Alternate Angles

When two parallel lines are crossed by a transversal, special pairs of angles are formed. Some of these are called alternate angles.

Alternate angles are easy to recognise once you remember one important idea:

They are on opposite sides of the transversal.

There are two types:

• Alternate interior angles — between the parallel lines
• Alternate exterior angles — outside the parallel lines

And when the two lines are parallel:$$\boxed{\text{Alternate angles are equal}}$$

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📘 Concept

Look at the diagram above.

• $l$ and $m$ are parallel lines.
• $t$ is the transversal.
• The transversal crosses both parallel lines.
• Eight angles are formed, numbered $1$ to $8$.

The parallel marks tell us:$$l\parallel m$$

Now we can identify the alternate-angle pairs.

The diagram uses matching colours to help us see which angles belong together.

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Alternate Interior Angles

📘 Concept

The word interior means inside.

So alternate interior angles are:

Angles that lie between the two parallel lines and on opposite sides of the transversal.

Look at the space between ll and mm.

The four interior angles are:$$\angle3,\quad \angle4,\quad \angle5,\quad \angle6$$

But not every pair of interior angles is an alternate pair. They must also be on opposite sides of transversal tt.

🟣 First Pair: Angles 3 and 5

Angle $3$ is:

• below line $l$
• between the parallel lines
• on the left of the transversal

Angle $5$ is:

• above line $m$
• between the parallel lines
• on the right of the transversal

They are inside the parallel lines and on opposite sides of $t$.

Therefore:$$\boxed{\angle3=\angle5}$$

The diagram shows both in purple to help you recognise the pair.

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🟢 Second Pair: Angles 4 and 6

Angle $4$ is inside the parallel lines on the right of the transversal.

Angle $6$ is inside the parallel lines on the left of the transversal.

Therefore they are also alternate interior angles:$$\boxed{\angle4=\angle6}$$

The diagram shows this pair in green.

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Alternate Exterior Angles

📘 Concept

The word exterior means outside.

So alternate exterior angles are:

Angles that lie outside the two parallel lines and on opposite sides of the transversal.

The exterior angles in our diagram are:$$\angle1,\quad\angle2,\quad\angle7,\quad\angle8$$

Again, we need to match angles on opposite sides of the transversal.

🔴 First Pair: Angles 1 and 7

Angle $1$ is:

• above line $l$
• outside the parallel lines
• on the right of the transversal

Angle $7$ is:

• below line $m$
• outside the parallel lines
• on the left of the transversal

Therefore:$$\boxed{\angle1=\angle7}$$

The diagram shows this pair in red.

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🔵 Second Pair: Angles 2 and 8

Angle $2$ is outside the parallel lines on the left of the transversal.

Angle $8$ is outside the parallel lines on the right of the transversal.

Therefore:$$\boxed{\angle2=\angle8}$$

The diagram shows this pair in blue.

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🧮 Formula / Rule

There isn’t really a formula to memorise. There is one very important angle rule:

When two parallel lines are cut by a transversal, alternate angles are equal.

For the diagram above:

Alternate Interior Angles

$$\boxed{\angle3=\angle5}$$$$\boxed{\angle4=\angle6}$$

Alternate Exterior Angles

$$\boxed{\angle1=\angle7}$$$$\boxed{\angle2=\angle8}$$

This works because:$$\boxed{l\parallel m}$$

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🧠 Easy Way to Remember

There are just two questions to ask.

Question 1: Are the angles inside or outside?

Inside the parallel lines:

Interior

Outside the parallel lines:

Exterior

Question 2: Are they on opposite sides of the transversal?

If yes, they may be an alternate pair.

So remember:

Inside + Opposite sides = Alternate Interior

Outside + Opposite sides = Alternate Exterior

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🔤 The Z-Shape Trick

Alternate angles are often remembered using a Z shape.

When you trace along the parallel lines and transversal, alternate interior angles can form the corners of a Z-like shape.

So a useful memory trick is:

Z → Alternate → Equal

But be careful! ⚠️

An exam may rotate or turn the diagram, so don’t depend only on finding a perfect letter Z.

The stronger method is:

Check whether the angles are on opposite sides of the transversal.

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💡 Exam Tip

Tip 1 — Check the Parallel Marks

Before saying alternate angles are equal, make sure the lines are parallel.

In our diagram:$$l\parallel m$$

The matching marks on $l$ and $m$ tell us this.

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Tip 2 — Find the Transversal

The line crossing both parallel lines is:$$t$$

That’s your transversal.

Now look on opposite sides of tt.

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Tip 3 — Decide Interior or Exterior

Imagine the space between $l$ and $m$ as the inside zone.

Angles:$$3,\ 4,\ 5,\ 6$$

are interior.

Angles:$$1,\ 2,\ 7,\ 8$$

are exterior.

This makes finding the correct pairs much easier.

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Tip 4 — Give the Geometrical Reason

If you calculate:$$x=68^\circ$$

and the question asks for a reason, write:

Alternate angles are equal.

Or, more specifically:

Alternate interior angles are equal.

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⚠️ Common Mistakes

Mistake 1 — Confusing Alternate with Corresponding Angles

Corresponding angles occupy the same relative position at the two intersections.

Alternate angles are on opposite sides of the transversal.

Remember:

Corresponding → same position

Alternate → opposite sides

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Mistake 2 — Choosing Two Interior Angles on the Same Side

For example:$$\angle3\text{ and }\angle6$$

are both interior angles, but they are not alternate interior angles because they lie on the same side of the transversal.

The correct partners are:$$\boxed{\angle3\leftrightarrow\angle5}$$

and:$$\boxed{\angle4\leftrightarrow\angle6}$$

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Mistake 3 — Mixing Up Interior and Exterior

Remember:

Interior = between the parallel lines

Exterior = outside the parallel lines

So:$$3,4,5,6=\text{interior}$$

and:$$1,2,7,8=\text{exterior}$$

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Mistake 4 — Adding Alternate Angles to $180^\circ$

Alternate angles are equal when the lines are parallel.

If:$$\angle3=70^\circ$$

then:$$\angle5=70^\circ$$

not $110^\circ$.

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✏️ Math

Example 1 — Alternate Interior Angles

Given:$$\angle3=65^\circ$$

Find $\angle5$.

Angles $3$ and $5$ are alternate interior angles.

Therefore:$$\angle3=\angle5$$

So:$$\boxed{\angle5=65^\circ}$$

Reason: Alternate angles are equal.

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Example 2 — Find Angle 6

Given:$$\angle4=118^\circ$$

Find $\angle6$.

From the diagram:$$\angle4=\angle6$$

because they are alternate interior angles.

Therefore:$$\boxed{\angle6=118^\circ}$$

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Example 3 — Alternate Exterior Angles

Given:$$\angle1=72^\circ$$

Find $\angle7$.

Angles $1$ and $7$ are alternate exterior angles.

Therefore:$$\angle1=\angle7$$

So:$$\boxed{\angle7=72^\circ}$$

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Example 4 — Find Angle 8

Given:$$\angle2=110^\circ$$

Find $\angle8$.

Angles $2$ and $8$ are alternate exterior angles.

Therefore:$$\boxed{\angle8=110^\circ}$$

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Example 5 — Algebra Question

Suppose:$$\angle3=3x+5^\circ$$

and:$$\angle5=80^\circ$$

Angles $3$ and $5$ are alternate interior angles, so:$$3x+5=80$$

Subtract $5$:$$3x=75$$

Divide by $3$:$$\boxed{x=25}$$

Check:$$3(25)+5=80^\circ$$

Correct. ✅

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📌 Quick Revision

For the diagram above:

🟣 Alternate Interior

$$\boxed{\angle3=\angle5}$$

🟢 Alternate Interior

$$\boxed{\angle4=\angle6}$$

🔴 Alternate Exterior

$$\boxed{\angle1=\angle7}$$

🔵 Alternate Exterior

$$\boxed{\angle2=\angle8}$$

⭐ The Rule to Remember

$$\boxed{\text{Alternate angles are equal when the lines are parallel.}}$$

And for recognising them:

Interior = Inside + Opposite sides

Exterior = Outside + Opposite sides

One final memory trick 📐

Z → Alternate → Equal

Once you can recognise inside/outside and opposite sides of the transversal, alternate-angle questions become much easier.