1. 1. We know that a closed figure formed by three intersecting lines is called a triangle(Tri means three).A triangle has three sides, three angles and three vertices. For e.g.-in Triangle ABC, denoted as ABC AB,BC,CA are the three sides, A,B,C are three angles and A,B,C are three vertices. A B C
2. 2. OBJECTIVES IN THIS LESSON 1 INEQUALITIES IN A TRIANGLE. 2 STATE THE CRITERIA FOR THE CONGRUENCE OF TWO TRIANGLES. 3 SOME PROPERTIES OF A TRIANGLE. 4 DEFINE THE CONGRUENCE OF TRIANGLE.
3. 3. DEFINING THE CONGRUENCE OF TRIANGLE:- Let us take ABC and XYZ such that corresponding angles are equal and corresponding sides are equal :- A B C X Y Z CORRESPONDING PARTS A=X B=Y C=Z AB=XY BC=YZ AC=XZ
4. 4. Now we see that sides of ABC coincides with sides of XYZ. A B C X Y Z SO WE GET THAT TWO TRIANGLES ARE CONGRUENT, IF ALL THE SIDES AND ALL THE ANGLES OF ONE TRIANGLE ARE EQUAL TO THE CORRESPONDING SIDES AND ANGLES OF THE OTHER TRIANGLE. Here, ABC XYZ
5. 5. This also means that:- A corresponds to X B corresponds to Y C corresponds to Z For any two congruent triangles the corresponding parts are equal and are termed as:- CPCT Corresponding Parts of Congruent Triangles
6. 6. CRITERIAS FOR CONGRUENCE OF TWO TRIANGLES SAS(side-angle-side) congruence Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of other triangle. ASA(angle-side-angle) congruence Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle. AAS(angle-angle-side) congruence Two triangles are congruent if any two pairs of angle and one pair of corresponding sides are equal. SSS(side-side-side) congruence If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. RHS(right angle-hypotenuse-side) congruence If in two right-angled triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.
7. 7. A B C P Q R S(1) AC = PQ A(2) C = R S(3) BC = QR Now If, Then ABC PQR (by SAS congruence)
8. 8. A B C D E F Now If, A(1) BAC = EDF S(2) AC = DF A(3) ACB = DFE Then ABC DEF (by ASA congruence)
9. 9. A B C P Q R Now If, A(1) BAC = QPR A(2) CBA = RQP S(3) BC = QR Then ABC PQR (by AAS congruence)
10. 10. Now If, S(1) AB = PQ S(2) BC = QR S(3) CA = RP A B C P Q R Then ABC PQR (by SSS congruence)
11. 11. Now If, R(1) ABC = DEF = 90 H(2) AC = DF S(3) BC = EF A B C D E F Then ABC DEF (by RHS congruence)
12. 12. PROPERTIES OF TRIANGLE A B C A Triangle in which two sides are equal in length is called ISOSCELES TRIANGLE. So, ABC is a isosceles triangle with AB = BC.
13. 13. Angles opposite to equal sides of an isosceles triangle are equal. B C A Here, ABC = ACB
14. 14. The sides opposite to equal angles of a triangle are equal. CB A Here, AB = AC
15. 15. Theorem on inequalities in a triangle If two sides of a triangle are unequal, the angle opposite to the longer side is larger ( or greater) 10 8 9 Here, by comparing we will get that- Angle opposite to the longer side(10) is greater(i.e. 90)
16. 16. In any triangle, the side opposite to the longer angle is longer. 10 8 9 Here, by comparing we will get that- Side(i.e. 10) opposite to longer angle (90) is longer.
17. 17. The sum of any two side of a triangle is greater than the third side. 10 8 9 Here by comparing we get- 9+8>10 8+10>9 10+9>8 So, sum of any two sides is greater than the third side.
18. 18. Angles An angle is formed when two lines meet. The size of the angle measures the amount of space between the lines. In the diagram the lines ba and bc are called the arms of the angle, and the point b at which they meet is called the vertex of the angle. An angle is denoted by the symbol .An angle can be named in one of the three ways: a c b . .Amount of space Angle
19. 19. 1. Three letters a b c . . Using three letters, with the centre at the vertex. The angle is now referred to as : abc or cba.
20. 20. 2. A number c b . .1 a Putting a number at the vertex of the angle. The angle is now referred to as 1.
21. 21. 3. A capital letter b . .B a c Putting a capital letter at the vertex of the angle. The angle is now referred to as B.
22. 22. Right angle A quarter of a revolution is called a right angle. Therefore a right angle is 90. Straight angle A half a revolution or two right angles makes a straight angle. A straight angle is 180. Measuring angles We use the symbol to denote a right angle.
23. 23. Acute, Obtuse and reflex Angles Any angle that is less than 90 is called an acute angle. An angle that is greater than 90 but less than 180 is called an obtuse angle. An angle greater than 180 is called a reflex angle.
24. 24. Angles on a straight line Angles on a straight line add up to 180. A + B = 180 . Angles at a point Angles at a point add up to 360. A+ B + C + D + E = 360 A B A B D E C
25. 25. Pairs of lines: Consider the lines L and K : .p L K L intersects K at p written : L K = {p} Intersecting
26. 26. Parallel lines L K L is parallel to K Written: LK Parallel lines never meet and are usually indicated by arrows. Parallel lines always remain the same distance apart.
27. 27. Perpendicular L is perpendicular to K Written: L K The symbol is placed where two lines meet to show that they are perpendicular L K
28. 28. Parallel lines and Angles 1.Vertically opposite angles When two straight lines cross, four angles are formed. The two angles that are opposite each other are called vertically opposite angles. Thus a and b are vertically opposite angles. So also are the angles c and d. From the above diagram: AB C D A+ B = 180 .. Straight angle B + C = 180 ... Straight angle A + C = B + C Now subtract c from both sides A = B
29. 29. 2. Corresponding Angles The diagram below shows a line L and four other parallel lines intersecting it. The line L intersects each of these lines. L All the highlighted angles are in corresponding positions. These angles are known as corresponding angles. If you measure these angles you will find that they are all equal.
30. 30. In the given diagram the line L intersects two parallel lines A and B. The highlighted angles are equal because they are corresponding angles. The angles marked with are also corresponding angles . A B L . . Remember: When a third line intersects two parallel lines the corresponding angles are equal.
31. 31. 3. Alternate angles The diagram shows a line L intersecting two parallel lines A and B. The highlighted angles are between the parallel lines and on alternate sides of the line L. These shaded angles are called alternate angles and are equal in size. Remember the Z shape. A B L
32. 32. Quadrilaterals A quadrilateral is a four sided figure. The four angles of a quadrilateral sum to 360. b a c d a + b + c + d = 360 (This is because a quadrilateral can be divided up into two triangles.) Note: Opposite angles in a cyclic quadrilateral sum to 180. a + c = 180 b + d = 180
33. 33. .. ... . Parallelogra m 1. Opposite sides are parallel 2. Opposite sides are equal 3. Opposite angles are equal 4. Diagonals bisect each other
34. 34. Rhombus 1. Opposite sides are parallel 2. All sides are equal 3. Opposite angles are equal .. ... . 4. Diagonals bisect each other 5. Diagonal intersects at right angles 6. Diagonals bisect opposite angles .. ... .
35. 35. Rectangl e 1. Opposite sides are parallel 2. Opposite sides are equal 3. All angles are right angles 4. Diagonals are equal and bisect each other
36. 36. Square 1. Opposite sides are parallel 2. All sides are equal 3. All angles are right angles 4. Diagonals are equal and bisect each other 5. Diagonals intersect at right angles 6. Diagonals bisect each angle .. .... ..
37. 37. Types of Triangles Equilateral Triangle . . . 3 equal sides 3 equal angles Isosceles Triangle a b 2 sides equal Base angles are equal a = b (base angles are the angles opposite equal sides) Scalene triangle 3 unequal sides 3 unequal angles
38. 38. Congruent triangles Congruent means identical. Two triangles are said to be congruent if they have equal lengths of sides, equal angles, and equal areas. If placed on top of each other they would cover each other exactly. The symbol for congruence is . For two triangles to be congruent (identical), the three sides and three angles of one triangle must be equal to the three sides and three angles of the other triangle. The following are the tests for congruency. a b c x y z abc xyz
39. 39. Case 1 = Three sides of the other triangle Three sides of one triangle SSS Three sides
40. 40. Case 2 Two sides and the included angle of one triangle Two sides and the included angle of one triangle = SAS (side, angle, side)
41. 41. Case 3 One side and two angles of one triangle Corresponding side and two angles of one triangle = ASA (angle, side, angle)
42. 42. Case 4 A right angle, the hypotenuse and the other side of one triangle A right angle, the hypotenuse and the other side of one triangle = RHS (Right angle, hypotenuse, side)
43. 43. Theorem: Vertically opposite angles are equal in measure. Given: To prove : Construction: Proof: Straight angle Straight angle 1=2 Label angle 3 1=2 Intersecting lines L and K, with vertically opposite angles 1 and 2. 1+3=180 2+3=180 Q.E.D. L K 1 2 1+3=3+2 .....Subtract 3 from both sides 3
44. 44. Theorem: The measure of the three angles of a triangle sum to 180. Given: To Prove: 1+2+3=180 Construction: Proof: 1=4 and 2=5 Alternate angles 1+2+3=4+5+3 But 4+5+3=180 Straight angle 1+2+3=180 The triangle abc with 1,2 and 3. 4 5 a b c 1 2 3 Q.E.D. Draw a line through a, Parallel to bc. Label angles 4 and 5.
45. 45. Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Given: A triangle with interior opposite angles 1 and 2 and the exterior angle 3. To prove: 1+ 2= 3 Construction: Label angle 4 Proof: 1+ 2+ 4=180 3+ 4=180 Three angles in a triangle 1+ 2+ 4= 3+ 4 Straight angle 1+ 2= 3 a b c 3 1 2 4 Q.E.D.
46. 46. a b c1 2 Theorem: If to sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. Given: The triangle abc, with ab = ac and base angles 1 and 2. To prove: 1 = 2 Construction: Draw ad, the bisector of bac. Label angles 3 and 4. Proof: ab = ac given 3 = 4 construction ad = ad common SAS 1 = 2 Corresponding angles d 3 4 abd acd Consider abd and acd: Q.E.D.
47. 47. Theorem: Opposite sides and opposite angles of a parallelgram are respectively equal in measure. Given: Parallelogram abcd a b c d To prove: Construction:Join a to c. Label angles 1,2,3 and 4. Proof: 1= 2 and 3= 4 Alternate angles ac = ac common ASA ab = dcand ad = bc Corresponding sides And abc = adc Corresponding angles Similarly, bad = bcd 1 23 4 ab = dc , ad = bc abc = adc, bad = bcd Consider abc and adc : abc adc Q.E.D.
48. 48. Theorem:A diagonal bisects the area of a parallelogram. a b c d Given: Parallelogram abcd with diagonal [ac]. To prove: Area of abc = area of adc. Proof: ab = dc Opposite sides ad = bc Opposite sides ac = ac Common SSS Consider abc and adc: abc adc area abc = area adc Q.E.D.
49. 49. Theorem: The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference, standing on the same arc. Given: Circle, centre o, containing points a, b and c. To prove: boc = 2 bac Construction: Join a to o and continue to d. Label angles 1,2,3,4 and 5. Proof: d a b c . o 1= 2 + 3 Exterior angle But 2 = 3 1 = 2 2 Similarly, 5 = 2 4 1+ 5 = 2 2 + 2 4 1 + 5 = 2(2 + 4) i.e. boc = 2 bac 1 2 3 4 5 Consider aob: Q.E.D. Base angles in an isosceles
50. 50. Deduction 1: All angles at the circumference on the same arc are equal in measure. To prove: bac = bdc Proof: 3 = 2 1 Angle at the centre is twice the angle on the circumference (both on the arc bc) 3 = 2 2 Angle at the centre is twice the angle on the circumference (both on arc bc) 2 1 = 2 2 1 = 2 i.e. bac = bdc Q.E.D. a b c d 3 1 2 .o
51. 51. Deduction 2: An angle subtended by a diameter at the circumference is a right angle. To prove: bac = 90 Proof: 2 = 2 1 Angle at the centre is twice the angle on the circumference (both on the arc bc) straight line. But 2 = 180 2 1 = 180 1 = 90 i.e. bac = 90 Q.E.D. a b c o 2 1 .
52. 52. Deduction 3: The sum of the opposite angles of a cyclic quadrilateral is 180. To prove: bad + bcd = 180 3 = 2 1Proof: Angle at the centre is twice the angle on the circumference. (both on minor arc bd) 4 = 2 2 Angle at the centre is twice the angle on the circumference. (Both on the major arc bd) 3 + 4 = 2 1 + 2 2 But 3 + 4 = 360 Angles at a point 2 1 + 2 2 = 360 1 + 2 = 180 i.e. bad + bcd = 180 Q.E.D. .o4 a b c d3 2 1
53. 53. a b c Ld . Theorem: A line through the centre of a circle perpendicular to a chord bisects the chord. Given: Circle, centre c, a line L containing c, chord [ab], such that L ab and L ab = d. To prove: ad = bd Construction:Label right angles 1 and 2. Proof: 1 = 2 = 90 Given ca = cb Both radii cd = cd common R H S Corresponding sides Consider cda and cdb: cda cdb ad = bd 1 2 Q.E.D.
54. 54. b a c d e f 2 1 2 3 1 3 Theorem: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Given : Two triangles with equal angles. To prove: |df| |ac| = |de| |ab| |ef| |bc| = Construction: On ab mark off ax equal in length to de. On ac mark off ay equal to df and label the angles 4 and 5. Proof: 1 = 4 [xy] is parallel to [bc] |ay| |ac| = |ax| |ab| As xy is parallel to bc. |df| |ac| = |de| |ab| Similarly |ef| |bc| = x y4 5 Q.E.D.
55. 55. Theorem: In a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the other two sides. Q.E.D. c b a c b a b a c b a c 1 2 3 45 To prove that angle 1 is 90 Proof: 3+ 4+ 5 = 180 Angles in a triangle But 5 = 90 => 3+ 4 = 90 => 3+ 2 = 90 Since 2 = 4 Now 1+ 2+ 3 = 180 Straight line => 1 = 180 - ( 3+ 2 ) => 1 = 180 - ( 90 ) Since 3+ 2 already proved to be 90 => 1 = 90
56. 56. SUMMARY 1.Two figures are congruent, if they are of the same shape and size. 2.If two sides and the included angle of one triangle is equal to the two sides and the included angle then the two triangles are congruent(by SAS). 3.If two angles and the included side of one triangle are equal to the two angles and the included side of other triangle then the two triangles are congruent( by ASA). 4.If two angles and the one side of one triangle is equal to the two angles and the corresponding side of other triangle then the two triangles are congruent(by AAS). 5.If three sides of a triangle is equal to the three sides of other triangle then the two triangles are congruent(by SSS). 6.If in two right-angled triangle, hypotenuse one side of the triangle are equal to the hypotenuse and one side of the other triangle then the two triangle are congruent.(by RHS) 7.Angles opposite to equal sides of a triangle are equal. 8.Sides opposite to equal angles of a triangle are equal. 9.Each angle of equilateral triangle are 60 10.In a triangle, angles opposite to the longer side is larger 11.In a triangle, side opposite to the larger angle is longer. 12.Sum of any two sides of triangle is greater than the third side.