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6 Triangles CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle. We denote a triangle by the symbol . In fig. ABC has (i) three vertices namely, A,B and C (ii) three sides namely, AB, BC, CA (iii) three angles namely, A , B and C . Types of Triangles on the basis of sides (i) Equilateral triangle. A triangle whose all the three sides are equal is called equilateral. In the figure ABC is an equilateral triangle in which AB = BC = CA (ii) Isosceles triangle. A triangle having two sides equal is called an isosceles triangle. In the figure, ABC is an isosceles triangle in which AB = AC. (iii) Scalene triangle. A triangle whose sides are of different lengths. In the figure ABC is a triangle in which AB BC CA. Types of Triangles on the Basis of Angles (i) Obtuse-angled triangle. A triangle in which one angle is an obtuse angle, is called an obtuse angled triangle. In figure, ABC is a triangle in which B 900 . (ii) Acute angle triangle a triangle in which all angles are less than 900 in measures is called acute angled triangle. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 85 Triangles (iii) A right angled triangle : A triangle in which one angle is of exact 900 is called right angle triangle. Some Other Important Terms of Triangles (a) Median. A median of a triangle is the line segment joining the mid-point of side with the opposite vertex. (b) Centroid. The point of intersection of all the three medians of a triangle is called its centroid. Characteristics of Centroid (i) Centroid is the point at which the medians of triangle meet. (ii) The medians of a triangle are concurrent. (iii) The centroid divides the medians in the ratio 2 : 1. (iv) The median of an equilateral triangle are equal. (v) The medians of an equilateral triangle coincide with the “altitudes”. (c) Altitudes. The altitude of a triangle is the perpendicular drawn from a vertex to the opposite side. (d) Orthocentre. The point of intersection of all the three altitudes of a triangle is called its orthocenter. Characteristics of Orthocentre (i) Orthocentre is the point at which the altitudes of a triangle meet. (ii) The altitudes of a triangle are concurrent. (iii) Orthocentre of an acute triangle lies in the interior of the triangle. (iv) Orthocentre of an obtuse triangle lies in the exterior of the triangle. Orthocentre of a right triangle lies on the vertex of the right angle. (e) (f) Angle bisectot: the angle bisector of an angle of a triangle is a line that divided the angle in two equal part Incentre of a triangle. The point of intersection of the bisectors the internal angles of a triangle in called its incentre. Characteristics of Incentre (i) The point at which the three angle bisectors of a triangle intersect is called the „incentre‟. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 86 Triangles (ii) The triangle may be acute, obtuse, or right, the angle bisectors of a triangle must meet at a point lying inside the triangle. (iii) The incentre of a triangle lies in the interior of the triangle. (iv) The bisectors of the angles of a triangle are concurrent. (v) From the incentre we can draw a on opposite sides. (vi) We can call this perpendicular as “inradius” (g) (h) Perpendicular bisector: the perpendicular bisector of a triangle is perpendicular drawn from the opposite vertex and divide the opposite side in two equal parts. Circumcentre of a triangle. The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumcentre. Characteristics of Circumcentre (i) The point at which the perpendicular bisectors of the sides of a triangle meet is called the cicumkcentre of the triangle. (ii) The right bisectors of the sides of a triangle are concurrent. (iii) With circumcentre as centre, we can drawn a circle passing through the vertices of a triangle. (iv) The distance from centre to the vertices is called the „circumradius‟. The circle thus drawn with circumentre as centre and circumradius as radius is called „circumcircle‟. (i) Perimeter. The sum of lengths of the sides of a figure is its perimeter. Perimeter of ABC = AB + BC + CA. Asseingment-1 1. Prove that the sum of the angles of a triangle is 1800. 2. In ABC , B 75 , C 32 find A . 3. In the figure, show that 0 0 A B C D E F 3600 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 87 Triangles 4. In the figure, prove that a b c 360 5. The angles of a triangle are in the ratio 3 : 5 : 10. Find the measures of each angle of the triangle. 6. The sum and difference of two angles of a triangle are 128 0 and 220 respectively. Find all the angles of the triangle. 7. If the bisector of the angle B and C of a ABC meet at a point O, then prove that BOC = 0 900 + 8. 1 A . 2 In ABC , B > C , if AM is the bisector of BAC and AN BC, prove that MAN = B C ). 1 ( 2 9. Fill in the blanks (a) The sum of three angles of a triangle is …….. (b) If two angles of a triangle are 510 and 380, the third angle is equal to ………. (c) If the angles of a triangle in the ratio 2 : 2 : 5, then the angles are…….. (d) The angles of a triangle are 3x – 5, 2x + 55 and 5x – 50 degrees then x is equal to………… (e) A triangle cannot have more than……………..right angles. (f) A triangle cannot have more than…………obtuse angle. 10. Which of the following statements are true or false? (a) An exterior angle of a triangle is less than either of its interior opposite angles. (b) Sum of the three angles of a triangle is 1800. (c) A triangle can have two right angles. (d) A triangle can have two acute angles. (e) A triangle can have two obtuse angles. (f) An exterior angle of a triangle is equal to the sum of the two interior opposite angles. Exterior angle of a Triangle: Definition. If the side BC of a triangle ABC is produced to ray BD, then ACD is called an exterior angle of triangle ABC at C, and is denoted by exterior ACD . A and B are called remote interior angles or interior opposite angles. Note : At each vertex there are two exterior angles Theorem . Given A To prove Proof. If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. ABC whose side BC has been produced to D forming exterior angle ACD . A B ACD In ABC ….(1) ….(2) (straight angle) A B ACB 1800 ACB ACD 1800 From (1) and (2), we get ACB ACD A B ACB ACD A B 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 88 Triangles A B ACD IX ACADEMIC QUESTIONS Subjective Assengment-2 1. In the figure, find BED . 2. In figure, prove that x A B C . 3. In the figure, find x and y, if AB || DF and AD || FG. 4. In the figure, prove that DE || BF. 5. In the figure, AB || DG, AC || DE, EDH = 250 and BAC = 200, Find x and y. 6. In the figure, CD AB, ABE = 1300 and BAC = 700. Find x and y. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 89 Triangles 7. If ABD = 1250 and ACE = 1300, then BAC = ……. Congruent Figures The geometric figures are said to be congruent if they are exactly of the same shape and size. (a) Congruent segments. Two segments are congruent if they are of the same length and conversely. Hence in Fig. AB CD A B C D (b) Congruent angles. Two angles are congruent if they have equal measures and conversely. Hence in fig. BAC FDE (c) Congruent circles. Two circles are congruent if they have equal radii and conversely. Hence in fig. If r1 = r2 Then c1 circle c2 circle Rules for Congruent triangles Rules 1. (SAS) When two sides and the included angle are given In ABC and DEF If AB = DE, A D , AC = DF Then ABC DEF . 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 90 Triangles Rule 2 (SSS). When three sides are given In ABC and DEF If AB = DE, BC = EF, CA = FD Then ABC DEF . Rule 3 (ASA). When two angles and the included side is given In ABC and DEF If B E , BC = EF, C F Then ABC DEF . Rule 4 (R.H.S). When Right Angle – Hypotenuse – Side are given In ABC and DEF If B E = 900, BC = EF, CA = FD Then ABC DEF . Congruence Relations of Triangles (i) Reflexive ABC ABC (congruence relation is reflexive) Every triangle is congruent to itself. (ii) Commutative If ABC DEF , then DEF ABC (congruence relation is commutative) (iii) Transitive If ABC DEF , and DEF PAR then ABC PAR (congruence relation is transitive) Note : Since the congruence relation is reflexive, commutative and transitive, it is an equivalence relation. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 91 Triangles IX ACADEMIC QUESTIONS Subjective Asseingment-3 1. Prove that ABC is isosceles if altitude AD bisects BC. 2. Prove that ABC perpendicular to BC. 3. In the fig. it is given that AB = CF, EF = BD and AFE = DBC . Prove that AFE CBD . 4. ABCD is a quadrilateral in which AD = BC and DAB CBA . Prove that : is isosceles if median AD is (i) ABD BAC (ii) BD = AC (iii) ABD BAC 5. In the fig. AC = AE and AB = AD and BAD = EAC . Prove that BC = DE. 6. In fig. l || m and M is the mid point of the line segment AB. Prove that M is also the mid-point of nay line segment CD having its end points on l and m respectively. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 92 Triangles 7. In fig. It is given that BC = CE and 1 2 . Prove that GCB DCE . 8. In fig. AD and BC are perpendicular to the line segment AB and AD = BC. Prove that O is the mid point of line segment AB and DC. 9. In the figure C is the mid point of AB BAD CBE ECA DCB Prove that (i) DAC EBC (ii) DA = EB 10. In the fig. BM and DN are both perpendiculars to the segments AC and BM = DN. Prove that AC bisects BD. 11. In fig., PS = PR, TPS QPR . Prove that PT = PQ. 12. In fig. AD = AE and D and E are points on BC such that BD = EC. Prove that AB = AC. 13. In the fig. AD CD and BC CD. If AQ = BP and DP = CQ. Prove that DAQ CBP . 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 93 Triangles 14. In the fig. AB = AC, ABD ACD Prove that BD = CD. 15. In fig. QPR PQR and M and N are respectively on sides QR and PR of PQR such that QM = PN. Prove that OP = OQ, where O is the point of intersection of PM and QN. 16. In the fig. AB = AC. BE and CF are respectively the bisectors of B and C . Prove that EBC FCB . 17. AD and BE are respectively altitude of ABC such that AE = BD. Prove that Ad = BE. 18. AD, BE and CF, the altitudes of ABC are equal. Prove that ABC is an equilateral triangle. 19. AD is the bisector of A of a triangle ABC, P is any point on AD. Prove that the perpendicular drawn from P on AB and AC are equal. 20. ABCD is a parallelogram, if the two diagonals are equal, find the measure of ABC . 21. Fill in the blanks in the following so that each of the following statements is true. (i) Sides opposite to equal angles of a triangle are…………. (ii) Angle opposite to equal sides of a triangle are………… (iii) In an equilateral triangle all angles are………….. (iv) In a ABC if A = C , then AB = ………. (v) If altitudes CE and BF of a triangle ABC are equal, then AB = ………. (vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …………. CE. (vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ABC …………… 22. Which of the following statements are true and which are false. (i) Sides opposite to equal angles of a triangle are unequal. (ii) Angles opposite to equal sides of a triangle are equal. (iii) The measure of each angle of an equilateral triangle is 600. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 94 Triangles (iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles. (v) The bisectors of two equal angles of a triangle are equal. (vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles. (vii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent. (viii) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle. 23. ABC and DBC are two isosceles triangles on the same base BC. Show that ABD ACD . 24. If ABC is an isosceles triangle with AB = AC. Prove that the perpendicular from the vertices B and C to their opposite sides are equal. 25. If the altitudes from two vertices of a triangle to the opposite sides are equal. Prove that the triangle is isosceles. 26. In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. Inequalities in a Triangle (i) If two sides of a triangle are unequal, the longer side has greater angle opposite to it. (ii) If two angles of a triangle are unequal, the greater angle has the longer side opposite to it. (iii) The sum of any two sides of a triangle is greater than the third side. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 95 Triangles IX ACADEMIC QUESTIONS Subjective Assignment-4 1. 2. 3. Prove that the difference of any two sides of a triangle is less than the third. Show that of all the line segments that can be drawn to a given line from a given point not lying on it, the perpendicular line segment is the shortest. In a right angled triangle, prove that the hypotenuse is the longest side. 4. In the figure, AD is the bisector of A , show that AB > BD. 5. In figure PR > PQ and PS bisects QPR . Prove that PSR PSQ . 6. In the fig. PQ > PR. QS and RS are the bisector of Q and R respectively. Prove that SQ > SR. 7. In the fig. x y , show that M N . 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 96 Triangles XI SCIENCE & DIP. ENTRANCE Subjective Assignment (MCQ) – 5 1. In the figure AB||CD, ABE 128 0 , BED 20 0 , what is the value of EDC ? (a) 128 o (b) 148 o (c) 108 o 2. 3. (d) 130 o In the given figure AB||CD, EFC 30 0 and ECF 100 0 , then BAF is equal to (a) 70 0 (b) 80 0 (c) 100 0 (d) 130 0 In ABC, when BC is produced on both ways, the exterior angles are 102 0 and 134 0 , what is the value of A (a) 36 0 (b) 56 0 (c) 106 0 (d) 1120 4. In the figure calculate the value of „y‟ (a) 45 0 (b) 50 0 0 (c) 60 (d) 90 0 5. In the figure the value of (a) 100 0 (c) 115 0 6. 7. is (b) 109 0 (d) 120 0 The angles of a triangle are 2x 10 , x 20 and triangle it is? (a) Equilateral Triangle (b) Right Angled Triangle (c) Acute Angle Triangle (d) Obtuse Angle Triangle 0 0 x 100 , which type of In the trapezium ABCD, EF||AD, what is the value of ACD ? (a) 70 0 (c) 50 0 8. x0 if x 5 0 (b) 60 0 (d) 40 0 In the figure CE is perpendicular to AB. ACE 20 0 and ABD 50 0 , what is the measure of BDA ? (a) 50 0 (b) 60 0 (c) 70 0 (d) 90 0 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 97 Triangles 9. In the figure what is the relation of x in term of a, b and c ? (a) x a b c (b) x a b c 90 (c) x 180 a b c (d) x 180 a b c 10. PN QR and PM is bisector of P in PQR, then ( Q R ) is equal to (a) MPN 1 (c) MPN 2 (b) 2MPN 1 (d) MPN 3 11. The bisector of exterior angles B and C meet at „O‟, what is BOC (a) 120 0 (b) 80 0 (c) 45 0 (d) 40 0 12. The bisectors of exterior angles of ABC intersect at „O‟ and form a BOC which is always (a) Acute Angle (b) Right Angle (c) Obtuse Angle (d) None of these 13. The bisectors of interior angles of a triangle forms an angle which is always (a) Acute Angle (b) Right Angle (c) Obtuse Angle (d) All of these 14. In a triangle ABC, the internal bisectors of angles B and C meet at „P‟ and the external bisectors of the angles B and C meet at Q, then the BPC BQC is equal to (a) 90 0 (b) 90 1 / 2A (c) 90 1 / 2A (d) 180 0 15. In the figure what is the value of a b c ? (a) 90 0 (b) 180 0 (c) 270 0 (d) 360 0 16. In the figure PM is bisector of P and PN is perpendicular on QR then the value of MPN is (a) 30 0 (b) 40 0 60 0 (d) 100 (c) 0 17. BM and CM are interior bisectors of B and B and C respectively. Which is correct? C while BN and CN are exterior bisectors of (a) BMC 110 0 (b) BNC 70 0 (c) BMC BNC 180 0 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 98 Triangles (d) All are correct 18. In a ABC, AB = 5cm, AC = 5cm and A = 50 o, then B = (a) 35 o (b) 65 o (c) 80 o (d) 40 o 19. If two sides of a triangle are unequal then opposite angle of larger side is (a) greater (b) less (c) equal (d) half 20. The sum of attitudes of a triangle is_______ than the perimeter of the triangle (a) greater (b) less (c) half (d) less 21. In the given figure, PQ = QR, QPR = 48 o, SRP = 18 o , then PQR = (a) 48 o (b) 84 o (c) 30 o (d) 36 o 22. In the given figure, PQR is an equilateral triangle and QRST is a square. Then PSR = (a) 30 o (b) 15 o (c) 90 o (d) 60 o 23. Can we draw a triangle ABC with AB = 3cm, BC = 3.5cm and CA = 6.5cm? (a) Yes (b) No (c) Can‟t be determined (d) None of these 24. Which of the following is not a criterion for congruence of triangles? (a) SSA (B) SAS (c) ASA (d) SSS 25. In the given figure, AB BE and EF BE. Also BC = DE and AB = EF. Then (a) BD = FEC (b) ABD = EFC (c) ABD = CMD (d) ABD = CEF 26. In quadrilateral ABCD, BM and DN are drawn perpendicular to AC such that BM = DN. If BR = 8cm, then BD is 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 99 Triangles (a) 4cm (b) 2cm (c) 12cm (d) 16cm 27. In the given figure PQ > PR, QS and RS are the bisectors of Q and R respectively. (a) SQ = SR (b) SQ > SR (c) SQ < SR (d) None of these 28. In the figure, PS is the median, bisecting angle P, then QPS is_______ (a) 110 o (b) 70 o (c) 45 o (d) 55 o 29. In the given figure x and y are (a) x = 70 o, y = 37 o (b) x = 37o, y = 70o (c) x + y = 117o (d) x – y = 100o 30. In the given figure BD AC, the measure of ABC is (a) 60 o (b) 30 o (c) 45 o 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 (d) 90 o 100 Triangles ANSWER Assignment – 1 2. A = 71 o 5. 30 o, 50 o , 100 o 9. (a) 180 o (b) 91 o (c) 40 o, 40 o, 100 o (d) 18 o (e) one (f) one 10. (a) False (b) True (c) False (d) True (e) False (f) True 6. 52 o, 53 o , 75 o Assignment – 2 1., BED = 82 o 3. x = 60 o, y = 55 o 5. x = 115 o , y = 20 o 6. x = 40 o, y = 20 o 7. 75 o Assignment – 3 20. 90 21. (i) equal (ii) equal (vi) Equal to (iii) 60 (iv) BC (v) AC (ii) True (iii) True (iv) True (v) True (vii) True (viii) True (vii) EFD 22. (i) False (vi) True Assignment – 5 1.c 2.d 3.b 4.b 5.d 6.b 7.a 8.b 9.c 10.d 11.c 12.a 13.c 14.d 15.d 16.a 17.d 18.b 19.a 20.b 21.b 22.b 23.b 24.a 25.a 26.d 27.b 28.c 29.b 30.d 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 101 Triangles