Question 1: State True or False:

(i) Two similar polygons are necessarily congruent – False

(ii) Two congruent polygons are necessarily similar – True

(iii) All equi-angular 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)

( )

Question 11: In and . Prove that:

(i)

(ii)

(iii)

Answer:

(i) Given and

Consider

(given)

(common angle)

Therefore (AAA postulate)

Hence

Therefore

(ii) Consider

(given)

(common angle)

Therefore (AAA postulate)

Hence

Therefore

(iii) Adding (i) and (ii)

Question 12: In . Prove that .

Answer:

.

Consider

(given)

(common angle)

Therefore (AAA postulate)

Hence

Therefore

Question 13: In .

(i) If

(ii) If

(iii) If

Answer:

Consider

(given)

Therefore (AAA postulate)

(i)

Therefore

(ii)

Therefore

(iii)

Therefore

Question 14: In the figure, is a parallelogram with and . is a point on such that . produced meets at and produced at . Find the lengths of and . **[1997]**

Answer:

In and

(vertically opposite angles)

(alternate angles)

Therefore (AAA postulate)

Therefore

In

(vertically opposite angles)

(alternate angles)

Therefore (AAA postulate)

Therefore

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

Answer:

Given … … … … (i)

In , because , by proportionality theorem

… … … … (i)

Therefore from (i) and (ii)

Similarly in

Also it is given that

Therefore . Therefore is a parallelogram.

Question 16: Given and . Prove that:

(i)

(ii)

Answer:

(i) Given and

In … … … … (i)

In … … … … (ii)

From (i) and (ii) you get

(ii) Consider

We have

and (common angle)

Therefore

Question 17: In and . Show that .

Answer:

Given

Also given

Therefore (SAS postulate)

Therefore

Adding

… … … (i)

We know … … … … (ii)

From (i) and (ii)

Question 18: In the given figure ; and .

(i) Name the three pairs of similar triangles

(ii) Find the lengths of

Answer:

Given , The three pairs of similar triangles are

Since

Since

Question 19: In the given figure , and . Prove that .

Answer:

Given , and

Using basic proportionality theorem

In

… … … … (i)

In

… … … … (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 and produced in . Prove that .

Answer:

Consider

(alternate angles)

(opposite angles)

(as M is the midpoint of CD)

Therefore

Therefore (corresponding angles)

(opposite sides of a parallelogram)

Therefore

Now consider

Therefore

Hence

Hence

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

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

(ii) In and in . Given , calculate length of . **[1999]**

Answer:

(i) Given

Also , applying basic proportionality theorem

(corresponding angles)

(corresponding angles)

Therefore (AAA postulate)

(ii) Given

(alternate angles)

Therefore (AAA postulate)

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

Answer:

(common)

(AAA postulate)

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

(i)

(ii) from (i)

Answer:

Given are medians

Therefore and

Applying converse of proportionality theorem

In and

(alternate angles)

(vertically opposite angles)

Therefore (AAA postulate)

Therefore

In and

(corresponding angles)

(common angle)

Therefore (AAA postulate)

Therefore

From 1,