RD Sharma 2020 solution class 9 chapter 11 Triangles and Its Angles VSAQS

VSAQS

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Question 1:

Define a triangle.

Answer 1:


A plane figure bounded by three lines in a plane is called a triangle. A triangle has three sides, three angles and three vertices. The figure below represents a ΔABC, with AB, BC and CA as the three sides; ∠A, ∠B and ∠C as the three angles; A, B and C as the three vertices. 

Question 2:

Write the sum of the angles of an obtuse triangle.

Answer 2:

In the given problem, ΔABC is an obtuse triangle, withas the obtuse angle.

So, according to “the angle sum property of the triangle”, for any kind of triangle, the sum of its angles is 180°. So,

Therefore, sum of the angles of an obtuse triangle is.

Question 3:

In Δ ABC, if u∠B = 60°, ∠C = 80° and the bisectors of angles ∠ABC and ∠ACB meet at a point O, then find the measure of ∠BOC.

Answer 3:

In ΔABC,,and the bisectors of and meet at O.

We need to find the measure of

Since,BO is the bisector of

Similarly,CO is the bisector of

Now, applying angle sum property of the triangle, in ΔBOC, we get,

Therefore,.

Question 4:

If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.

Answer 4:

In the given problem, angles of ΔABC are in the ratio 2:1:3

We need to find the measure of the smallest angle.

Let,

According to the angle sum property of the triangle, in ΔABC, we get,

Thus, 

Since, the measure of is the smallest of all the three angles.

Therefore, the measure of the smallest angle is .

Question 5:

State exterior angle theorem.

Answer 5:

Exterior angle theorem states that, if a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

Thus, in ΔABC

Question 6:

The sum of two angles of a triangle is equal to its third angle. Determine the measure of the third angle.

Answer 6:

In the given problem, the sum of two angles of a triangle is equal to its third angle.

We need to find the measure of the third angle.

Thus, it is given, in

                                ........(1)

Now, according to the angle sum property of the triangle, we get,

(Using 1)

Therefore, the measure of the third angle is.

Question 7:

In the given figure, if AB || CD, EF || BC, ∠BAC = 65° and ∠DHF = 35°, find ∠AGH.

Answer 7:

In the given figure,,,and

We need to find

Here, GF and CD are straight lines intersecting at point H, so using the property, “vertically opposite angles are equal”, we get,

Further, asand AC is the transversal

Using the property, “alternate interior angles are equal”

Further applying angle sum property of the triangle

In ΔGHC

Hence, applying the property, “angles forming a linear pair are supplementary”

As AGC is a straight line

Therefore,

Question 8:

In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
 

Answer 8:

In the given figure,,,and

We need to find the value of x and y

Here, asand BD is the transversal, so according to the property, “alternate interior angles are equal”, we get

Similarly, as and DF is the transversal

(Using 1)

Further, EGH is a straight line. So, using the property, angles forming a linear pair are supplementary



Also, using the property, “an exterior angle of a triangle is equal to the sum of the two opposite interior angles”, we get,

In with as its exterior angle

Thus,

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Question 9:

If the angles A, B and C of ΔABC satisfy the relation B − A = C − B, then find the measure of ∠B.

Answer 9:

In the given ΔABC, 

,and satisfy the relation

We need to fine the measure of.

As,

         ........(1)

Now, using the angle sum property of the triangle, we get,

(Using 1)

Therefore,

Question 10:

In ΔABC, if bisectors of ∠ABC and ∠ACB intersect at O at angle of 120°, then find the measure of ∠A.

Answer 10:

In the given ΔABC,, the bisectors of and meet at O and

We need to find the measure of

So here, using the corollary, “if the bisectors of and of a meet at a point O, then”

Thus, in ΔABC

Thus,

Question 11:

If the side BC of ΔABC is produced on both sides, then write the difference between the sum of the exterior angles so formed and ∠A.

Answer 11:

In the given problem, we need to find the difference between the sum of the exterior angles and.

Now, according to the exterior angle theorem

              .........(1)

Also,

             .........(2)

Further, adding (1) and (2)

             .........(3)

Also, according to the angle sum property of the triangle, we get,

       .........(4)

Now, we need to find the difference between the sum of the exterior angles and.

Thus, 

(Using 4)

Therefore,

Question 12:

In a triangle ABC, if AB =  AC and AB is produced to D such that BD =  BC, find ∠ACD: ∠ADC.

Answer 12:

In the given ,and AB is produced to D such that

We need to find

Now, using the property, “angles opposite to equal sides are equal”

As

                ........(1)

Similarly,

As

                 ........(2)

Also, using the property, “an exterior angle of the triangle is equal to the sum of the two opposite interior angle” 

In ΔBDC

(Using 2)

From (1), we get

               .......(3)

Now, we need to find

That is, 

(Using 3)

(Using 2)

Eliminating from both the sides, we get 3:1 

Thus, the ratio of is

Question 13:

In the given figure, side BC of ΔABC is produced to point D such that bisectors of ∠ABC and ∠ACD meet at a point E. If ∠BAC = 68°, find ∠BEC.
 

Answer 13:

In the given figure, bisectors of and meet at E and

We need to find

Here, using the property: an exterior angle of the triangle is equal to the sum of the opposite interior angles.

In ΔABC with as its exterior angle

    ........(1)

Similarly, in ΔBE with as its exterior angle

(CE and BE are the bisectors of and)

   ........(2)

Now, multiplying both sides of (1) by

We get, 

         ........(3)

From (2) and (3) we get,

Thus,

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