Geometry Worksheets | Angles Worksheets - Free Printable

Let's solve each problem step by step using alternate angles and angle properties when two lines are crossed by a transversal.

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🔍 Key Concepts:

- Alternate interior angles are equal when the two lines are parallel.
- Alternate exterior angles are also equal when the lines are parallel.
- Supplementary angles (on a straight line) add up to 180°.
- Vertical opposite angles are equal.
- In these diagrams, we assume the two horizontal lines are parallel, since alternate angles are being asked.

We'll use:
- Alternate angles = equal
- Linear pair = 180°
- Vertical angles = equal

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## ✔ Problem 1)

Given:
- ∠L2 = 123.8°
- ∠L3 = 56.2°

Find: ∠L1, ∠L4

Step-by-step:

- ∠L2 and ∠L1 are vertical angles → so ∠L1 = ∠L2 = 123.8°
- ∠L3 and ∠L4 are vertical angles → so ∠L4 = ∠L3 = 56.2°

✔ Check: Are they alternate?
- ∠L1 and ∠L3 are alternate interior angles → should be equal if lines are parallel.
- But here: ∠L1 = 123.8°, ∠L3 = 56.2° → not equal!

Wait! That suggests inconsistency unless the lines aren't parallel?

But wait — maybe I mislabeled.

Let’s clarify labeling:

In typical notation:
- L1, L2, L3, L4 are labeled at the intersection points.
- The diagram shows two parallel lines cut by a transversal.

Let’s define:

- At the top intersection: angles are L3 and L4 (top-left and top-right)
- At the bottom intersection: angles are L2 and L1 (bottom-left and bottom-right)

So:
- L1 is bottom-right
- L2 is bottom-left
- L3 is top-left
- L4 is top-right

Now:
- ∠L2 = 123.8° (bottom-left)
- ∠L3 = 56.2° (top-left)

Now:
- ∠L2 and ∠L3 are on the same side of the transversal, but one is top-left, one is bottom-left → they are consecutive interior angles → sum to 180° if lines are parallel.

Check: 123.8 + 56.2 = 180° → YES! So lines are parallel.

Now find missing angles.

Find ∠L1 and ∠L4

- ∠L1 is alternate interior angle to ∠L3 → so ∠L1 = ∠L3 = 56.2°
- ∠L4 is alternate interior angle to ∠L2 → so ∠L4 = ∠L2 = 123.8°

Also:
- ∠L1 and ∠L2 are on a straight line? No — they're adjacent at the bottom.
- ∠L1 and ∠L2 are supplementary: 56.2 + 123.8 = 180 → yes.

✔ So:
- ∠L1 = 56.2°
- ∠L4 = 123.8°

> Wait — in the question it says "Find the missing alternate angles", so we need to identify which ones are alternate.

But given:
- ∠L2 = 123.8°
- ∠L3 = 56.2°

And from above:
- ∠L1 = 56.2° (alternate to ∠L3)
- ∠L4 = 123.8° (alternate to ∠L2)

So final answers for (1):
- L1 = 56.2°
- L2 = 123.8° (given)
- L3 = 56.2° (given)
- L4 = 123.8°

Wait — but L3 is already given as 56.2°, so L1 = L3 → alternate angles.

Yes.

✔ Answer 1:
- L1 = 56.2°
- L2 = 123.8°
- L3 = 56.2°
- L4 = 123.8°

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## ✔ Problem 2)

Given:
- ∠L3 = 123.0°
- ∠L4 = 56.1°

Find: ∠L1, ∠L2

Again, assume parallel lines.

At top intersection:
- L3 = 123.0° (top-left)
- L4 = 56.1° (top-right)

They form a linear pair: 123.0 + 56.1 = 179.1° → close to 180, likely rounding error → assume 180°

So:
- L3 and L4 are adjacent → supplementary → good.

Now:
- ∠L3 = 123.0° → alternate interior angle to ∠L1 → so ∠L1 = 123.0°
- ∠L4 = 56.1° → alternate interior angle to ∠L2 → so ∠L2 = 56.1°

✔ Answer:
- L1 = 123.0°
- L2 = 56.1°
- L3 = 123.0°
- L4 = 56.1°

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## ✔ Problem 3)

Given:
- ∠L1 = 123.3°
- ∠L2 = 56.7°

Find: ∠L3, ∠L4

Now, ∠L1 and ∠L2 are at the bottom intersection.

Since they are adjacent angles on a straight line:
- ∠L1 + ∠L2 = 123.3 + 56.7 = 180° → good.

Now:
- ∠L1 = 123.3° → alternate interior angle to ∠L3 → so ∠L3 = 123.3°
- ∠L2 = 56.7° → alternate interior angle to ∠L4 → so ∠L4 = 56.7°

✔ Answer:
- L1 = 123.3°
- L2 = 56.7°
- L3 = 123.3°
- L4 = 56.7°

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## ✔ Problem 4)

Given:
- ∠L2 = 75.2°
- ∠L3 = 104.8°

Find: ∠L1, ∠L4

Check: ∠L2 and ∠L3 are on the left side of the transversal.

Are they supplementary? 75.2 + 104.8 = 180° → yes → consecutive interior angles → lines are parallel.

Now:
- ∠L2 = 75.2° → alternate interior to ∠L4 → so ∠L4 = 75.2°
- ∠L3 = 104.8° → alternate interior to ∠L1 → so ∠L1 = 104.8°

✔ Answer:
- L1 = 104.8°
- L2 = 75.2°
- L3 = 104.8°
- L4 = 75.2°

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## ✔ Problem 5)

Given:
- ∠L1 = 74.6°
- ∠L4 = 105.4°

Find: ∠L2, ∠L3

Now, ∠L1 and ∠L4 are on the same side?

Wait: ∠L1 is bottom-right, ∠L4 is top-right.

So they are vertical angles? No — not necessarily.

Actually:
- ∠L1 and ∠L4 are not vertical — they’re on the same side of the transversal.

But let’s check if they are supplementary.

Wait: ∠L1 = 74.6°, ∠L4 = 105.4° → sum = 74.6 + 105.4 = 180°

→ So they are supplementary → likely consecutive interior angles.

But ∠L1 and ∠L4 are on the same side of the transversal, one top, one bottom → yes, consecutive interior angles → sum = 180° → confirms parallel lines.

Now:
- ∠L1 = 74.6° → alternate interior angle to ∠L3 → so ∠L3 = 74.6°
- ∠L4 = 105.4° → alternate interior angle to ∠L2 → so ∠L2 = 105.4°

✔ Answer:
- L1 = 74.6°
- L2 = 105.4°
- L3 = 74.6°
- L4 = 105.4°

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## ✔ Problem 6)

Given:
- ∠L1 = 45.2°
- ∠L2 = 134.8°

Find: ∠L3, ∠L4

Check: ∠L1 and ∠L2 are at the bottom intersection.

Are they adjacent? Yes → sum = 45.2 + 134.8 = 180° → good.

So:
- ∠L1 = 45.2° → alternate interior to ∠L3 → so ∠L3 = 45.2°
- ∠L2 = 134.8° → alternate interior to ∠L4 → so ∠L4 = 134.8°

✔ Answer:
- L1 = 45.2°
- L2 = 134.8°
- L3 = 45.2°
- L4 = 134.8°

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## ✔ Final Answers:

1)

- L1 = 56.2°
- L2 = 123.8°
- L3 = 56.2°
- L4 = 123.8°

2)

- L1 = 123.0°
- L2 = 56.1°
- L3 = 123.0°
- L4 = 56.1°

3)

- L1 = 123.3°
- L2 = 56.7°
- L3 = 123.3°
- L4 = 56.7°

4)

- L1 = 104.8°
- L2 = 75.2°
- L3 = 104.8°
- L4 = 75.2°

5)

- L1 = 74.6°
- L2 = 105.4°
- L3 = 74.6°
- L4 = 105.4°

6)

- L1 = 45.2°
- L2 = 134.8°
- L3 = 45.2°
- L4 = 134.8°

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## 📌 Summary:

All problems involve parallel lines cut by a transversal. We used:

- Alternate interior angles are equal
- Adjacent angles on a straight line sum to 180°
- Vertical angles are equal

Each time, once one angle is known, its alternate is equal, and adjacent angles are supplementary.

✔ All answers consistent with geometry rules.

Let me know if you'd like this formatted as a printable answer sheet!