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1 Triangles Chapter Flowchart The Chapter Flowcharts give you the gist of the chapter flow in a single glance. Triangle A plane figure bounded by three line segments is called a triangle. Types of Triangles Scalene triangle : A triangle in which all the sides are of different lengths. Isosceles triangle : A triangle whose two sides are equal. Equilateral triangle: A triangle having all sides equal. Right-angled triangle : A triangle with one angle as a right angle. Angle sum property of a triangle The sum of the three angles of a triangle is Exterior Angle property If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. Some Properties of a Triangle Angles opposite to equal sides of a triangle are equal. Sides opposite to equal angles of a triangle are equal. Each angle of an equilateral triangle is of In an isosceles triangle altitude from the vertex bisects the base. Conversely, if the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles. A point equidistant from two given points lies on the perpendicular bisector of the line segment joining the two points. A point equidistant from two intersecting lines lies on the bisectors of the angles formed by the two lines. Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 1 of 26

2 Advanced NTSE Olympiad JEE - Mains & Congruent Figures Two figures are congruent, if they are of the same shape and of the same size. If two triangles ABC and PQR are congruent under the correspondence A P, B Q and C R, the symbolically, it is expressed as ABC PQR. Examples: Two squares of the same sides are congruent. Two circles of the same radii are congruent. Criteria for Congruence SAS congruence criterion: If two sides and included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. ASA congruence criterion. Two triangles are congruent if two angles and the induced side of one triangle are equal to two angles and the included side of the other triangle AAS congruence criterion: Two triangles are congruent if two angles and one side of one triangle are equal of two angles and the corresponding side of other triangle. SSS congruence criterion: Two triangles are congruent if three sides of one triangle are equal to three sides of other triangle. RHS congruence criterion: Two right-angled triangles are congruent if hypotenuse and one side of a triangle are equal to hypotenuse and one side of other triangle Inequalities in a Triangle In a triangle, angle opposite to the longer side is larger. In a triangle, side opposite to the larger angle is longer. Sum of any two sides of a triangle is greater that the third side. Of all the line segments that can be drawn to a given line, from a point, not lying on it, the perpendicular line segment is the shortest. Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 2 of 26

3 Revision Question Bank 1. In the given figure 1, find the values of x and y. 2. In the given figure 2, PQ > PR, QS and RS are the bisectors of Q and R, respectively. Then, prove that SQ > SR 3. In the given figure3, BA AC,DE DF such that BA=DE and BF = EC. Show that ABC DEF 4. In figure 5, AB = DE, BC=EF and median AP = median DQ. Prove that B = E. 5. Show that a median of a triangle divides it into two triangles of equal areas. 6. In the given figure 6, XYZ and PYZ are two isosceles triangles on the base YZ with XY = XZ and PY= PZ. If P=120 and XYP=40, then find YXZ. 7. In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. 8. In the given figure 8, RS = QT and QS=RT. Prove that PQ =PR. 9. ABC is a triangle in which B = 2 C. D is a point on side BC such that AD bisects BAC and AB = CD. Prove that BAC= In the given figure 9, if E > A, C > D, then AD > EC. Is it true? Answers 1. X = 50 0, y = True Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 3 of 26

4 Previous Years Question Bank 1 1. In the figure 1, of ABC, AE is the bisector of BAC and AD BC. Show that DAE C B 2 [CBSE Schools ] Fig. 1 Fig l and m are two parallel lines intersected by another pair of parallel lines p and q as shown in the figure 2. Show that ABC CDA. [CBSE Schools ] 3. ABCD is a square and ABE is an equilateral triangle outside the square. Prove that ACE = 1 ABE. 2 [CBSE Schools ] 4. In a triangle PQR, QO and RO are the bisector of PQR and PRQ, such that RQO = QRO and PQO = PRO. Show that PQR = PRQ. [CBSE Schools ] 5. ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that triangle is isosceles. [CBSE Schools ] 6. In the given figure 6, AB = CD and BC = AD. Prove that ADC CBA and D B. [CBSE Schools ] Fig. 6 Fig Two equal pillars AB and CD are standing on either side of the roas as shown in the figure 7. If AF = CE, then prove that BE = FD. [CBSE Schools ] 8. Prove that angles opposite to equal sides of an isosceles triangle are equal. [CBSE Schools ] 9. In the given figure 9, AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Show that A > C and B > D. [CBSE Schools ] Fig. 9 Fig. 10 Fig. 11 Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 4 of 26

5 10. In the given figure 10, X is a point in the interior of square ABCD. AXYZ is also a square. DY = 3 cm and AZ = 2 cm, then find BY. [CBSE Schools ] 11. In the given figure 11, 1 3 and Show that: A C [CBSE Schools ] 12. AB is a line segment and line l is its perpendicular bisector. If a point P lies on l. show that P is equidistant from A and B. [CBSE Schools ] 13. In the given figure 13, ABCD is a square and EF is parallel to diagonal BD. If EM = FM, prove that: [CBSE Schools ] Fig. 13 Fig. 15 Fig. 16 (i) DF = BE. (ii) AM bisects BAD. 14. Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle. 15. In Fig 15, AC = BC, DCA = ECB and DBC = EAC. Prove that triangles DBC and EAC are congruent, and hence DC = EC and BD = AE. 16. In the given figure 16, if 1 2 and 3 = 4, then prove that BC = CD. 17. If figure 17, D and E are points on base BC of ABC such that BD = CE, AD = AE and ADE = AED prove that ABE ACD Fig. 17 Fig. 18 Fig Shyam lal has two triangular plots connected to each other as shown in figure. He thought to give the bigger triangular part to his daughter and son equally. What value he is exhibiting by doing so? How a triangle can be divided into two parts of equal area? Also find the value of WZX. (see fig. 18) 19. ABC is an isosceles triangle with AC = BC. Side AC is produced to D so that AC = CD. Prove that ABD is a right angle. (see fig. 19) Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 5 of 26

6 20. In figure 20, AO and DO are the bisectors of A and D respectively of the quadrilateral ABCD. Prove that AOD = 1 ( B C) 2 Fig. 20 Fig In the figure 21, two sides AB and BC and median AM of ABC are respectively equal to sides DE and DF and the median DN of DEF. Prove that ABC DEF. 22. ABC and DBC are isosceles triangles on the same base BC. Show that ABD = ACD. (see fig. 22) Fig. 22 Fig In the given figure 23, B < A and C < D show that AD < BC. 24. In the given figure 24, ABC = BAC = 65 0 and ADC = Write the inequality relation between AD and AC. Fig. 24 Fig In the given fig. 25, 1= 2 and 3= 4, then prove that BC = CD. [CBSE Schools ] 26. Prove that any two, sides of a triangle are together greater than twice the median drawn to the third side. [CBSE Schools ] 27. ABC and MBC are two isosceles triangles on the same base BC and vertices A and M are on the same side of BC. If M is extended to intersect BC at L, show that (see fig. 27) [CBSE Schools ] Fig. 27 Fig. 28 (i) ABM ACM (ii) ABL ACL (iii) AM bisects A as well as M. Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 6 of 26

7 28. In the given fig. 28, CB AB, ED AD. IF AB = ED and AC=EC. Prove that [CBSE Schools ] (i) ABC EDC (ii) BC = CD 29. Prove that apposite to equal sides of an isosceles triangle are equal. [CBSE Schools ] 30. In the given fig. 30, BP AC, DQ AC such that BP = DQ and CP = AQ. Show that APB CQD. [CBSE Schools ] Fig. 30 Fig In figure 31, two sides AB and BC and median AM of ABC are respectively equal to sides DE and DF and the median DN of DEF. Prove that ABC DEF. [CBSE Schools ] 32. In fig. 32, AC>AB and D point on AC such that AB=AD. Prove that CD<BC. [CBSE Schools ] Fig. 32 Fig LMN is straight in which altitudes MP and NQ to sides LN and LM respectively are equal. (see fig. 33) Show that (i) LMP LNQ (ii) LM = LN ie LMN is an isosceles triangle. [CBSE Schools ] 34. Triangle RST and VSU are shown below, which, [CBSE Schools ] R= V, and RT = VU. Which additional condition is sufficient to prove that RST VSU? Also prove that RST VSU. Triangular regions shown above are the part of land given by an industrialist for a hospital and a school. What value is he exhibiting by doing so? (see fig. 34) 35. Prove that, Two triangles are congruent, if two angles and included side of one triangle are equal to two angles and the include side other triangle. [CBSE Schools ] 36. In fig. 36, ACB is a right angle and AC = CD. And CDEF is a parallelogram. If ECF=10 0 then calculate BDE. [CBSE Schools ] Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 7 of 26

8 37. ABC is an isosceles triangle with AB = AC. Side BA is produced to D such that AB = AD. Prove that BCD is fight angle. [CBSE Schools ] 38. In the given fig. 38, D is a point on side BC of ABC such that AD = AC. Show that AB > AD. [CBSE Schools ] 39. In the given fig. 39, two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of PQR. Show that: [CBSE Schools ] (i) ABM PQN (ii) ABC PQR Fig. 39 Fig In the given fig. 40, AB CD, ECD = 24 0, EDC = 42 0 and AC = CE. Find x, y and z. [CBSE Schools ] For Solutions: /latestupdates Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 8 of 26

9 Maximum Marks: 30 Chapter Test Maximum Time: 1 hr. 1. BE and CF are two equal altitudes of a ABC. Using RHS congruence rule, prove that ABC is an isosceles triangle. [2] 2. In the figure, ABCD is a quadrilateral in which AD = BC and DAB CBA. Prove that [2] (i) ABD BAC (ii) BD = AC 3. In figure, AB = AC, D is a point in the interior of ABC such that DBC = DCB. Prove that AD bisects BAC of ABC. [3] 4. In ABC, if AD is the bisector of A, then show that AB > BD and AC > DC. [3] 5. In the given figure, PQR is an equilateral triangle and QRST is a square. Prove that (i) PT = PS (ii) PSR = 15 [3] 6. In figure, S is any point on the side QR of PQR. Prove that PQ + QR + RP > 2PS. [4] 7. In the figure, ABCD is a quadrilateral in which AD = BC and DAB = CBA. [3] Prove that (i) ABD BAC (ii) BD = AC Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 9 of 26

10 8. In a ABC, AB = AC and AD is bisector of BAC. Anju has to prove that: ABD ACD, which as follows: Here, AB = AC [given] BAD = CAD [given] and B = C [corresponding angles of a equal side are equal) ABD ACD [by ASA congruence rule] Further Anju shows this proof to his classmate Amita and she find that there is some errors in the proof. (i) Write the correct proof. (ii) What is the mistake in Anju's proof? (iii) Which value is depicted from this action? [4] 9. A plot is in the form of AEF. Owner of this plot wants to build old age home, health centre and dispensary for elderly people as shown in figure. In which ABCD is a parallelogram and AB = BE and AD = DF. (i) Prove that BEC DCF. (ii) What values are depicted here? [4] 10. ABCD is a square. P and Q are points on DC and BC respectively, such that AP = DQ, prove that (i) ADP DCQ (ii) DMP = 90 [4] 11. In figure, S is any point on the side QR of PQR. Prove that PQ + QR+ RP>2PS. [4] Answers 4. (ii) Anju has used the result B = C for providing ABD ACD, which is wrong. Because for providing B = C, firstly we prove that ABD ACD. (iii) The value depicted from this action, it is cooperative and learning among students without any religion bias. 8. (ii) Kindness and respectful for elder people For Solutions: /latestupdates Pioneer Education SCO 320, Sector 40 D, Chandigarh , Page 10 of 26

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