Chapter 6 : Triangles
Q. Fill in the blanks using the correct word given in brackets :
(i) All circles are _______. (congruent, similar)
(ii) All squares are ________. (similar, congruent)
(iii) All _______ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are _______ and (b) their corresponding sides are ______. (equal, proportional)
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Q. Give two different examples of pair of:
(i) similar figures. (ii) non-similar figures.
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Q. State whether the above quadrilaterals are similar or not:

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Q. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
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Q. The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO. Show that ABCD is a trapezium.
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Q. In Fig., DE||AC and DF||AE. Prove that BFFE=BEEC.

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Q. In Fig., (i) and (ii), DE||BC. Find EC in (i) and AD in (ii).

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Q. In Fig., if LM||CB and LN||CD, prove that AMAB=ANAD.

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Q. In given figure, A, B and C are points on OP, OQ and OR respectively such that AB||PQ and AC||PR. Show that BC||QR.

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Q. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Theorem 6.2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
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Q. ABCD is a trapezium in which AB||DC and its diagonals intersect each other at the point O. Show that AOBO=CODO.
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Q. In Fig., DE||OQ and DF||OR. Show that EF||QR.

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Q. E and F are points on the sides PQ and PR respectively of a PQR. For each of the following cases, state whether EF||QR :
(i) PE=3.9 cm, EQ=3 cm, PF=3.6 cm and FR=2.4 cm
(ii) PE=4 cm, QE=4.5 cm, PF=8 cm and RF=9 cm
(iii) PQ=1.28 cm, PR=2.56 cm, PE=0.18 cm and PF=0.36 cm
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Q. S and T are points on sides PR and QR of PQR such that P=RTS. Show that RPQRTS.
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Q. In Fig, ODCOBA, BOC=125o and CDO=70o. Find DOC, DCO and OAB.

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Q. State which pairs of triangles in Fig. are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :

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Q. In Fig, altitudes AD and CE of ABC intersect each other at the point P. Show that:
(i) AEPCDP
(ii) ABDCBE