Geometry Class 3: Congruence

CONGRUENT GEOMETRICAL FIGURES Abstract. This is the third of a series of classes organized in memory of Late Prof. P. C. Joseph. The presentation is based on his book Logical Foundations of Geometry. 1. Let’s Recollect Figure 1. Segment, Angle, Triangle 1.1. Segments, Angles, Triangles. 1.2. Quadrilaterals. 1.3. Radian Measure of an Angle. Suppose \AOB is an angle that is not straight. Draw a circle with center O and radius 1. Then the length of the arc of the circle in the interior of the angle is defined as the measure of the angle. 2. Introduction When we say that two figures are congruent, we mean that they have the same shape and the same size, as the figures given below: We assert that (1) two segments are congruent if they have the same length, and (2) two angles are congruent if they have the same measure. 12 CLASS 3 Figure 2. Quadrilateral Figure 3. Measure of an AngleCLASS 3 3 Figure 4. Congruent Figures Next, we shall define congruence for figures consisting of line segments and angles only. Definition 2.1. Congruent Figures. Two figures are congruent if there is a one-one correspondence of the parts of the figures such that corresponding parts are congruent. 3. Congruence of Triangles Definition 3.1. Congruent Triangles. Two triangles 4ABC and 4DEF are said to be congruent if there is a one-one correspondence ABC ! DEF such that AB = DE,BC = EF,CA = FD. 1 and \A = \D,\B = \E,\C = \F. The notation 4ABC = 4DEF means that the 1Here AB stands for distAB. Hereafter we may stick to this practice when there is no room for confusion4 CLASS 3 Figure 5. Congruent Segments and Angles Figure 6. Congruent Triangles correspondence ABC ! DEF is a congruence. Hence the equivalence: 4ABC = 4DEF , AB = DE,AC = DF,BC = EF,\A = \D,\B = \E,\C = \F The six equations on the right of the above equivalence are not independent of each other.CLASS 3 5 Axiom 3.1. S.A.S. Congruence Axiom A correspondence between two triangles is given. If two sides and the included angle of one triangle are equal to the corresponding parts of the other, then the correspondence is a congruence. [Axiom 7.1 in NCERT IX] Figure 7. S.A.S. Congruence Axiom Given. ABC ! DEF and AB = DE,AC = DF,\A = \D Conclusion. 4ABC = 4DEF Note.S.A.S stands for ‘side-angle-side.’ Theorem 3.2. A.S.A Congruence Theorem. A correspondence between the vertices of two triangles is given. If two angles and the included side of one triangle are equal to the corresponding parts of the other, the correspondence is a congruence. [Theorem 7.1 in NCERT IX] Given. ABC ! DEF,\A = \D,\B = \E,AB = DE Conclusion. 4ABC = 4DEF Proof [Figure 8] (1) Let P be a point on ray AC such that AP = DF.6 CLASS 3 Figure 8. Theorem 3.2 (2) Consider the correspondence ABP ! DEF. (3) Since AB = DE,\A = \D,AP = DF, by S.A.S Axiom 3.1 4ABP = 4DEF (4) \ABP = \E [Corresponding angles] (5) Since \E = \B, \ABP = \ABC (6) If AP < AC [Figure 8(a)], then ray BP divides internally \ABC, and hence \ABP < \ABC, a contradiction of (5). (7) If AP > AC [Figure 8(c)], then ray BC divides internally \ABP, and hence \ABP > \ABC, again a contradiction of (5). (8) By steps (6) and (7), AP = AC ) P = C (9) The conclusion follows from (3) and (8). Before concluding this section, we state without proof, a few more theorems. Theorem 3.3. A.A.S Congruence Theorem. A correspondence between the vertices of two triangles is given. If two angles and any side of one triangle are equal to the corresponding parts of the other, the correspondence is a congruence. Theorem 3.4. Angles opposite to equal sides of an isosceles triangle are equal. [Theorem 7.2 in NCERT IX] Theorem 3.5. Converse of Theorem 3.4 The sides opposite to equal angles of a triangle are equal.CLASS 3 7 [Theorem 7.3 in NCERT IX] Theorem 3.6. S.S.S. Congruence Theorem. A correspondence between two triangles is given. If the corresponding sides are equal, the correspondence is a congruence. [Theorem 7.4 in NCERT IX] Theorem 3.7. R.H.S Congruence Theorem. A correspondence between two right triangles is given. If the hypotenuse and one side of a triangle are equal to the hypotenuse and the corresponding side of the other, the correspondence is a congruence. [Theorem 7.5 in NCERT IX] 4. Congruent figures -a general definition Congruent figures can be given a general definition as follows: Two figures are congruent if (1) there is a one-one correspondence of the figures regarded as sets of points, and (2) the distance between any two points of the first figure is equal to the distance between the corresponding points of the second figure. If two figures are congruent, corresponding sub-figures are congruent. In particular we choose segments and angles as sub-figures. Correspondence is a fundamental concept in mathematics. Congruence is correspondence with equality of corresponding parts. If a congruence of two figures is given or established, any sub-congruence (as equality of segments, angles) can be written without any further examination of the figures. 5. Point for Discussion How can we define congruence of two continuous curves in the complex plane?