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Practise Class 7 Maths Chapter 5 MCQs Set 4 on angles, angle relationships, intersecting lines, and geometrical reasoning with detailed answers and step-by-step solutions.
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Chapter Information
- Class: 7
- Subject: Mathematics
- Chapter: 5 – Parallel and Intersecting Lines
- Practice Set: 4
- Board: CBSE
- Textbook Reference: NCERT Ganita Prakash
- Question Type: Multiple-Choice Questions (MCQs)
Introduction
Parallel and intersecting lines help us understand the relationships between angles in geometry. When a straight line intersects another line, the angles formed follow specific mathematical rules. These rules allow us to calculate unknown angles and understand geometric figures.
This practice set focuses on angle relationships, algebraic expressions involving angles, and the properties of intersecting lines. The questions and explanations are independently written for educational practice.
What Will You Learn?
After completing this practice set, students will be able to:
- Apply angle properties to solve problems.
- Calculate unknown angles using algebraic expressions.
- Understand adjacent angles on a straight line.
- Identify equal angles formed by intersecting lines.
- Check mathematical answers through substitution.
Why Is This Topic Important?
Understanding angle relationships improves logical thinking and mathematical accuracy. These properties are useful when solving problems involving triangles, polygons, parallel lines, and geometrical constructions.
Exam Relevance
Questions from this topic may involve:
- Calculating unknown angles.
- Solving simple equations involving angles.
- Applying the properties of intersecting lines.
- Identifying adjacent and vertically opposite angles.
- Verifying whether calculated angle values are correct.
MCQs with Detailed Solutions
Question 1: Finding an Angle Using Algebra
Two vertically opposite angles measure $$(5x-20)^\circ$$ and $$(3x+30)^\circ$$. What is the value of $$x$$?
A. $$15$$
B. $$20$$
C. $$25$$
D. $$30$$
Answer: C. $$25$$
Concept: Vertically opposite angles.
Explanation:
Vertically opposite angles are equal.
Therefore:
$$5x-20=3x+30$$
Subtract $$3x$$ from both sides:
$$2x-20=30$$
Add $$20$$ to both sides:
$$2x=50$$
Divide both sides by $$2$$:
$$x=25$$
Check the angle measurements:
$$5(25)-20=105^\circ$$
$$3(25)+30=105^\circ$$
Both angles are equal.
Correct Answer: C. $$25$$
Question 2: Adjacent Angles on a Straight Line
Two adjacent angles form a straight line. Their measures are $$3x^\circ$$ and $$(2x+30)^\circ$$. Find the value of $$x$$.
A. $$20$$
B. $$25$$
C. $$30$$
D. $$35$$
Answer: B. $$30$$
Concept: Angles forming a linear pair.
Formula:
$$\text{Angle 1}+\text{Angle 2}=180^\circ$$
Solution:
Since the angles form a straight line:
$$3x+(2x+30)=180$$
Remove the brackets:
$$3x+2x+30=180$$
Combine like terms:
$$5x+30=180$$
Subtract $$30$$:
$$5x=150$$
Divide by $$5$$:
$$x=30$$
The two angles are:
$$3x=3(30)=90^\circ$$
$$2x+30=2(30)+30=90^\circ$$
Correct Answer: C. $$30$$
Question 3: Finding the Larger Angle
Two angles form a linear pair. The larger angle is twice the smaller angle. What is the measure of the larger angle?
A. $$60^\circ$$
B. $$90^\circ$$
C. $$120^\circ$$
D. $$150^\circ$$
Answer: C. $$120^\circ$$
Concept: Linear pair and algebraic reasoning.
Explanation:
Let the smaller angle be $$x^\circ$$.
The larger angle is $$2x^\circ$$.
Since they form a linear pair:
$$x+2x=180$$
Combine like terms:
$$3x=180$$
Divide by $$3$$:
$$x=60$$
Therefore, the larger angle is:
$$2x=2(60)$$
$$2x=120^\circ$$
Correct Answer: C. $$120^\circ$$
Question 4: Identifying Equal Angles
Two straight lines intersect at point O. If one angle measures $$143^\circ$$, which statement about its vertically opposite angle is correct?
A. It measures $$37^\circ$$.
B. It measures $$47^\circ$$.
C. It measures $$90^\circ$$.
D. It measures $$143^\circ$$.
Answer: D. $$143^\circ$$
Concept: Vertically opposite angles.
Explanation:
When two straight lines intersect, vertically opposite angles are equal.
Therefore, the angle directly opposite the angle measuring $$143^\circ$$ also measures:
$$143^\circ$$
The adjacent angles would each measure:
$$180^\circ-143^\circ=37^\circ$$
However, the question asks for the vertically opposite angle, not an adjacent angle.
Correct Answer: D. $$143^\circ$$
Question 5: Finding an Unknown Angle Around a Point
Three angles around a point measure $$105^\circ$$, $$85^\circ$$, and $$x^\circ$$. Find the value of $$x$$.
A. $$160^\circ$$
B. $$170^\circ$$
C. $$180^\circ$$
D. $$190^\circ$$
Answer: B. $$170^\circ$$
Concept: Angles around a point.
Formula:
$$\text{Sum of angles around a point}=360^\circ$$
Solution:
The three angles add up to $$360^\circ$$.
$$105^\circ+85^\circ+x=360^\circ$$
Add the known angles:
$$190^\circ+x=360^\circ$$
Subtract $$190^\circ$$:
$$x=360^\circ-190^\circ$$
$$x=170^\circ$$
Correct Answer: B. $$170^\circ$$
Important Formulas & Concepts
1. Vertically Opposite Angles
Vertically opposite angles are equal.
$$\angle A=\angle C$$
2. Linear Pair
Two adjacent angles forming a straight line add up to $$180^\circ$$.
$$\angle 1+\angle 2=180^\circ$$
3. Angles Around a Point
The sum of all angles around a point is $$360^\circ$$.
$$\angle 1+\angle 2+\angle 3+\cdots=360^\circ$$
4. Solving Equations Involving Angles
If two vertically opposite angles are represented by algebraic expressions, equate the expressions.
For example:
$$3x+10=5x-30$$
5. Checking an Answer
Substitute the calculated value of the variable into the original expressions to verify the result.
Related Practice Questions
Try solving these questions independently.
- Two vertically opposite angles measure $$(4x+5)^\circ$$ and $$(6x-35)^\circ$$. Find $$x$$.
- Two angles form a linear pair. If they measure $$4x^\circ$$ and $$(2x+30)^\circ$$, find $$x$$.
- Two supplementary angles are in the ratio $$2:3$$. Find the larger angle.
- Three angles around a point measure $$90^\circ$$, $$115^\circ$$, and $$x^\circ$$. Find $$x$$.
- One angle of a linear pair measures $$154^\circ$$. Find the other angle.
Mini Quiz Challenge
Question 1
If two vertically opposite angles are $$(2x+15)^\circ$$ and $$(4x-25)^\circ$$, what is the value of $$x$$?
A. $$15$$
B. $$20$$
C. $$25$$
D. $$30$$
Answer: B. $$20$$
Question 2
Two angles form a linear pair. If one angle is $$68^\circ$$, what is the other angle?
A. $$22^\circ$$
B. $$102^\circ$$
C. $$112^\circ$$
D. $$122^\circ$$
Answer: C. $$112^\circ$$
Question 3
What is the sum of all angles around a point?
A. $$90^\circ$$
B. $$180^\circ$$
C. $$270^\circ$$
D. $$360^\circ$$
Answer: D. $$360^\circ$$
Exam Tips
- Vertically opposite angles are equal.
- Angles forming a straight line add up to $$180^\circ$$.
- Angles around a point add up to $$360^\circ$$.
- Form an equation before calculating an unknown angle.
- Check your answer by substituting it into the original equation.
- Distinguish vertically opposite angles from adjacent angles.
Quick Revision Notes
- Vertically opposite angles are equal.
- A linear pair adds up to $$180^\circ$$.
- The sum of angles around a point is $$360^\circ$$.
- Algebra can be used to find unknown angle measurements.
- Substitution helps verify algebraic answers.
Common Mistakes Students Make
- Treating adjacent angles as vertically opposite angles.
- Forgetting to add both angles in a linear-pair equation.
- Using $$180^\circ$$ instead of $$360^\circ$$ for a complete turn around a point.
- Making sign errors when solving equations.
- Failing to verify the final value of the variable.
Key Takeaways
The properties of intersecting lines provide reliable methods for finding unknown angles. Vertically opposite angles are equal, a linear pair adds up to $$180^\circ$$, and angles around a point total $$360^\circ$$. Algebraic equations help apply these properties to more challenging problems.
FAQs
1. What are vertically opposite angles?
They are the opposite angles formed when two straight lines intersect. Each pair is equal.
2. What is the sum of a linear pair?
The sum is $$180^\circ$$.
3. How many degrees are there around a point?
There are $$360^\circ$$ around a point.
4. How can algebra help with angle problems?
Algebra represents unknown angle measurements using variables and equations.
5. How can an angle calculation be checked?
Substitute the calculated variable into the original expressions and verify that the required angle relationship holds.
Conclusion
This practice set provides additional practice with vertically opposite angles, linear pairs, angles around a point, and equations involving unknown angles. Students should work through each solution carefully to strengthen their understanding of geometric reasoning.
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