**GSEB Solutions for Class 9 Mathematics – Some Primary Concepts in Geometry : 2 (English Medium)**

GSEB SolutionsMathsScience

**Exercise 8:**

**Solution 1:**

**Solution 2:**

**Solution 3:**

**Solution 4:**

**Solution 5:**

**Solution 6:**

Given, m ∠XOY : m ∠YOZ = 2 : 3

Let m ∠XOY = 2k, then m ∠YOZ = 3k.

∠XOY and ∠YOZ form a linear pair.

∴ m ∠XOY + m ∠YOZ = 180

∴ 2k + 3k = 180

∴ 5k = 180

∴ k =180/5

∴ k = 36

m ∠XOY = 2k = 2 × 36 = 72

m ∠YOZ = 3k= 3 × 36 = 108

Thus, the measure of ∠XOY is 72 and ∠YOZ that of 108.

**Solution 7(1):**

c. rays

An angle is a union of __rays__.

**Solution 7(2):**

d. 0 and 180

The measure of an angle always lies between 0 and 180.

**Solution 7(3):**

d. 9

Measure of the complementary angle of ∠A

= 90 – m ∠A

= 90 – 81

= 9

**Solution 7(4):**

**Solution 7(5):**

**Solution 7(6):**

b. skew

If two lines cannot lie in the same plane, they are called skew lines.

**Solution 7(7):**

c. 113

The measure of the complementary angle of complementary angle of an angle having measure 23 = 90 – 23 = 67

The measure of the supplementary angle of complementary angle of an angle having measure 23 = 180 – 67 = 113

**Solution 7(8):**

a. -(x – 60)

The measure of the complementary angle of an angle having measure x + 30

= 90 – (x + 30)

= 60 – x – 30

= -(x – 60)

**Solution 7(9):**

c. obtuse

If one angle of a linear pair is acute, then other angle is __obtuse__.

**Solution 7(10):**

a. supplementary

If t is a transversal for two parallel lines l and m, interior angles on the same side of the transversal are __supplementary__.

**Solution 7(11):**

b. 15

We know that the sum of measures of angles forming a linear pair is 180.

∴ (6y + 30) + 4y = 180

∴ 10y + 30 = 180

∴ 10y = 180 – 30∴ 10y = 150

∴ y = 15

**Solution 7(12):**

**Exercise 8.1:**

**Solution 1:**

Any one of the following three conditions uniquely determines a plane:

- Three non-collinear points.
- A line and a point not lying on it.
- Two distinct intersecting lines.

**Solution 2:**

**Solution 3:**

**Solution 4:**

- Coplanar and non-coplanar points

**Coplanar points**: If there exists a plane containing all the given points, the points are said to be coplanar.

**Non-coplanar points**: If there does not exist a plane containing all the given points, we say they are non-coplanar. - Coplanar and non-coplanar lines

**Coplanar lines:**If there exists a plane containing all the given lines, we say the lines are coplanar.

**Non-coplanar points:**If there does not exist a plane containing all the given lines, such lines are called non-coplanar lines.

**Solution 5:**

**Solution 6:**

**Exercise 8.2:**

**Solution 1:**

**Solution 2:**

**Coplanar lines**: If there exists a plane containing all the given lines, we say the lines are coplanar.**Skew lines**: The lines which are not coplanar are called skew lines.**Coplanar points**: If there exists a plane containing all the given points, the points are said to be coplanar.**A closed half plane**: The union of any half plane formed by line l and the line l itself is a closed half plane.

**Solution 3:**

**Solution 4:**

**Solution 5:**

**Exercise 8.3:**

**Solution 1:**

**Solution 2:**

Pairs of adjacent angles are given below:

- ∠DOA and ∠DOC
- ∠DOA and ∠DOB
- ∠COB and ∠COD
- ∠COB and ∠COA

Linear pairs of angles are given below:

- ∠DOA and ∠DOB
- ∠COB and ∠COA

**Solution 3:**

**Solution 4:**

**Solution 5:**

**Solution 6(1):**

**Solution 6(2):**

**Solution 6(3):**

The measure of the supplementary angle of the angle having measure 120° = 180° – 120° = 60°.

The measure of the complementary angle of the supplementary angle of the angle having measure 120° = 90° – 60° = 30°.

**Solution 6(4):**

- The measure of the complementary angle of the angle with measure 42° = 90° – 42° = 48°.
- The measure of the complementary angle of the angle with measure 37° = 90° – 37° = 53°.
- The measure of the complementary angle of the angle with measure (10 + x)° = 90° – (10 + x)° = 90 – 10 – x = (80 – x)°.
- The measure of the complementary angle of the angle with measure 81° = 90° – 81° = 9°.

**Solution 6(5):**

- The measure of the supplementary angle of the angle with measure 100° = 180° – 100° = 80°.
- The measure of the supplementary angle of the angle with measure 89° = 180° – 89° = 91°.
- The measure of the supplementary angle of the angle with measure (y – 30)° = 180° – (y – 30)° = 180 – y + 30 = (210 – y)°.
- The measure of the supplementary angle of the angle with measure 49° = 180° – 49° = 131°.

**Exercise 8.4:**

**Solution 1:**

**Solution 2:**

From the figure, m ⃦ n. Line t is the transversal of line m and line l.

∴ m∠EFB = m∠DGF (corresponding angles)

But, m∠EFB = 65° …[Given]

∴ m∠DGF = 65° … [∵ m∠EFB = m∠DGF]

Now, ∠DGF and ∠CGF form a linear pair.

∴ m∠DGF + m∠CGF = 180°

∴ 65° + m∠CGF = 180° … [∵ m∠DGF = 65°

∴ m∠CGF = 180° – 65°

∴ m∠CGF = 115°

So, m∠CGF = 115° and m∠DGF = 65°.

**Solution 3:**

**Solution 4:**

**Solution 5:**

From the figure, ∠CQD and ∠PQE are vertically opposite angles.

∴ m∠CQD = m∠PQE

∴ m∠PQE = 85° [∵ m∠CQD = 85°]

m∠APB = 85° …[Given]

∴ m∠APB = m∠PQE

∴ ∠APB @ ∠PQE

∠APB and ∠PQE are corresponding angles formed by transversal t to lines l and m.

Thus, l ⃦ m.

**Exercise 8.5:**

**Solution 1:**

Alternate angles are given below:

- ∠MQR and ∠QRB
- ∠NQR and ∠QRA

Corresponding angles are given below:

- ∠PQM and ∠QRA
- ∠PQN and ∠QRB
- ∠MQR and ∠ARS
- ∠NQR and ∠BRS

**Solution 2:**

**Solution 3:**

**Solution 4:**

**Solution 5:**

Given: Line t_{1} and t_{2} are transversals to the lines l, m and n.

m∠ABE = m∠BCD and m∠FEI = m∠GFH

To Prove : l ⃦ n

Proof:

m∠ABE = m∠BCD

∴ ∠ABE ≌ ∠BCD

∠ABE and ∠BCD are corresponding angles formed by transversal t_{1} to lines l and m and they are congruent.

∴ l ⃦ m …[1]

m∠FEI = m∠GFH

∴ ∠FEI ≌ ∠GFH

∠FEI and ∠GFH are corresponding angles formed by transversal t_{2} to lines m and n and they are congruent.

∴ m ⃦ n …[2]

From [1] and [2],

l ⃦ n.

**Solution 6:**

**Solution 7:**