Welcome, students! Geometry can be one of the most rewarding branches of Mathematics once you master its fundamental concepts. Today, we are exploring the step-by-step solutions for Class 9 Maths Chapter 6 (Lines and Angles) Exercise 6.1. These solutions follow the MP Board and NCERT curriculum, designed to help you construct flawless proofs and secure high marks in your exams.
Q1. In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.
Step-by-step Solution:
- Since AB is a straight line, the sum of all linear angles on it is 180°:
∠AOC + ∠COE + ∠BOE = 180° - Rearranging the terms, we get:
(∠AOC + ∠BOE) + ∠COE = 180° - Substitute the given value of ∠AOC + ∠BOE = 70°:
70° + ∠COE = 180°
∠COE = 110° - Now, calculate the reflex angle of ∠COE:
Reflex ∠COE = 360° - ∠COE = 360° - 110° = 250° - Next, CD is also a straight line, so:
∠COE + ∠BOE + ∠BOD = 180° - Substitute ∠COE = 110° and the given value of ∠BOD = 40°:
110° + ∠BOE + 40° = 180°
150° + ∠BOE = 180°
∠BOE = 180° - 150° = 30°
Final Answer: ∠BOE = 30° and reflex ∠COE = 250°.
Q2. In the given figure, lines XY and MN intersect at O. If ∠POY = 90° and a : b = 2 : 3, find c.
Step-by-step Solution:
- XY is a straight line. Therefore, ∠XOP + ∠POY = 180° (Linear pair).
- Given ∠POY = 90°, we find:
∠XOP + 90° = 180° ⇒ ∠XOP = 90° - From the figure, ∠XOP = a + b. Thus, a + b = 90°.
- Given the ratio a : b = 2 : 3, let a = 2x and b = 3x:
2x + 3x = 90°
5x = 90° ⇒ x = 18° - Calculate values of a and b:
a = 2 × 18° = 36°
b = 3 × 18° = 54° - Since MN is a straight line, angle b and c form a linear pair:
b + c = 180°
54° + c = 180°
c = 180° - 54° = 126°
Final Answer: c = 126°.
Q3. In the given figure, ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT.
Proof:
- SQR is a straight line. Thus, ∠PQS and ∠PQR form a linear pair:
∠PQS + ∠PQR = 180° — (Equation 1) - Similarly, QRT is a straight line. Thus, ∠PRT and ∠PRQ form a linear pair:
∠PRT + ∠PRQ = 180° — (Equation 2) - Equating Equation 1 and Equation 2 because both equal 180°:
∠PQS + ∠PQR = ∠PRT + ∠PRQ - Given that ∠PQR = ∠PRQ, we can subtract this identical value from both sides:
∠PQS = ∠PRT
Hence Proved!
Q4. In the given figure, if x + y = w + z, then prove that AOB is a line.
Proof:
- The sum of all angles around a point is 360°:
x + y + w + z = 360° - It is given that x + y = w + z. Substituting (x + y) in place of (w + z):
(x + y) + (x + y) = 360° - Simplify the equation:
2(x + y) = 360°
x + y = 180° - Since the adjacent angles x and y sum up to 180°, they form a linear pair. According to the axiom of geometry, if adjacent angles form a linear pair, their non-common arms lie on a straight line. Therefore, AOB is a straight line.
Hence Proved!
Q5. In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 1/2(∠QOS - ∠POS).
Proof:
- Since OR is perpendicular to PQ, we know:
∠ROP = 90° and ∠ROQ = 90° - We can express ∠QOS as:
∠QOS = ∠ROQ + ∠ROS = 90° + ∠ROS — (Equation 1) - Similarly, we can express ∠POS as:
∠POS = ∠ROP - ∠ROS = 90° - ∠ROS — (Equation 2) - Subtracting Equation 2 from Equation 1:
∠QOS - ∠POS = (90° + ∠ROS) - (90° - ∠ROS) - Expanding the brackets:
∠QOS - ∠POS = 90° + ∠ROS - 90° + ∠ROS
∠QOS - ∠POS = 2∠ROS - Rearranging for ∠ROS:
∠ROS = 1/2(∠QOS - ∠POS)
Hence Proved!
Q6. It is given that ∠XYZ = 64° and XY is produced to point P. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.
Step-by-step Solution:
- Since XY is produced to P, XYP is a straight line:
∠XYZ + ∠ZYP = 180° (Linear pair) - Substitute ∠XYZ = 64°:
64° + ∠ZYP = 180°
∠ZYP = 180° - 64° = 116° - Since ray YQ bisects ∠ZYP, it divides the angle into two equal parts:
∠ZYQ = ∠QYP = 116° / 2 = 58° - Now, find ∠XYQ:
∠XYQ = ∠XYZ + ∠ZYQ = 64° + 58° = 122° - Find reflex ∠QYP:
Reflex ∠QYP = 360° - ∠QYP = 360° - 58° = 302°
Final Answer: ∠XYQ = 122° and reflex ∠QYP = 302°.
ЁЯТб Aptitude Corner (For NTSE & Olympiads)
Theorem on Angle Bisectors of Linear Pairs:
Did you know? The angle bisectors of a linear pair of angles are always perpendicular to each other. That is, the angle between the two bisectors is always 90°.
Quick Question: If rays OA and OB bisect a pair of adjacent supplementary angles, then find ∠AOB.
Answer: Since supplementary angles add up to 180°, their halved sum is always 180° / 2 = 90°.
Excellent progress! Geometry becomes much easier when you visualize the lines and solve with structured, logical steps. For more such clean chapter solutions and extra exam-prep tricks, keep practicing and stay curious!
Academic Discussion & Doubts