# Parallel and Intersecting Lines

## Overview

This chapter introduces students to the different relationships between lines—how they intersect, when they are perpendicular or parallel, and how transversals interact with these lines. It helps learners understand geometrical reasoning over visual approximation, highlights angle properties, and strengthens foundational concepts used in drawing, architecture, and problem-solving.

## Key Topics Covered

### 1. Intersecting Lines

* **Definition**: Two lines that meet at a single point.
* **Properties**:

* Four angles are formed at the point of intersection.
* Opposite angles (called **vertically opposite angles**) are equal.
* Adjacent angles form **linear pairs** and add up to 180°.

### 2. Perpendicular Lines

* **Definition**: Two lines that intersect and form four right angles (90°).
* **Identification**: All four angles formed are equal.

### 3. Parallel Lines

* **Definition**: Two lines on the same plane that never intersect, no matter how far extended.
* **Real-life Examples**: Railway tracks, opposite edges of a notebook, and walls.

### 4. Identifying Relationships Between Lines

* **Using Paper Folding**:

* Opposite edges are **parallel**.
* Adjacent edges are **perpendicular**.
* **Line Notations**:

* Parallel lines: shown using arrows (>, >>).
* Perpendicular lines: marked with a small square (□) at the angle.

### 5. Transversals

* **Definition**: A line that intersects two or more lines at different points.
* **Angle Types Formed**:

* **Vertically opposite angles**
* **Corresponding angles**
* **Alternate angles**
* **Interior angles on the same side of transversal**

### 6. Corresponding Angles

* **Definition**: Angles that occupy the same relative position at each intersection.
* **Property**: Equal when the transversal intersects **parallel lines**.

### 7. Alternate Angles

* **Definition**: Angles that lie on opposite sides of the transversal but inside the two lines.
* **Property**: Equal when the lines are **parallel**.

### 8. Interior Angles on the Same Side

* **Definition**: Angles that lie on the same side of the transversal and inside the two lines.
* **Property**: Add up to 180° when the lines are **parallel**.

### 9. Drawing Parallel Lines

* **Tools Used**: Ruler, Set Square, Protractor
* **Method**:

* Draw a line.
* Use a set square to draw two perpendicular lines to it.
* These two perpendiculars are **parallel** to each other.



## New Terms and Simple Definitions

| Term | Simple Definition |
| ------------------------------- | -------------------------------------------------------------------------------- |
| **Line** | A straight path that extends in both directions with no endpoints. |
| **Intersecting Lines** | Lines that cross each other at one point. |
| **Angle** | The space between two lines that meet at a point. |
| **Linear Pair** | Two angles that are next to each other and add up to 180°. |
| **Vertically Opposite Angles** | Angles across from each other when two lines intersect, always equal. |
| **Perpendicular Lines** | Lines that meet at a right angle (90°). |
| **Parallel Lines** | Lines that never meet, even if extended far. |
| **Transversal** | A line that cuts across two or more lines. |
| **Corresponding Angles** | Angles in the same position on different lines cut by a transversal. |
| **Alternate Angles** | Angles on opposite sides of the transversal, but between the same pair of lines. |
| **Interior Angles (Same Side)** | Angles on the same side of the transversal between two lines. |

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## Practice Questions with Answers and Explanations

### 🔹 Easy (3 Questions)

**Q1.** What is the sum of angles in a linear pair?
**Answer:** 180°
**Explanation:** Linear pairs always add up to 180°.

**Q2.** Two lines intersect to form four angles. If one of the angles is 70°, what is the opposite angle?
**Answer:** 70°
**Explanation:** Vertically opposite angles are equal.

**Q3.** Identify whether the following pair of lines is parallel or not: Railway tracks.
**Answer:** Parallel
**Explanation:** Railway tracks do not meet and lie on the same plane.

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### 🔸 Medium (2 Questions)

**Q4.** Two lines intersect and form angles of 120° and 60°. What is the measure of the other two angles?
**Answer:** 120°, 60°
**Explanation:** Opposite angles are equal. So, two 120° and two 60° angles are formed.

**Q5.** A transversal intersects two lines. One pair of corresponding angles is equal. Are the lines parallel?
**Answer:** Yes
**Explanation:** Equal corresponding angles mean the lines are parallel.

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### 🔺 Difficult (3 Questions)

**Q6.** In a diagram, ∠a and ∠b are corresponding angles formed by a transversal. ∠a = 110°. What is ∠b?
**Answer:** 110°
**Explanation:** Corresponding angles are equal when the lines are parallel.

**Q7.** Two parallel lines are intersected by a transversal. One interior angle is 135°. What is its adjacent interior angle on the same side?
**Answer:** 45°
**Explanation:** Interior angles on the same side add up to 180°. So, 180° – 135° = 45°.

**Q8.** Line l is parallel to line m. A transversal t intersects both lines. If ∠3 = 50°, what is the measure of ∠6 on the same side?
**Answer:** 130°
**Explanation:** ∠3 and ∠6 are interior angles on the same side. 50° + ∠6 = 180° ⇒ ∠6 = 130°.

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### 🔻 Very Difficult (2 Questions)

**Q9.** In a figure, lines AB || CD and AD || BC. ∠DAC = 65°, ∠ADC = 60°. Find ∠CAB.
**Answer:** ∠CAB = 55°
**Explanation:** ∠DAB = 120° (because 60° + ∠DAB = 180°); ∠CAB = 120° – 65° = 55°.

**Q10.** A transversal cuts two lines. ∠a = 120°, ∠f = 70°. Are the lines parallel?
**Answer:** No
**Explanation:** Corresponding angles must be equal for lines to be parallel. 120° ≠ 70°, so lines are not parallel.

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