CONGRUENCE OF TRIANGLES

Transcription

1 Finish Line & eyond ONGRUENE OF TRINGLES 1. Two figures are congruent, if they are of the same shape and of the same size. 2. Two circles of the same radii are congruent. 3. Two squares of the same sides are congruent. 4. If two triangles and PQR are congruent under the correspondence P, -Q and -R, then symbolically, it is expressed as Δ Δ PQR. 5. SS ongruence Rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. (xiom: This result cannot be proved with the help of previously known results.) 6. S ongruence Rule: If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent (S ongruence Rule). onstruction: Two triangles are given as follows, where EF FE. Sides =E To Prove: EF and Proof: EF (given) = E = F (Sides opposite to corresponding angles are in the same ratio as ratio of angles) Hence, by SS congruence rule EF is proved.

2 Finish Line & eyond E F 7. S ongruence Rule: If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent. This theorem can be proved in similar way as the previous one. 8. ngles opposite to equal sides of a triangle are equal. 9. Sides opposite to equal angles of a triangle are equal. 10. Each angle of an equilateral triangle is of SSS ongruence Rule: If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent. 12. RHS ongruence Rule: If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS ongruence Rule). 13. In a triangle, angle opposite to the longer side is larger (greater). 14. In a triangle, side opposite to the larger (greater) angle is longer. 15. Sum of any two sides of a triangle is greater than the third side. Theorem: ngles opposite to equal sides of an isosceles triangle are equal. Theorem: The sides opposite to equal angles of a triangle are equal. Theorem: If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater). Theorem: In any triangle, the side opposite to the larger (greater) angle is longer. Theorem: The sum of any two sides of a triangle is greater than the third side.

3 Finish Line & eyond EXERISE 1 1. In quadrilateral, = and bisects. Show that Δ Δ. nswer: In & = ( is bisecting ) = (common side in both triangles) So, by SS axiom it is proved that; 2. is a quadrilateral in which = and =. Prove that (i) Δ Δ (ii) = (iii) =. nswer: (i) In & = = (common) So, by SS rule (ii) Since,, so = (third corresponding sides of respective triangles). (iii) In congruent triangles all corresponding angles re always equal, so is proved 3. and are equal perpendiculars to a line segment. Show that bisects. nswer: In O & O = (given) O O (Right ngle) O O (Opposite angles of intersecting lines So, by S rule O O O O and it is proved that bisects. O

4 Finish Line & eyond 4. l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that Δ Δ. nswer: In & = (l and m are parallel) = ( and are parallel) m (ngles on the same side of transversal ) m (lternate ngles are equal) So, So, by SS rule p q l m 5. Line l is the bisector of an angle and is any point on l. P and Q are perpendiculars from to the arms of. Show that: (i) Δ P Δ Q (ii) P = Q or is equidistant from the arms of. Q P l P & Q nswer: In = (ommon side) P Q ( is bisector of QP Q P (Right ngle) P Q So, by S rule nd Q=P ) E 6. In the given figure, = E, = and = E. Show that = E. nswer: In & E = (given) =E (given) Since, E So, E Or, E So, by SS rule E E proved E 7. is a line segment and P is its mid-point. and E are points on the same side of such that = E and EP = P P

5 Finish Line & eyond Show that (i) Δ P Δ EP (ii) = E nswer: In P EP E (given) EP P (given) So, EP EP P EP Or, P EP P=P (Since P is mid point) So, by S rule P EP So, =E M 8. In right triangle, right angled at, M is the mid-point of hypotenuse. is joined to M and produced to a point such that M = M. Point is joined to point. Show that: (i) Δ M Δ M (ii) is a right angle. (iii) Δ Δ (iv) M = 2 1 nswer: In M & M M=M (M is midpoint) M=M (given) M M (opposite angles) So, M M Hence, = So, (alternate angles are equal) So, = Right ngle) (internal angles are complementary in ase of transversal of parallel lines) & = (proved earlier) = (ommon side) (proved earlier) So, So, = So, M=M=M=M

6 Finish Line & eyond So, M= 2 1 EXERISE 2 1. In an isosceles triangle, with =, the bisectors of and intersect each other at O. Join to O. Show that : (i) O = O (ii) O bisects nswer: In O O O (they are half of angles & ) So, O=O ( Sides opposite to equal angles) O In O & O = (given) O=O (proved earlier) O O (they are half of angles & ) So, O O (SS Rule) So, O O It means that O bisects 2. In Δ, is the perpendicular bisector of. Show that Δ is an isosceles triangle in which =. nswer: In & = (common side) = (given) (right angle) So, So, =, which proves that is isosceles 3. is an isosceles triangle in which altitudes E and F are drawn to equal sides and respectively. Show that these altitudes are equal. F E nswer: In E & F = (given) E F (common to both triangles) F E (right angles)

7 Finish Line & eyond So, E F ( S Rule ) So, E=F 4. is a triangle in which altitudes E and F to sides and are equal. Show that (i) Δ E Δ F (ii) =, i.e., is an isosceles triangle. nswer: This can be solved like previous question. 5. and are two isosceles triangles on the same base. Show that =. nswer: So, 6. Δ is an isosceles triangle in which =. Side is produced to such that =. Show that is a right angle. nswer: In & = = In Δ, 180

8 (1) Similarly in Δ, (2) s is a straight line, so 180 So, adding equations (1)&(2) we get ( ) 2( ) is a right angled triangle in which = 90 and =. Find and. Finish Line & eyond nswer: If = then angles opposite to these sides will be equal. s you know the sum of all angles of a triangle is equal to 180, So, + + =180 Or, =180 Or, + = =90 Or, = =90 8. Show that the angles of an equilateral triangle are 60 each. nswer: s angles opposite to equal sides of a triangle are always equal. So, in case of equilateral triangle all angles will be equal. So they will measure one third of 180, which is equal to 60 EXERISE 3 1. Δ and Δ are two isosceles triangles on the same base and vertices and are on the same side of. If is extended to intersect at P, show that (i) Δ Δ (ii) Δ P Δ P (iii) P bisects as well as. (iv) P is the perpendicular bisector of. nswer: In & = = = So, Δ Δ (SSS Rule) P In P & P = P=P

9 Finish Line & eyond P P (ngle opposite to equal sides) So, Δ P Δ P (SS Rule) Since Δ P Δ P So, P P So, P is bisecting Similarly P & P can be proved to be ongruent and as a result it can be proved that P is bisecting P 2. is an altitude of an isosceles triangle in which =. Show that (i) bisects (ii) bisects. M Q N R nswer: This can be solved like previous question. 3. Two sides and and median M of one triangle are respectively equal to sides PQ and QR and median PN of Δ PQR. Show that: (i) Δ M Δ PQN (ii) Δ Δ PQR M & PQN nswer: In =PQ M=PN M=QN (median bisects the base) So, M PQN & PQR In =PQ =QR =PR (Equal medians means third side will be equal) PQR So, 4. E and F are two equal altitudes of a triangle. Using RHS congruence rule, prove that the triangle is isosceles. F E nswer: In E & F E=F (Perpendicular) = (Hypotenuse) So, E F 5. is an isosceles triangle with =. raw to show that =. nswer: fter drawing In & =

10 Finish Line & eyond = So, So, EXERISE 4 1. Show that in a right angled triangle, the hypotenuse is the longest side. nswer: In a right angled triangle, the angle opposite To the hypotenuse is 90, while other two angles are lways less than 90. s you know that the side opposite to the largest angle is always the largest in a triangle. 2. In the given triangle sides and of Δ are extended to points P and Q respectively. lso, P < Q. Show that >. P Q nswer: 180 P 180 O Since P O So, s you know side opposite to the larger angle is larger than the side opposite to the smaller angle. Hence, > O 3. In the given figure < and <. Show that <. nswer: O<O (Side opposite to smaller angle) O<O (Side opposite to smaller angle) So, O+O<O+O Or, < 4. and are respectively the smallest and longest sides of a quadrilateral. Show that > and >. nswer: Let us draw two diagonals and as shown in the figure. In Δ Sides << So, (1) ngle opposite to smaller side is smaller In Δ Sides << So, (2)

11 Finish Line & eyond dding equation (1) & (2) Similarly in (3) In (4) dding equations (3) & (4) 5. In following figure, PR > PQ and PS bisects QPR. Prove that PSR > PSQ. 3 P 4 5 Q 1 2 S R nswer: For convenience let us name these angles as follows: PQR 1 PRQ 2 QPR 3 QPS 4 RPS 5 PSQ 6 PSR 7 Since, PR>PQ, so 1 2 In PQS

12 Finish Line & eyond In PRS In both these triangles So, for making the sum total equal to 180 the following will always be true: 6 7