proof : Assume that DE is not parallel to BC, then we draw a line DF || BC So, But by equation (1) and (2) But AF = AC - FC and AE = AC - EC } .............(4 ) AC - FC AC - EC FC EC = FC = EC It is only possible when F and E coincide each other which So DF is nothing but DE So DE is Parallel to BC i.e. DE || BC by 3 and 4 {by BPT}.............(1) {Given}.............(2) .............(3) .............(5) In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the Given :- ? ABC such that AB 2 + BC 2 = AC 2 To prove :- ? ABC is right angled at angle B Construction :- Construct a right angle ? PQR, right angle at Q such that PQ = AB and QR = BC Proof :- In ? PQR, Since Q = 90 0 PQ + QR = PR .......... (Pythagoras therom) or AB + BC = PR ........ (Construction).........(1) But AB + BC = AC ......(Given).........(2) From equations (1) and (2) , we get PR = AC i.e. PR = AC ...... (3) In ?PQR and ?ABC, PQ = AB ........ (Construction) QR = BC ......... (Construction) AC = PR ........ (From equation 3) ? ABC ? PQR .......(by SSS criterion for Congruency) B = Q .........................(Corresponding angle of congruent triangle) ~ = But Q = 90 0 (by construction) B = 90 Hence, ? ABC is a right angle triangle, right angled at B P Q R A B C . . . . . . ConstructionRead More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

1 videos|13 docs