Lines and Transversals

Pair of Lines and Transversals: Definition

In geometry, we often study the relationships between lines. When a third line intersects two other lines, a specific configuration arises, creating various angles with special names and properties. The intersecting line is called a transversal. This section introduces the concept of a pair of lines and formally defines what a transversal is.

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Pair of Lines

We begin by considering two lines lying in the same plane. Such a pair of lines can exist in one of two states:

Intersecting Lines:

Two lines in a plane are called intersecting lines if they cross each other at exactly one point. This was discussed in an earlier section.

Lines $l_1$ and $l_2$ in this figure are intersecting lines.

Parallel Lines:

Two lines in a plane are called parallel lines if they never intersect, no matter how far they are extended in either direction. A key property of parallel lines is that the perpendicular distance between them remains constant throughout.

Lines $m_1$ and $m_2$ in this figure are parallel lines ($m_1 || m_2$).

The concept of a transversal applies to both intersecting and parallel pairs of lines, but the angle relationships that arise are particularly significant when the lines are parallel.

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Definition of a Transversal

A transversal is a line that intersects two or more distinct lines at distinct points. The key here is that the transversal must cross each of the other lines at a different location.

Let's say we have two lines, $l$ and $m$. A third line, $t$, is a transversal if it intersects line $l$ at a point P and intersects line $m$ at a different point Q (where P and Q are distinct points).

The lines $l$ and $m$ that the transversal $t$ intersects can be either parallel to each other or they can be intersecting lines themselves (though the most studied case involves parallel lines).

In the diagram, line $t$ is the transversal. It intersects line $l$ at point P and line $m$ at point Q. Since P and Q are distinct points, line $t$ is a transversal to lines $l$ and $m$.

Non-examples of Transversals:

• A line that intersects two other lines at the same point is not a transversal to those two lines.
• A line that is parallel to two other lines is not a transversal to them.

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Angles Formed by a Transversal

When a transversal intersects two lines, it creates a total of eight angles. Four angles are formed around the first point of intersection, and four angles are formed around the second point of intersection.

Consider a transversal $t$ intersecting line $l$ at point P and line $m$ at point Q.

As shown in the figure, the eight angles formed are labelled $\angle 1, \angle 2, \angle 3, \angle 4$ at the intersection point P, and $\angle 5, \angle 6, \angle 7, \angle 8$ at the intersection point Q.

These eight angles are situated in specific positions relative to the transversal and the two lines ($l$ and $m$). We classify these angles into different pairs based on these positions. Understanding these classifications is absolutely crucial for studying the properties that arise when the lines intersected by the transversal are parallel. The main types of angle pairs are:

• Interior Angles
• Exterior Angles
• Corresponding Angles
• Alternate Interior Angles
• Alternate Exterior Angles
• Consecutive Interior Angles (or Co-interior Angles)

We will define and explore these specific pairs of angles in the next section, followed by their properties when the intersected lines are parallel.

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Example

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Angles Formed by a Transversal (Corresponding, Alternate Interior/Exterior, Consecutive Interior)

When a transversal line intersects two other lines, a set of eight angles is created at the two points of intersection. These angles are named based on their relative positions to the two lines and the transversal. Understanding these specific names and locations is crucial, as these angle pairs have special properties, particularly when the two lines being intersected are parallel.

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Classification of Angles by Position

Let's consider two lines, $l$ and $m$, intersected by a transversal line $t$. The two intersection points create a total of eight angles, as shown in the diagram below, numbered from 1 to 8 for easy reference.

These eight angles can be broadly classified based on whether they lie between the two lines ($l$ and $m$) or outside them.

1. Interior Angles

The angles that lie in the region between the two lines ($l$ and $m$) are called interior angles.

In the diagram, angles $\angle 3, \angle 4, \angle 5$, and $\angle 6$ are the interior angles.

2. Exterior Angles

The angles that lie in the region outside the two lines ($l$ and $m$) are called exterior angles.

In the diagram, angles $\angle 1, \angle 2, \angle 7$, and $\angle 8$ are the exterior angles.

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Pairs of Angles

More significantly, the angles formed by a transversal are grouped into pairs based on their relative positions. These specific pairings have important properties when the intersected lines are parallel.

1. Corresponding Angles

Corresponding angles are pairs of angles that are in the same relative position at each intersection where the transversal crosses the two lines. They are located on the same side of the transversal, with one angle being an exterior angle and the other being an interior angle.

Looking at the diagram, the pairs of corresponding angles are:

• $\angle 1$ and $\angle 5$: Both are on the upper side of lines $l$ and $m$ respectively, and on the left side of the transversal $t$.
• $\angle 2$ and $\angle 6$: Both are on the upper side of lines $l$ and $m$ respectively, and on the right side of the transversal $t$.
• $\angle 4$ and $\angle 8$: Both are on the lower side of lines $l$ and $m$ respectively, and on the left side of the transversal $t$.
• $\angle 3$ and $\angle 7$: Both are on the lower side of lines $l$ and $m$ respectively, and on the right side of the transversal $t$.

Illustration of Corresponding Angles:

2. Alternate Interior Angles

Alternate interior angles are a pair of interior angles that are located on opposite sides of the transversal. They are situated between the two lines.

Looking at the diagram, the pairs of alternate interior angles are:

• $\angle 4$ and $\angle 5$: Both are interior, and $\angle 4$ is on the left of $t$ while $\angle 5$ is on the right of $t$.
• $\angle 3$ and $\angle 6$: Both are interior, and $\angle 3$ is on the right of $t$ while $\angle 6$ is on the left of $t$.

Illustration of Alternate Interior Angles:

3. Alternate Exterior Angles

Alternate exterior angles are a pair of exterior angles that are located on opposite sides of the transversal. They are situated outside the two lines.

Looking at the diagram, the pairs of alternate exterior angles are:

• $\angle 1$ and $\angle 8$: Both are exterior, and $\angle 1$ is on the left of $t$ while $\angle 8$ is on the right of $t$.
• $\angle 2$ and $\angle 7$: Both are exterior, and $\angle 2$ is on the right of $t$ while $\angle 7$ is on the left of $t$.

Illustration of Alternate Exterior Angles:

4. Consecutive Interior Angles (or Co-interior Angles / Allied Angles)

Consecutive interior angles (also known as co-interior angles or allied angles) are a pair of interior angles that are located on the same side of the transversal. They are situated between the two lines.

Looking at the diagram, the pairs of consecutive interior angles are:

• $\angle 4$ and $\angle 6$: Both are interior and on the left side of the transversal $t$.
• $\angle 3$ and $\angle 5$: Both are interior and on the right side of the transversal $t$.

Illustration of Consecutive Interior Angles:

Other Angle Pairs:

Remember that at each intersection point, we also have pairs of adjacent angles and vertically opposite angles, as discussed in previous sections.

• Adjacent angles at the upper intersection: ($\angle 1, \angle 2$), ($\angle 2, \angle 3$), ($\angle 3, \angle 4$), ($\angle 4, \angle 1$).
• Adjacent angles at the lower intersection: ($\angle 5, \angle 6$), ($\angle 6, \angle 7$), ($\angle 7, \angle 8$), ($\angle 8, \angle 5$).
• Linear pairs: The four adjacent pairs listed above also form linear pairs, summing to $180^\circ$.
• Vertically opposite angles at the upper intersection: ($\angle 1, \angle 3$), ($\angle 2, \angle 4$). These pairs are equal.
• Vertically opposite angles at the lower intersection: ($\angle 5, \angle 7$), ($\angle 6, \angle 8$). These pairs are equal.

The terms corresponding, alternate interior/exterior, and consecutive interior angles are specific angle pairs formed by a transversal intersecting two lines, and their properties are particularly important when those two lines are parallel.

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Example

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Parallel Lines and a Transversal: Properties of Angles

When a transversal intersects two lines, it creates various angle pairs with specific names based on their position. The relationships between the measures of these angle pairs become particularly significant and predictable when the two intersected lines are parallel. These relationships are fundamental theorems in Euclidean geometry and are widely used in solving problems and proving other geometric results.

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Properties when Transversal Intersects Parallel Lines

Consider two parallel lines, $l$ and $m$, intersected by a transversal line $t$. The eight angles formed at the two intersection points are numbered 1 through 8, as shown in the diagram:

When lines $l$ and $m$ are parallel ($l || m$), the following properties hold true for the angle pairs formed by the transversal $t$:

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1. Corresponding Angles Property (Axiom/Postulate)

If a transversal intersects two parallel lines, then each pair of corresponding angles is equal in measure.

This property is often taken as an axiom or postulate (a statement accepted as true without proof) in many geometry systems, particularly equivalent forms of Euclid's Parallel Postulate.

Based on the diagram, the pairs of corresponding angles are:

• $\angle 1$ and $\angle 5$
• $\angle 2$ and $\angle 6$
• $\angle 4$ and $\angle 8$
• $\angle 3$ and $\angle 7$

Thus, if $l || m$, then:

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2. Alternate Interior Angles Property (Theorem)

If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal in measure.

Based on the diagram, the pairs of alternate interior angles are:

• $\angle 4$ and $\angle 6$
• $\angle 3$ and $\angle 5$

Thus, if $l || m$, then $\text{m}\angle 4 = \text{m}\angle 6$ and $\text{m}\angle 3 = \text{m}\angle 5$.

Proof for $\text{m}\angle 4 = \text{m}\angle 6$ (assuming Corresponding Angles Axiom):

Given: Line $l$ is parallel to line $m$ ($l || m$), and transversal $t$ intersects $l$ and $m$.

To Prove: $\text{m}\angle 4 = \text{m}\angle 6$.

Proof:

Thus, alternate interior angles $\angle 4$ and $\angle 6$ are equal. Similarly, we can prove that $\text{m}\angle 3 = \text{m}\angle 5$.

Hence Proved.

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3. Alternate Exterior Angles Property (Theorem)

If a transversal intersects two parallel lines, then each pair of alternate exterior angles is equal in measure.

Based on the diagram, the pairs of alternate exterior angles are:

• $\angle 1$ and $\angle 7$
• $\angle 2$ and $\angle 8$

Thus, if $l || m$, then $\text{m}\angle 1 = \text{m}\angle 7$ and $\text{m}\angle 2 = \text{m}\angle 8$.

Proof for $\text{m}\angle 1 = \text{m}\angle 7$ (assuming Corresponding Angles Axiom):

Given: Line $l$ is parallel to line $m$ ($l || m$), and transversal $t$ intersects $l$ and $m$.

To Prove: $\text{m}\angle 1 = \text{m}\angle 7$.

Proof:

Thus, alternate exterior angles $\angle 1$ and $\angle 7$ are equal. Similarly, we can prove that $\text{m}\angle 2 = \text{m}\angle 8$.

Hence Proved.

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4. Consecutive Interior Angles Property (Theorem)

If a transversal intersects two parallel lines, then each pair of consecutive interior angles (also called Same-Side Interior Angles or Co-interior Angles) is supplementary (their sum is $180^\circ$).

Based on the diagram, the pairs of consecutive interior angles are:

• $\angle 4$ and $\angle 5$
• $\angle 3$ and $\angle 6$

Thus, if $l || m$, then $\text{m}\angle 4 + \text{m}\angle 5 = 180^\circ$ and $\text{m}\angle 3 + \text{m}\angle 6 = 180^\circ$.

Proof for $\text{m}\angle 4 + \text{m}\angle 5 = 180^\circ$ (assuming Corresponding Angles Axiom):

Given: Line $l$ is parallel to line $m$ ($l || m$), and transversal $t$ intersects $l$ and $m$.

To Prove: $\text{m}\angle 4 + \text{m}\angle 5 = 180^\circ$.

Proof:

Thus, consecutive interior angles $\angle 4$ and $\angle 5$ are supplementary. Similarly, we can prove that $\text{m}\angle 3 + \text{m}\angle 6 = 180^\circ$.

Hence Proved.

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Summary of Angle Properties with Parallel Lines

When two parallel lines are intersected by a transversal:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Consecutive interior angles are supplementary (sum to $180^\circ$).

These properties are fundamental tools for finding unknown angles in geometric figures involving parallel lines.

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Answer:

Criteria for Parallel Lines

In the previous section, we learned that if two lines are parallel, then specific relationships hold true for the angles formed by a transversal (corresponding angles are equal, alternate interior angles are equal, consecutive interior angles are supplementary, etc.). These properties are not only consequences of parallelism but also serve as conditions or tests to determine if two lines are parallel. These conditions are essentially the conversations of the theorems we discussed.

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Conditions for Two Lines to be Parallel

Consider two lines, $l$ and $m$, intersected by a transversal line $t$. If any one of the following conditions involving the angles formed by the transversal is met, then you can conclude that the lines $l$ and $m$ are parallel ($l || m$).

Using the angle numbering from the diagram:

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1. Converse of Corresponding Angles Axiom (or Postulate)

If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel.

Example: If $\text{m}\angle 1 = \text{m}\angle 5$, then line $l$ is parallel to line $m$ ($l || m$). This holds true for any pair of corresponding angles: if $\text{m}\angle 2 = \text{m}\angle 6$, or $\text{m}\angle 3 = \text{m}\angle 7$, or $\text{m}\angle 4 = \text{m}\angle 8$, then $l || m$.

This converse is often considered a fundamental postulate itself, equivalent to the Parallel Postulate, and is frequently used as the basis to prove the converses of other angle properties.

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2. Converse of Alternate Interior Angles Theorem

If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

Example: If $\text{m}\angle 3 = \text{m}\angle 6$, then line $l$ is parallel to line $m$ ($l || m$). Similarly, if $\text{m}\angle 4 = \text{m}\angle 5$, then $l || m$.

Proof (If $\text{m}\angle 4 = \text{m}\angle 5$, then $l || m$, assuming Converse of Corresponding Angles Axiom):

Given: Lines $l$ and $m$ are intersected by transversal $t$, such that $\text{m}\angle 4 = \text{m}\angle 5$.

To Prove: $l || m$.

Proof:

Now, observe the positions of $\angle 2$ and $\angle 5$. $\angle 2$ is an exterior angle, $\angle 5$ is an interior angle, both are on the right side of the transversal $t$, and are in the same relative position. Thus, $\angle 2$ and $\angle 5$ are a pair of corresponding angles.

Since a pair of corresponding angles ($\angle 2$ and $\angle 5$) is equal, by the Converse of Corresponding Angles Axiom, the lines $l$ and $m$ must be parallel.

Hence Proved.

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3. Converse of Alternate Exterior Angles Theorem

If a transversal intersects two lines such that a pair of alternate exterior angles is equal, then the two lines are parallel.

Example: If $\text{m}\angle 1 = \text{m}\angle 7$, then line $l$ is parallel to line $m$ ($l || m$). Similarly, if $\text{m}\angle 2 = \text{m}\angle 8$, then $l || m$.

Proof (If $\text{m}\angle 1 = \text{m}\angle 7$, then $l || m$, assuming Converse of Corresponding Angles Axiom):

Given: Lines $l$ and $m$ are intersected by transversal $t$, such that $\text{m}\angle 1 = \text{m}\angle 7$.

To Prove: $l || m$.

Proof:

Now, observe the positions of $\angle 3$ and $\angle 7$. $\angle 3$ is an interior angle, $\angle 7$ is an exterior angle, both are on the right side of the transversal $t$, and are in the same relative position. Thus, $\angle 3$ and $\angle 7$ are a pair of corresponding angles.

Since a pair of corresponding angles ($\angle 3$ and $\angle 7$) is equal, by the Converse of Corresponding Angles Axiom, the lines $l$ and $m$ must be parallel.

Hence Proved.

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4. Converse of Consecutive Interior Angles Theorem

If a transversal intersects two lines such that a pair of consecutive interior angles (or Co-interior/Allied angles) is supplementary (their sum is $180^\circ$), then the two lines are parallel.

Example: If $\text{m}\angle 4 + \text{m}\angle 6 = 180^\circ$, then line $l$ is parallel to line $m$ ($l || m$). Similarly, if $\text{m}\angle 3 + \text{m}\angle 5 = 180^\circ$, then $l || m$.

Proof (If $\text{m}\angle 4 + \text{m}\angle 6 = 180^\circ$, then $l || m$, assuming Converse of Corresponding Angles Axiom):

Given: Lines $l$ and $m$ are intersected by transversal $t$, such that $\text{m}\angle 4 + \text{m}\angle 6 = 180^\circ$.

To Prove: $l || m$.

Proof:

Subtract $\text{m}\angle 6$ from both sides:

Now, observe the positions of $\angle 4$ and $\angle 5$. They are alternate interior angles. Since a pair of alternate interior angles ($\angle 4$ and $\angle 5$) is equal, by the Converse of Alternate Interior Angles Theorem (which we proved using the Converse of Corresponding Angles Axiom), the lines $l$ and $m$ must be parallel.

Hence Proved.

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Summary of Criteria for Parallel Lines

Two lines intersected by a transversal are parallel if and only if any of the following conditions are met:

• A pair of corresponding angles is equal.
• A pair of alternate interior angles is equal.
• A pair of alternate exterior angles is equal.
• A pair of consecutive interior angles is supplementary.

These criteria are interchangeable. If one is true, all of them are true, and the lines are parallel. If none of them are true for any pair, the lines are not parallel (they will intersect somewhere).

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