Question 1: In the adjoining figure, and . Prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

Now consider

Given (angles opposite to equal sides of a triangle are equal)

Therefore

. Hence proved.

Question 2: In the adjoining figure, . and are respectively the bisectors of and . Prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

Therefore we can also say that

Since

Now consider and

is common

Therefore (by A.S.A theorem). Hence proved.

Question 3: If is an isosceles triangle with . Prove that perpendiculars from the vertices and to their opposite sides are equal.

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

Now consider and

is common

(right angles)

Therefore (by A.S.A theorem). Hence proved.

Question 4: In the adjoining figure, it is given that and . Prove that .

Answer:

Given

and (vertically opposite angles)

Now consider and

(given)

(by A.S.A theorem). Hence proved.

Question 5: In , and the bisector of and intersect at . Prove that and the is the bisector of .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

(since sides opposite to equal angles are equal)

Now consider and

(given)

Therefore (by S.A.S theorem)

Question 6: In the adjoining figure, it is given and . Prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

We have

Consider and

(given)

Therefore (by S.A.S. theorem)

Question 7: In the adjoining figure, line is the bisector of and is any point on . and are perpendiculars from to the arms of . Show that: (i) (ii) or is equidistant from the arms of .

Answer:

Since is the bisector of

(perpendicular lines)

is common.

Therefore (by A.S.A theorem)

Question 8: In the adjoining figure is a median and are perpendiculars drawn from and respectively on and produced. Prove that .

Answer:

Consider and

(vertically opposite angles)

(right angles)

is the median

Therefore (A.A.S theorem)

Question 9: In the adjoining figure, and . Also . Prove that bisects .

Answer:

Consider and

(right angles)

(vertically opposite angles)

and (given)

Therefor (by A.A.S theorem)

. Hence proved.

Question 10: In the adjoining figure, is an isosceles triangle with and are two medians of the triangle. Prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

Also,

Consider and

is common

Therefore (by S.A.S theorem)

Therefore . Hence proved.

Question 11: In the adjoining figure, , . Prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

Consider and

(given)

Therefore

Therefore

Question 12: In the adjoining figure, and are two triangles on the same base such that and . Prove that .

Answer:

Consider

Given … … … … … (i) (angles opposite to equal sides of a triangle are equal)

Similarly in

Given … … … … … (ii) (angles opposite to equal sides of a triangle are equal)

Subtracting (ii) from (i) we get

Question 13: In the adjoining figure, . is the midpoint of . Show that (i) (ii) is also the midpoint of .

Answer:

(i) Consider and

(given)

Since

also (vertically opposite angles)

Therefore (by A.A.S theorem)

(ii) Since

is the midpoint of

Question 14: In a , it is given that and the bisectors of and intersect at . If is a point on produced, prove that .

Answer:

Consider

Given (angles opposite to equal sides of a triangle are equal)

In , we have

. Hence proved.

Question 15: is a point on the bisector of an . If the line through parallel to meets at , prove that the is isosceles.

Answer:

Since

(alternate angles)

Also since ,

Hence

(sides opposite equal angles in a triangle are equal).

Hence is an isosceles triangle.

Question 16: is a triangle in which . is a point on such that bisects and . Prove that .

Answer:

In

Let , where

is the bisector of . So

Let be the bisector of .

In , we have

In and , we have

(given) and

Therefore (by S.A.S theorem)

and

and

In , we have

In , we have

Hence