Consider the following statements in respect of three straight lines A, B and C on a plane: 1. If A and C are parallel and B and C are parallel, then A and B are parallel 2. If A is perpendicular to C and B is perpendicular to C, then A and B are parallel 3. If the acute angle between A and C is equal to the acute angle between B and C; then A and B are parallel
1 and 2 only
Understanding Line Relationships on a Plane
This question asks us to evaluate three statements about the relationships between three straight lines, A, B, and C, on a plane. The relationships involve parallelism and perpendicularity. Let's analyze each statement carefully based on fundamental geometry principles.
Analyzing Statement 1: Parallelism Transitivity
The first statement says: "If A and C are parallel and B and C are parallel, then A and B are parallel".
- If line A is parallel to line C (\(A \parallel C\)), it means they lie in the same plane and never intersect.
- If line B is parallel to line C (\(B \parallel C\)), it means they also lie in the same plane and never intersect C.
- Consider line C as a reference. If both A and B maintain the same distance from C everywhere, then they must maintain the same distance from each other everywhere (assuming they are distinct lines).
- This is a fundamental property in Euclidean geometry: if two lines are parallel to the same line, they are parallel to each other.
Conclusion for Statement 1: This statement is correct.
Analyzing Statement 2: Perpendicularity to a Common Line
The second statement says: "If A is perpendicular to C and B is perpendicular to C, then A and B are parallel".
- If line A is perpendicular to line C (\(A \perp C\)), the angle between A and C is 90 degrees.
- If line B is perpendicular to line C (\(B \perp C\)), the angle between B and C is also 90 degrees.
- Imagine line C is horizontal. Any line perpendicular to a horizontal line must be vertical.
- If both line A and line B are vertical lines on the same plane, they must be parallel to each other.
- This is another fundamental property: two lines perpendicular to the same line are parallel to each other.
Conclusion for Statement 2: This statement is correct.
Analyzing Statement 3: Equal Acute Angles
The third statement says: "If the acute angle between A and C is equal to the acute angle between B and C; then A and B are parallel".
- Let the acute angle between A and C be \(\theta\). So, \(0^\circ < \theta \le 90^\circ\).
- Let the acute angle between B and C also be \(\theta\).
- Consider line C as the x-axis. Line A could make an angle \(\theta\) with the x-axis. Its slope would be \(\tan(\theta)\).
- Line B could also make an angle \(\theta\) with the x-axis. Its slope could be \(\tan(\theta)\) (in which case A and B would be parallel, or the same line if they share a point), OR its slope could be \(\tan(-\theta) = -\tan(\theta)\).
- If the slope of A is \(m\) and the slope of B is \(-m\) (where \(m = \tan(\theta)\)), and \(m \neq 0\), then lines A and B are not parallel; they will intersect. For example, if \(\theta = 45^\circ\), line A could be \(y = x\) and line B could be \(y = -x\). Both make a 45-degree acute angle with the x-axis (line C), but they are not parallel; they are perpendicular in this specific case, intersecting at the origin.
- Therefore, having equal acute angles with a third line C does not guarantee that lines A and B are parallel. They could be intersecting lines that are symmetric with respect to line C.
Conclusion for Statement 3: This statement is incorrect.
Summary of Statement Analysis
| Statement | Condition | Conclusion | Correct? |
|---|---|---|---|
| 1 | A || C and B || C | A || B | Yes |
| 2 | A ⊥ C and B ⊥ C | A || B | Yes |
| 3 | Acute angle between A and C equals acute angle between B and C | A || B | No |
Based on our analysis, statements 1 and 2 are correct, while statement 3 is incorrect.
Revision Table: Key Line Relationships
| Relationship | Description | Property with a Third Line C |
|---|---|---|
| Parallel Lines | Lines in the same plane that never intersect. | If A || C and B || C, then A || B. |
| Perpendicular Lines | Lines that intersect at a 90-degree angle. | If A ⊥ C and B ⊥ C, then A || B. |
| Intersecting Lines | Lines that cross at a single point. | Acute angles with C being equal does not imply A || B. |
Additional Information on Plane Geometry
In plane geometry, also known as Euclidean geometry, the relationships between lines are fundamental. Lines can be parallel, perpendicular, or intersecting. These relationships are defined by the angles they form with each other or with transversal lines (lines that intersect two or more other lines).
- Parallel Postulate: A key concept related to parallel lines is the parallel postulate. In Euclidean geometry, it states that through a point not on a given line, there is exactly one line parallel to the given line.
- Transversal Lines: When a transversal line intersects two other lines, it creates various angles (corresponding angles, alternate interior angles, consecutive interior angles, etc.). The properties of these angles help determine if the two lines are parallel. For instance, if corresponding angles are equal, the lines are parallel.
- Angles between Lines: The angle between two non-parallel lines is typically defined as the smaller (acute or right) angle formed at their intersection.
Understanding these basic concepts helps in analyzing the relative positions and orientations of lines on a plane.
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Two parallel lines are intersected by a transversal, the corresponding angles are:
If l, m, n are lines such that, l is parallel to n and m is parallel to n, then, ______
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