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

MCQS

Page-11.23

Question 1:

Mark the correct alternative in each of the following:

If all the three angles of a triangle are equal, then each one of them is equal to

(a) 90°

(b) 45°

(c) 60°

(d) 30°

Answer 1:

In a given ΔABC we are given that the three angles are equal. So,

According to the angle sum property of a triangle, in ΔABC

 

Therefore, all the three angles of the triangle are equal to

So, the correct option is (c).

Question 2:

If two acute angles of a right triangle are equal, then each acute is equal to

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Answer 2:

In the given problem, we have a right angled triangle and the other two angles are equal.

So, In ΔABC

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

()

Therefore, the correct option is (b).

Question 3:

An exterior angle of a triangle is equal to 100° and two interior opposite angles are equal. Each of these angles is equal to

(a) 75°

(b) 80°

(c) 40°

(d) 50°

Answer 3:

In the ΔABC, CD is the ray extended from the vertex C of ΔABC. It is given that the exterior angle of the triangle is and two of the interior opposite angles are equal.

So, and_.

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

In ΔABC

Therefore, each of the two opposite interior angles is

So, the correct option is (d).

Question 4:

If one angle of a triangle is equal to the sum of the other two angles, then the triangle is

(a) an isosceles triangle

(b) an obtuse triangle

(c) an equilateral triangle

(d) a right triangle

Answer 4:

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

Thus, 

     ..........(1)

Now, according to the angle sum property of the triangle

In ΔABC

                  .........(2)

Further, using (2) in (1),

Thus,

Therefore, the correct option is (d).

Question 5:

Side BC of a triangle ABC has been produced to a point D such that ∠ACD = 120°. If ∠B = $\frac{1}{2}$ ∠A is equal to

(a) 80°

(b) 75°

(c) 60°

(d) 90°

Answer 5:

In the given problem, side BC of ΔABC has been produced to a point D. Such that and. Here, we need to find

Given

We get,

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

In ΔABC

Also, (Using 1)

Thus,

Therefore, the correct option is (a).

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Question 6:

In ΔABC, ∠B = ∠C and ray AX bisects the exterior angle ∠DAC. If ∠DAX = 70°, then ∠ACB =

(a) 35°

(b) 90°

(c) 70°

(d) 55°

Answer 6:

In the given ΔABC, . D is the ray extended from point A. AX bisectsand

Here, we need to find

As ray AX bisects

Thus,

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

Thus,

Therefore, the correct option is (c).

Question 7:

In a triangle, an exterior angle at a vertex is 95° and its one of the interior opposite angle is 55°, then the measure of the other interior angle is

(a) 55°

(b) 85°

(c) 40°

(d) 9.0°

Answer 7:

In the given ΔABC, and

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

So,

Therefore, the correct option is (c).

Question 8:

If the sides of a triangle are produced in order, then the sum of the three exterior angles so formed is

(a) 90°

(b) 180°

(c) 270°

(d) 360°

Answer 8:

In the given ΔABC, all the three sides of the triangle are produced. We need to find the sum of the three exterior angles so produced.

Now, according to the angle sum property of the triangle

        .......(1)

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

            ......(2)

Similarly,

           .......(3)

Also, 

           .......(4)

Adding (2) (3) and (4)

We get,

Thus, 

Therefore, the correct option is (d).

Question 9:

In ΔABC, if ∠A = 100°, AD bisects ∠A and AD ⊥ BC. Then, ∠B =

(a) 50°

(b) 90°

(c) 40°

(d) 100°

Answer 9:

In the given ΔABC,, AD bisects and .

Here, we need to find.

As, AD bisects,

We get, 

Now, according to angle sum property of the triangle

In ΔABD

Hence,

Therefore, the correct option is (c).

Question 10:

An exterior angle of a triangle is 108° and its interior opposite angles are in the ratio 4 : 5. The angles of the triangle are

(a) 48°, 60°, 72°

(b) 50°, 60°, 70°

(c) 52°, 56°, 72°

(d) 42°, 60°, 76°

Answer 10:

In the given ΔABC, an exterior angle and its interior opposite angles are in the ratio 4:5.

Let us take,

Now using the property, “exterior angle of a triangle is equal to the sum of two opposite interior angles”

We get,

Thus, 

Also, using angle sum property in ΔABC

Thus, 

Therefore, the correct option is (a).

Question 11:

In a ΔABC, if ∠A = 60°, ∠B = 80° and the bisectors of ∠B and ∠C meet at O, then ∠BOC =

(a) 60°

(b) 120°

(c) 150°

(d) 30°

Answer 11:

In the given ΔABC,and . Bisectors of and meet at O. 

We need to find

Since, OB is the bisector of.

Thus,

Now, using the angle sum property of the triangle

In ΔABC, we get,

Similarly, in ΔBOC

Hence,

Therefore, the correct option is (b).

Question 12:

Line segments AB and CD intersect at O such that AC || DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =

(a) 100°

(b) 80°

(c) 90°

(d) 135°

Answer 12:

In the given problem, line segment AB and CD intersect at O, such that,and .

We need to find

As

Applying the property, “alternate interior angles are equal”, we get,

    .......(1)

Now, using the angle sum property of the triangle

In ΔODB, we get,

(using 1)

Thus,

Therefore, the correct option is (b).

Question 13:

In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is

(a) 20°

(b) 50°

(c) 60°

(d) 70°

Answer 13:

In the given figure,,and . We need to find.

Here, and CD is the transversal, so using the property, “corresponding angles are equal”, we get

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

Thus,

Therefore, the correct option is (b).

Question 14:

In the given figure, x + y =

(a) 270

(b) 230

(c) 210

(d) 190°
 

Answer 14:

In the given figure, we need to find

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

Further, applying the property, “an exterior angle of a triangle is equal to the sum of the two opposite interior angles”, in ΔAOC, we get,

Similarly, in ΔBOD

Thus,

Therefore, the correct option is (b).

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Question 15:

If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?

(a) 25°

(b) 30°

(c) 45

(d) 60°

Answer 15:

In the given figure, measures of the angles of ΔABC are in the ratio. We need to find the measure of the smallest angle of the triangle.

Let us take,

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

Substituting the value of x in,and

Since, the measure of is the smallest

Thus, the measure of the smallest angle of the triangle is

Therefore, the correct option is (c).

Question 16:

In the given figure, if AB ⊥ BC. then x =

(a) 18

(b) 22

(c) 25

(d) 32
 

Answer 16:

In the given figure,

We need to find the value of x.

Now, since AB and CD are straight lines intersecting at point O, using the property, “vertically opposite angles are equal”, we get,

Further, applying angle sum property of the triangle 

In ΔBOC 

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

In ΔEOC

Further solving for x, we get,

Thus,

Therefore, the correct option is (b).

Question 17:

In the given figure, what is z in terms of x and y?

(a) x + y + 180

(b) x + y − 180

(c) 180° − (x + y) 

(d) x + y + 360°
 

Answer 17:

In the given ΔABC, we need to convert z in terms of x and y 

Now, BC is a straight line, so using the property, “angles forming a linear pair are supplementary”

Similarly,

Also, using the property, “vertically opposite angles are equal”, we get,

Further, using angle sum property of the triangle

Thus,

Therefore, the correct option is (b).

Question 18:

In the given figure, for which value of x is l₁ || l₂?

(a) 37

(b) 43

(c) 45

(d) 47

Answer 18:

In the given problem, we need to find the value of x if

Here, if , then using the property, “if the two lines are parallel, then the alternate interior angles are equal”, we get,

Further, applying angle sum property of the triangle

In ΔABC

Thus,

Therefore, the correct option is (d).

Question 19:

In the given figure, what is y in terms of x?

(a) $\frac{3}{2}x$

(b) $\frac{4}{3}x$

(c) x

(d) $\frac{3}{4}x$
 

Answer 19:

In the given figure, we need to find y in terms of x

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

In ΔABC

                          ..........(1)

Similarly, in ΔOCD

(using 1)

Thus,

Therefore, the correct option is (a).

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Question 20:

In the given figure, what is the value of x?

(a) 35

(b) 45

(c) 50

(d) 60
 

Answer 20:

In the given figure, we need to find the value of x.

Here, DBA is a straight line, so using the property, “angles forming a linear pair are supplementary”, we get,

Now, applying the value of y inand

Also,

Further, applying angle sum property of the triangle

In ΔABC

Thus,

Therefore, the correct option is (d).

Question 21:

In the given figure, the value of x is

(a) 65°

(b) 80°

(c) 95°

(d) 120°
 

Answer 21:

In the given figure, we need to find the value of x

Here, according to the angle sum property of the triangle

In ΔABD

Also, ABC is a straight line. So, using the property, “angles forming a linear pair are supplementary”, we get,

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

Thus,

Therefore, the correct option is (d).

Question 22:

In the given figure, if BP || CQ and AC = BC, then the measure of x is

(a) 20°

(b) 25°

(c) 30°

(d) 35°

Answer 22:

In the given figure,and

We need to find the measure of x

Here, we draw a line RS parallel to BP, i.e

Also, using the property, “two lines parallel to the same line are parallel to each other”

As,

Thus,

Now, and BA is the transversal, so using the property, “alternate interior angles are equal”

Similarly, and AC is the transversal

                              ........(2)

Adding (1) and (2), we get

Also, as

Using the property,”angles opposite to equal sides are equal”, we get

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

In ΔABC

Thus,

Therefore, the correct option is (c).

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Question 23:

In the given figure, AB and CD are parallel lines and transversal EF intersects them at P and Q respectively. If ∠APR = 25°, ∠RQC = 30° and ∠CQF = 65°, then

(a) x = 55°, y = 40°

(b) x = 50°, y = 45°

(c) x = 60°, y = 35°

(d) x = 35°, y = 60°

Answer 23:

In the given figure,,,and

We need to find the value of x and y

Here, we draw a line ST parallel to AB, i.e

Also, using the property, “two lines parallel to the same line are parallel to each other”

As,

Thus,

Now, and EF is the transversal, so using the property, ”alternate interior angles are equal”, we get,

Similarly,and EF is the transversal

                          .......(2)

Adding (1) and (2), we get

Further,FPE is a straight line

Applying the property, angles forming a linear pair are supplementary

Also, applying angle sum property of the triangle

In ΔPRQ

Thus,

Therefore, the correct option is (a).

Question 24:

The base BC of triangle ABC is produced both ways and the measure of exterior angles formed are 94° and 126°. Then, ∠BAC =

(a) 94°

(b) 54°

(c) 40°

(d) 44°

Answer 24:

In the given problem, the exterior angles obtained on producing the base of a triangle both ways areand. So, let us draw ΔABC and extend the base BC, such that:

Here, we need to find

Now, since BCD is a straight line, using the property, “angles forming a linear pair are supplementary”, we get

Similarly, EBS is a straight line, so we get,

Further, using angle sum property in ΔABC

Thus,

Therefore, the correct option is (c).

Question 25:

If the bisectors of the acute angles of a right triangle meet at O, then the angle at O between the two bisectors is

(a) 45°

(b) 95°

(c) 135°

(d) 90°

Answer 25:

In the given problem, bisectors of the acute angles of a right angled triangle meet at O. We need to find .

Now, using the angle sum property of a triangle

In ΔABC

Now, further multiplying each of the term by in (1)

Also, applying angle sum property of a triangle

In ΔAOC

Thus,

Therefore, the correct option is (c).

Question 26:

The bisects of exterior angle at B and C of ΔABC meet at O. If ∠A = x°, then ∠BOC =

(a) $90°+\frac{x°}{2}$

(b) $90°-\frac{x°}{2}$

(c) $180°+\frac{x°}{2}$

(d) $180°-\frac{x°}{2}$

Answer 26:

In the given figure, bisects of exterior anglesand meet at O and

We need to find

Now, according to the theorem, “if the sides AB and AC of a ΔABC are produced to P and Q respectively and the bisectors of and intersect at O, therefore, we get,

Hence, in ΔABC

Thus,

Therefore, the correct option is (b).

Question 27:

In a ΔABC, ∠A = 50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACD meet at E, then ∠E =

(a) 25°

(b) 50°

(c) 100°

(d) 75°

Answer 27:

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”, we get,

In ΔABC with as its exterior angle

         ........(1)

Similarly, in 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,

Therefore, the correct option is (a).

Question 28:

The side BC of ΔABC is produced to a point D. The bisector of ∠A meets side BC in L. If ∠ABC = 30° and ∠ACD = 115°, then ∠ALC =

(a) 85°

(b) $72\frac{1}{2}°$

(c) 145°

(d) none of these

Answer 28:

In the given problem, BC of ΔABC is produced to point D. bisectors of meet side BC at L, and

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

In ΔABC

Now, as AL is the bisector of

Also, is the exterior angle of ΔALC

Thus,

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

In

Thus,

Therefore the correct option is (b).

Question 29:

In the given figure, if l₁ || l₂, the value of x is

(a) $22\frac{1}{2}$

(b) 30

(c) 45

(d) 60

Answer 29:

In the given problem,

We need to find the value of x

Here, as, using the property, “consecutive interior angles are supplementary”, we get

               ..........(1)

Further, applying angle sum property of the triangle 

In ΔABC

(using 1)

Now, AB is a straight line, so using the property, “angles forming a linear pair are supplementary”, we get,

Thus,

Therefore, the correct option is (c).

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Question 30:

In ΔRST (See figure), what is the value of x?

(a) 40

(b) 90°

(c) 80°

(d) 100

Answer 30:

In the given problem, we need to find the value of x.

Here, according to the corollary, “if bisectors of and of a ΔABC meet at a point O, then

In ΔRST

Further solving for x, we get,

Thus,

Therefore, the correct option is (d).