Geometry worksheet for calculating angles on parallel lines with clues and reasoning.

Let's solve each of the problems on the worksheet step by step, using angle properties involving parallel lines, triangles, and angle sums. We’ll use geometric reasoning to find each missing angle and give a valid reason.

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🔹 Question 1: Angle a

Given:
- Two parallel lines (indicated by arrows).
- A transversal cuts them.
- One angle is $72^\circ$, and we need to find $a^\circ$.

Observation:
The $72^\circ$ angle and angle $a^\circ$ are interior angles on the same side of the transversal (also known as consecutive interior angles or co-interior angles).

> ✔ Rule: Co-interior angles on parallel lines add up to $180^\circ$.

So:
$$
a + 72^\circ = 180^\circ \\
a = 180^\circ - 72^\circ = 108^\circ
$$

✔ Answer:
- Angle a: $108^\circ$
- Reason: Co-interior angles on parallel lines are supplementary (add to $180^\circ$).

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🔹 Question 2: Angle b

Given:
- A parallelogram-like shape with two parallel sides.
- One angle is $67^\circ$, and we need to find $b^\circ$.

Observation:
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary.

But here, it’s not labeled as a parallelogram, but the opposite sides have arrows indicating they are parallel.

So, since the lines are parallel, and the angles are adjacent along the transversal:

- The $67^\circ$ and angle $b^\circ$ are alternate interior angles? Wait — let’s look carefully.

Actually, the $67^\circ$ and $b^\circ$ are on the same side of the transversal, between the parallel lines → co-interior angles.

Wait — no! Let's see: the $67^\circ$ and $b^\circ$ are on the same side of the transversal, but one is inside, one is outside?

Wait — actually, in this figure, both angles are inside the parallel lines, on the same side of the transversal → so again, co-interior angles.

Thus:
$$
b + 67^\circ = 180^\circ \Rightarrow b = 113^\circ
$$

✔ Answer:
- Angle b: $113^\circ$
- Reason: Co-interior angles on parallel lines are supplementary (add to $180^\circ$).

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🔹 Question 3: Angle c

Given:
A quadrilateral with angles $126^\circ$, $108^\circ$, $52^\circ$, and an exterior angle $c^\circ$ marked at the bottom right.

We need to find $c^\circ$, which is an exterior angle at the bottom-right vertex.

First, let's find the interior angle at that vertex.

Sum of interior angles in a quadrilateral: $360^\circ$

So:
$$
\text{Interior angle} = 360^\circ - (126^\circ + 108^\circ + 52^\circ) = 360^\circ - 286^\circ = 74^\circ
$$

Now, angle $c^\circ$ is the exterior angle adjacent to this $74^\circ$ interior angle.

So:
$$
c = 180^\circ - 74^\circ = 106^\circ
$$

✔ Answer:
- Angle c: $106^\circ$
- Reason: Exterior angle is supplementary to the adjacent interior angle (they form a straight line).

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🔹 Question 4: Angle d

Given:
A triangle with angles $68^\circ$ and $75^\circ$. There’s a line extending from the top vertex, forming angle $d^\circ$ with the top horizontal line.

The top and bottom lines are parallel, and the triangle is between them.

We can find the third angle of the triangle first:

Sum of angles in a triangle = $180^\circ$

So:
$$
\text{Third angle} = 180^\circ - 68^\circ - 75^\circ = 37^\circ
$$

Now, angle $d^\circ$ is formed where the triangle's top side meets the upper parallel line. Since the upper line is parallel to the lower one, and the triangle's side acts as a transversal, angle $d^\circ$ is corresponding to the $37^\circ$ angle?

Wait — let's think.

Actually, the $37^\circ$ angle is at the top vertex of the triangle. The line extends horizontally from there.

So angle $d^\circ$ is the angle between the extension of the top line and the triangle's side, forming a straight line with the $37^\circ$ angle?

No — actually, angle $d^\circ$ is external to the triangle and lies on the top parallel line.

Wait — looking closely: the triangle has a side going down from the top, and the top line continues to the left. So angle $d^\circ$ is the angle between the extended top line and the triangle’s side.

But the triangle's top angle is $37^\circ$, and the top line is straight. So angle $d^\circ$ is adjacent to the $37^\circ$ angle.

So:
$$
d + 37^\circ = 180^\circ \Rightarrow d = 143^\circ
$$

Alternatively, perhaps it's corresponding or alternate?

Wait — no. The key is: the triangle’s top angle is $37^\circ$, and the top line is straight. So the angle between the triangle’s side and the top line is $37^\circ$. Then the other side of the triangle forms an angle with the top line — but $d^\circ$ is the angle above the triangle’s side.

Wait — better way: the top line is straight, and the triangle’s side makes a $37^\circ$ angle with it. Then the angle between the side and the extension is $180^\circ - 37^\circ = 143^\circ$.

So yes.

✔ Answer:
- Angle d: $143^\circ$
- Reason: Angles on a straight line add to $180^\circ$. The interior angle of the triangle at the top is $37^\circ$, so $d = 180^\circ - 37^\circ = 143^\circ$.

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🔹 Question 5: Angle e and Angle f

This diagram has a triangle with angles $39^\circ$, $f^\circ$, and another angle $81^\circ$ marked at the top.

Also, there’s a transversal crossing two parallel lines, and angle $e^\circ$ is formed at the intersection.

Let’s break it down.

#### Step 1: Find angle $f$

In the triangle, we know:
- One angle is $39^\circ$
- Another is $f^\circ$
- The third angle is $81^\circ$?

Wait — is $81^\circ$ part of the triangle?

Yes — the top angle is $81^\circ$, and the bottom-left is $39^\circ$. So sum of angles in triangle:
$$
f = 180^\circ - 81^\circ - 39^\circ = 60^\circ
$$

So $f = 60^\circ$

✔ Angle f: $60^\circ$
✔ Reason: Sum of angles in a triangle is $180^\circ$.

#### Step 2: Find angle $e$

Now, angle $e^\circ$ is formed where the diagonal line crosses the parallel lines.

Looking at the diagram: the two lines are parallel (arrows), and the diagonal is a transversal.

Angle $e^\circ$ is vertically opposite to the angle formed between the transversal and the lower line.

But also, note that angle $f = 60^\circ$ is at the bottom-right of the triangle.

Now, the transversal intersects the lower line at an angle. Is there a corresponding or alternate angle?

Wait — the angle $e^\circ$ is between the transversal and the upper parallel line, and it appears to be alternate to the $60^\circ$ angle at the bottom.

Yes — because the transversal crosses the two parallel lines, and angle $e^\circ$ is alternate interior to the $60^\circ$ angle.

So:
$$
e = 60^\circ
$$

✔ Angle e: $60^\circ$
✔ Reason: Alternate interior angles are equal when lines are parallel.

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🔹 Question 6: Angle g

Given:
A triangle with angles $74^\circ$, $21^\circ$, and an external angle $g^\circ$ at the top.

We need to find $g^\circ$, which is an exterior angle at the top vertex.

First, find the interior angle at the top of the triangle.

Sum of angles in triangle = $180^\circ$

So:
$$
\text{Top interior angle} = 180^\circ - 74^\circ - 21^\circ = 85^\circ
$$

Now, angle $g^\circ$ is the exterior angle at that vertex, so it is supplementary to the $85^\circ$ angle.

So:
$$
g = 180^\circ - 85^\circ = 95^\circ
$$

Alternatively, you can use the exterior angle theorem: the exterior angle equals the sum of the two opposite interior angles.

$$
g = 74^\circ + 21^\circ = 95^\circ
$$

✔ Answer:
- Angle g: $95^\circ$
- Reason: Exterior angle of a triangle equals the sum of the two opposite interior angles.

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## ✔ Final Answers Summary:

| Angle | Value | Reason |
|------|-------|--------|
| a | $108^\circ$ | Co-interior angles on parallel lines are supplementary |
| b | $113^\circ$ | Co-interior angles on parallel lines are supplementary |
| c | $106^\circ$ | Exterior angle is supplementary to adjacent interior angle |
| d | $143^\circ$ | Angles on a straight line add to $180^\circ$ |
| e | $60^\circ$ | Alternate interior angles are equal (parallel lines) |
| f | $60^\circ$ | Sum of angles in a triangle is $180^\circ$ |
| g | $95^\circ$ | Exterior angle equals sum of two opposite interior angles |

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