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DIGITAL TEXT Digital Text Page 1 CONGRUENT TRIANGALE This chapter focuses on students ability to equalize different measurement of two figures with same shape and it leads to the congruency of figures and thereby the formation of the general principles and geometrical constructions. The main aim of this chapter is to develop the concepts and aptitude which enable know about congruency the students should acquire to know about congruency. Digital Text Page 2 Look at these pictures Figure (1) Figure(2) Notice the difference In figure (1) stack of coins are placed on over the other. In figure (2) books are placed one over the other but they are not fit exactly. Each one has different sizes of note books so they will not fit exactly. In geometry, figures such as triangles, rectangles, circles and so on, which can be placed one over the other so as to fit exactly are said to be congruent figures . Digital Text Page 3 TRIANGLE MATCH Take a look at these triangles A B C P Q To specify the equal sides and angles, let’s name the triangles as ∆PQR Equal sides R ∆ ABC and Equal angles AB = PQ <ACB = <QRP BC = PR < BAC = < PQR AC = QR < ABC = <QPR From the table, we see that the pair of sides and the pair of angles opposite to these sides are equal . If two triangles are congruent, then the sides and angles of one are equal to the sides and angles of the other , angles opposite to equal sides are equal and sides opposite to equal angles are equal . Assignment Cut out a pair of eerkkil bits , each 3 c.m long and another pair , each 2 c.m long . How many different kinds of rectangles can you make with these ? Digital Text Page 4 WHEN SIDES ARE EQUAL 4CM 5CM 5CM 6CM 4CM 6CM Make a copy of one of these on a piece of tracing paper and place it over the others . These triangles are all congruent . Here we see If the three sides of a triangle are equal to the three sides of another triangle , then three triangles are congruent. Assignment (1) Check the two triangles given below are congruent and why ? 6CM 8CM 7CM 6CM 7CM 8CM EVEN IF ANGLES ARE EQUAL We can draw triangles with the same three angles in different size , can’t we ? 600 300 □ Digital Text 600 300 Page 5 If the sides of two triangles are equal , their angles are also equal , but simply because angles are equal , the sides need not be equal . Look at this picture Assignment (1) Check the two triangles given below are congruent .why? 1000 500 1000 Digital Text 300 500 300 Page 6 TWO SIDES AND AN ANGLE We can draw a triangle ,if two side and the angle included between them are specified We can draw in different ways 5CM 600 3CM 3CM 3CM 600 600 5CM 5CM Make a copy of one of these on a piece of tracing paper and place it over the others . These triangles are congruent . Here we see If two sides of a triangle and their included angle are equal to two sides of another triangle and their included angle , then these triangles are congruent . Look at this pictures 300 300 4cm 4cm 5cm 5cm These are not congruent. Just because two sides and some angles of a triangle are equal to two sides and some angles of another triangle , the two triangles need not be congruent. Digital Text Page 7 Assignment (1) check the two triangle given below are congruent? 700 5CM 5CM 3CM 700 (2) Check the two triangles given below are congruent and why ? 2CM 920 920 4CM 2CM 4CM ONE SIDE AND TWO ANGLES We can draw a triangle , if one of the sides and the two angles on it are specified Some such triangles are shown below 4cm 450 300 300 450 4cm From these we see If one side and the two angles on it of a triangles are equal tone side and the angles on it of another triangle then these triangles are congruent . Digital Text Page 8 Assignment (1) Check the two the triangles given below are congruent. Why ? 4cm 300 4CM 300 450 450 (2) Check the two triangles given below are congruent .why? 300 600 5CM Digital Text 300 600 6CM Page 9 RIGHT ANGLED TRIANGLES P Q A R B C In the right angled triangles ABC ,PQR shown below PR = AB and PQ = BC Let’s take a look at the third sides . Is there any relation between the sides QR and AC ? AB is the hypotenuse of the right angled triangle ABC . So by Pythagoras Theorem AC2 = AB2 –BC2 Similarly , since PR is the hypotenuse of the right angled triangle PQR we get QR2 = PR2 –PQ2 = AB2 – BC2 = AC2 From this we find QR = AC Then the three sides of triangle PQR are equal to the sides of triangle ABC .So ∆ ABC = ∆ PQR General result we have here If the hypotenuse and one other side of a right angled triangle are equal to the hypotenuse and one other side of another right angled triangle then these two triangles are congruent. Digital Text Page 10 Assignment (1) Check the two triangles are congruent . Why? 3C .M Digital Text 5 C.M 5CM 3C.M Page 11