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

Answer:

In

Consider

(alternate angles)

(alternate angles)

Therefore (AAA postulate)

Hence

Question 3: Given , and . Find the length of the segments

Answer:

Consider

(Given)

Therefore (AAA postulate)

Substituting

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

Answer:

Consider

(Given)

Therefore (AAA postulate)

Therefore

Question 5: In the given figure, and are right angled at and respectively. Given and . (i) Prove (ii) Find and . **[2012]**

Answer:

In

(common angle)

Therefore (AAA postulate)

Since

Given AC=10 cm, AP = 15 cm and PM= 12 cm

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

Answer:

In

(opposite angles)

(alternate angles)

Therefore (AAA postulate)

Therefore

Question 7: Given are altitudes of . Prove that (i) (ii)

Answer:

(i) Consider

(Given, altitudes)

Therefore (AAA postulate)

(ii)

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

Answer:

In

(vertically opposite angles)

(alternate angles)

Therefore (AAA postulate)

Therefore

( is a Rhombus)

Question 9: Given . Prove

Answer:

and .

Therefore

Consider

(Given)

Therefore (AAA postulate)

Question 10: In and the bisector of meets at point . Prove that (i) (ii)

Answer:

Given

Therefore

Consider

(Given)

Therefore (AAA postulate)

( )