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1 The six elements of a triangle are its three angles and the threesides. The line segment joining a vertex of a triangle to the mid point of itsopposite side is called a median of the triangle. A triangle has3 medians. The perpendicular line segment from a vertex of a triangle to itsopposite side is called an altitude of the triangle. A triangle has3 altitudes. An exterior angle of a triangle is formed, when a side of a triangle isproduced. The measure of any exterior angle of a triangle is equal to the sum ofthe measures of its two interior opposite angles.

2 The sum of the three angles of a triangle is 180 . A triangle is said to be equilateral, if each of its sides has the samelength. In an equilateral triangle, each angle has measure 60 . A triangle is said to be isosceles if at least two of its sides are of samelength. The sum of the lengths of any two sides of a triangle is always greaterthan the length of the third side. The difference of the lengths of any two sides of a triangle is alwayssmaller than the length of the third In a right-angled triangle, the side opposite to the right angle iscalled the hypotenuse and the other two sides are called its legs orarms.

3 In a right-angled triangle, the square of the hypotenuse is equal tothe sum of the squares on its legs. Two plane figures, say, F1 and F2 are said to be congruent, if thetrace-copy of F1 fits exactly on that of F2. We write this as F1 F2. Two line segments, say ABandCD, are congruent, if they have equallengths. We write this as ABCD. However, it is common to write itas =ABCD. Two angles, say ABC and PQR, are congruent, if their measuresare equal. We write this as ABC PQR or as m ABC = m PQR orsimply as ABC = PQR.

4 Under a given correspondence, two triangles are congruent, if thethree sides of the one are equal to the three sides of the other (SSS). Under a given correspondence, two triangles are congruent if twosides and the angle included between them in one of the trianglesare equal to the two sides and the angle included between them ofthe other triangle (SAS). Under a given correspondence, two triangles are congruent if twoangles and the side included between them in one of the trianglesare equal to the two angles and the side included between them ofthe other triangle (ASA).

5 Under a given correspondence, two right-angled triangles arecongruent if the hypotenuse and a leg (side) of one of the trianglesare equal to the hypotenuse and one of the leg (side) of the othertriangle (RHS). In Examples 1 to 5, there are four options, out of which only one iscorrect. Write the correct 1:In Fig. , side QR of a PQR has been produced to thepoint PRS = 115 and P = 45 ,then Q is equal to,(a) 70 (b)105 (c)51 (d)80 15-04-2018 Solution:Correct answer is (a).Example 2:In an equilateral triangle ABC (Fig.)

6 , AD is an 4AD2 is equal to(a)2BD2(b)BC2(c)3AB2(d)2DC2 Solution:Correct answer is (c).Example 3:Which of the following cannot be the sides of a triangle?(a)3cm, 4cm, 5cm(b)2cm, 4cm, 6cm(c) , , (d) , , :Correct answer is (b).Fig. word equilateral contains the roots equi,which means equal, and lateral, whichmeans of the side. What do you supposean equilateral is? Greek prefix poly means many, andthe root gon means angle. What do yousuppose a polygon is?15-04-2018 Example 4:Which one of the following is not a criterion forcongruence of two triangles?

7 (a)ASA(b)SSA(c)SAS(d) SSSS olution:Correct answer is (b).Example 5:In Fig. , PS is the bisector of P and PQ = PR. Then PRS and PQS are congruent by the criterion(a)AAA(b)SAS(c)ASA (d) both (b) and (c)Fig. : Correct answer is (b).In examples 6 to 9, fill in the blanks to make the statements 6:The line segment joining a vertex of a triangle to themid-point of its opposite side is called its :medianExample 7:A triangle is said to be _____, if each one of its sideshas the same :equilateralExample 8:In Fig. , PRS = QPR + _____Fig.

8 :PQR15-04-2018 Example 9:Let ABC and DEF be two triangles in which AB = DE,BC = FD and CA = EF. The two triangles are congruentunder the correspondenceABC _____Solution:EDFIn Examples 10 to 12, state whether the statements are True or 10:Sum of any two sides of a triangle is not less than thethird :FalseExample 11:The measure of any exterior angle of a triangle is equalto the sum of the measures of its two interior :TrueExample 12:If in ABC and DEF, AB = DE, A = D and BC = EFthen the two triangle ABC and DEF are congruent bySAS :False Example 13In Fig.

9 , find x and : Understand and Explore the Problem What all are given? ABD = 60 , BAD = 30 and ACD = 45 What are to be found? ADC and XAC, which are respectively exterior anglesfor ABD and Plan a Strategy Find ADC using exterior angle property for ABD. Find y using exterior angle property for ABC. Solve x = ADC = DBA + BAD (In ABD) = 60 + 30 = 90 y = XAC = ABC + ACB ( In ABC) = 60 + 45 = 105 Revise Verify your answer by using some other properties of ABD, ADB = 180 (30 +60 ) = 90 (Angle sum propertyof a triangle) x= ADC = 180 ADB= 180 90 = 90 , Hence, ADC = 90 verified.

10 DAC = 180 (x + 45 ) = 180 135 = 45 At point A on BAX , 30 + DAC + y = 180 Hence for verifying value of y, 30 + 45 + y = 180 or y = 180 75 = 105 AD = DC? Why? given problem, can B be 85 instead of 60 ? If yes find the values ofx and y in that type of triangle is ADC?15-04-2018 In each of the questions 1 to 49, four options are given, out of whichonly one is correct. Choose the correct sides of a triangle have lengths (in cm) 10, and a, where a isa whole number. The minimum value that a can take is(a)6(b)5(c)3(d) DEF of Fig.