Question 1: State True or False:

(i) Two similar polygons are necessarily congruent – False

(ii) Two congruent polygons are necessarily similar – True

(iii) All equiangular triangles are similar – True

(iv) All isosceles triangles are similar – False

(v) Two isosceles right angles triangles are similar – True

(vi) Two isosceles triangles are similar if an angle of one is congruent to the corresponding angle of the other – True

(vii) The diagonals of the trapezium, divide each into proportional segments – True

Question 2: , where are points on respectively. Prove that Also find the length of

Answer:

Question 3: , Find the length of the segments

Answer:

Question 4: is a point of side of such that Prove that

Answer:

Question 5: In the given figure, are right angled at respectively.

(i) Prove (ii) Find **[2012]**

Answer:

Question 6: are points in sides respectively of parallelogram If diagonal and segment intersect at ; prove that:

Answer:

Question 7: are altitudes of Prove that

Answer:

(Given, altitudes)

Question 8: is a rhombus, are straight lines. Prove that

Answer:

Answer:

Question 10: and the bisector of meets at point Prove that

Answer:

( )

Question 11: Prove that:

Answer:

(iii) Adding (i) and (ii)

Question 12: Prove that

Answer:

Question 13:

Answer:

Question 14: In the figure, is a parallelogram with is a point on such that

produced meets at produced at Find the lengths of **[1997]**

Answer:

Question 15: In quadrilateral , Diagonal intersect at point such that: Show that is a parallelogram.

Answer:

.. … … … (i)

.. … … … (i)

Therefore from (i) and (ii)

Similarly

Also it is given that

Question 16: Prove that:

Answer:

.. … … … (i)

.. … … … (ii)

and

Question 17: Show that

Answer:

Also

(SAS postulate)

Adding

.. … … (i)

We k.. … … … (ii)

From (i) and

Question 18: In the given figure ;

(i) Name the three pairs of similar triangles

(ii) Find the lengths of

Answer:

, The three pairs of similar triangles are

Answer:

Using basic proportionality theorem

.. … … … (i)

.. … … … (ii)

From (i) and (ii) we get

Question 20: Through the mid point of the side of a parallelogram , the line is drawn intersecting diagonal produced Prove that

Answer:

(as M is the midpoint of CD)

Question 21: In the figure given below, is a point on such that

(i) Calculate the ratio , giving reasons for your answer.

, calculate length of **[1999]**

Answer:

Also , applying basic proportionality theorem

Question 22: In the right angled is the altitude. Given that , calculate the value of **[2000]**

Answer:

Question 23: In the figure given below, median of the meet at Prove that:

Answer:

are medians

Applying converse of proportionality theorem