Worksheet on Euclid's Geometry and Lines and Angles for Class 9

Straight lines and angles create beautiful geometric patterns everywhere. Students work with printable grade 9 geometry lines and angles practice questions that explore different types of angles and their relationships. These activities help learners understand parallel lines, intersecting lines, and angle properties through visual examples. Every exercise builds spatial reasoning skills effectively. Students can download a free Class 9 geometry lines and angles PDF worksheet for practice.

Solved Questions on Euclid’s Geometry and Lines and Angles

1. Which of the following axioms states that if a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the reverse is also true?

a) Adjacent angles axiom
b) Straight angles axiom
c) Linear pair axiom
d) Supplementary angles axiom

Answer: c) Linear pair axiom

Explanation: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the reverse is also true. This property is known as the linear pair axiom.

2. Which of the following is NOT true for Euclid's postulates?

Postulate I: It is possible to draw a circle with any centre and diameter.
Postulate II: A straight line can be drawn from any given point to another point.
Postulate III: Lines that coincide with each other are equal.
Postulate IV: There is only one line that goes through both of them if there are two different points.
Postulate V: A terminated line can be extended infinitely.

a) Only I and II
b) Only II and III
c) Only III and IV
d) Only IV and V

Answer: c) Only III and IV

Explanation: III and IV are Euclid's axioms, not postulates.

3. What is the value of angle z if AB || CD || EF and the angle y is one-third of angle x?

a) 120°
b) 128°
c) 135°
d) 144°

Answer: c) 135°

Explanation: In the given figure, x and z are alternate angles (AB || EF) and y and z are co-interior angles (CD || EF).

∠x = ∠z (alternate angles, AB || EF)

According to the question, ∠y = 1/3 ∠x
⇒∠y = ¹⁄₃ ∠z 

∠y + ∠z = 180° (co-interior angles, CD || EF)
⇒ ¹⁄₃ ∠z + ∠z = 180°
⇒ 4/3 ∠z = 180°
⇒ ∠z = 180° × 3/4
∴ ∠z = 135°

Hence, ∠x = ∠z = 135°

4. Which of the following is true if AB || DE?

a) ∠ABC + ∠BCD + ∠CDE = 180º 
b) ∠ABC + ∠BCD − ∠CDE = 180º 
c) ∠ABC + ∠BCD + ∠CDE = 270º 
d) ∠ABC + ∠CDE − ∠BCD = 180º 

Answer: b) ∠ABC + ∠BCD − ∠CDE = 180°

Explanation: Draw a line CF through point C, making it parallel to both AB and DE.

Hence, AB || CF || DE. BC and CD are transversals.

The labelled diagram is shown as:

∠1 + ∠2 = 180° (Co-interior angles, AB || CF) ……………….(i)
∠3 = ∠4 (Alternate angles, CF || DE) ………………….(ii)
Adding (i) and (ii), we get:
∠1 + ∠2 + ∠3 = 180° + ∠4
⇒ ∠1 + (∠2 + ∠3) = 180° +  ∠4  
⇒ ∠ABC + ∠BCD = 180° +  ∠CDE
∴ ∠ABC + ∠BCD − ∠CDE = 180º

5. What is the value of x, y and z if LI is parallel to ON, angle x is three-sevenths of the angle y and angle y is 1$\frac{\frac{2}{}}{\frac{5}{}}$ times of angle z?

a) ∠x = 36°; ∠y = 72°; ∠z = 45°
b) ∠x = 36°; ∠y = 72°; ∠z = 60°
c) ∠x = 36°; ∠y = 84°; ∠z = 45°
d) ∠x = 36°; ∠y = 84°; ∠z = 60°

Answer: d) ∠x = 36°; ∠y = 84°; ∠z = 60°

Explanation: Angle x is three-sevenths of the angle y.
⇒ ∠x = 3/7 × ∠y
Angle y is 1 ²⁄₅ times of angle z.
⇒ ∠y = 1 ²⁄₅ × ∠z
⇒ ∠y = 7/5 × ∠z
∴ ∠z = 5/7 × ∠y

∠x + ∠y + ∠z = 180° (Co-interior angles, LI || ON)
⇒ (3/7 × ∠y) + ∠y + (5/7 × ∠y) = 180°  [Put values of ∠x and ∠z in terms of ∠y]
⇒ (3/7 + 1 + 5/7) × ∠y = 180°                [Take ∠y as common]
⇒ (3 + 7 + 5)/7 × ∠y = 180°                   [Take LCM as 7]
⇒ 15/7 × ∠y = 180°
⇒ ∠y = 180° × 7/15
∴ ∠y = 84°

∠x = 3/7 × ∠y
⇒ ∠x = 3/7 × 84°
∴ ∠x = 36°

∠z = 5/7 × ∠y
⇒ ∠z = 5/7 × 84°
∴ ∠z = 60°