Find Angles in Triangles and on Parallel Lines Worksheet - EdPlace - Free Printable

Let’s solve this step by step.

We have two parallel lines (shown by the arrow marks on them), and a triangle drawn between them. The angles inside the triangle are labeled: 48°, 58°, and c°. Also, there are other angles marked as a°, b°, and d° outside the triangle but connected to it.

First, let’s find angle c° — that’s the third angle inside the triangle.

In any triangle, the three interior angles add up to 180°.

So:

> 48° + 58° + c° = 180°
> 106° + c° = 180°
> c° = 180° - 106° = 74°

✔ So, c = 74

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Now, look at angle b°. It is next to the 48° angle and forms a straight line with it along the bottom parallel line.

Angles on a straight line add up to 180°.

So:

> b° + 48° = 180°
> b° = 180° - 48° = 132°

✔ So, b = 132

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Next, angle a°. It is next to angle c° (which we found is 74°) and also lies on the same straight line (the bottom parallel line).

So:

> a° + c° = 180°
> a° + 74° = 180°
> a° = 180° - 74° = 106°

✔ So, a = 106

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Finally, angle d°. This one is trickier. Notice that d° is formed where the slanted line meets the top parallel line. Since the two horizontal lines are parallel, and the slanted line is a transversal, we can use alternate interior angles or corresponding angles.

Look at angle a° (which is 106°). Angle d° is actually equal to angle a° because they are corresponding angles — both are on the same side of the transversal and in matching positions relative to the parallel lines.

Wait — let me double-check that.

Actually, looking again: angle d° is adjacent to the 58° angle on the top line. But more directly — since the lines are parallel, the angle d° should be equal to the angle that is “inside” and opposite — which is angle a°, because they are alternate exterior? Let’s think differently.

Alternatively, notice that the slanted line creates a Z-shape with the parallel lines. The angle inside the triangle at the top right is 58°, and angle d° is next to it on the top line.

But here’s a better way: consider the triangle and the parallel lines.

Angle d° and angle a° are corresponding angles — yes! Because the transversal cuts the two parallel lines, and d° is above the top line, a° is below the bottom line, but on the same side — so they are corresponding → therefore equal.

Wait — actually, no. Corresponding angles are in the same relative position. If you imagine sliding the bottom line up to match the top line, angle a° would land exactly where angle d° is. So yes — d° = a° = 106°

But wait — let’s verify using another method.

The angle next to d° on the top line is 58°, and together they form a straight line? No — not necessarily. Actually, the 58° is inside the triangle, and d° is outside, adjacent to it along the top line.

So:

On the top line, at the vertex where the 58° and d° meet, those two angles form a straight line? Let’s see — the triangle’s side goes down from that point, so yes — the 58° and d° are adjacent angles forming a straight line along the top parallel line.

Is that correct?

Looking at the diagram description: the 58° is inside the triangle, and d° is outside, sharing the same vertex on the top line. So yes — they are adjacent and form a straight line.

Therefore:

> 58° + d° = 180°
> d° = 180° - 58° = 122°

Wait — now I’m confused. Earlier I thought d° = a°, but that might be wrong.

Let me re-express.

Actually, let’s label the points mentally.

Call the top-left vertex of the triangle P, top-right Q, bottom vertex R.

At point Q (top-right), the triangle has an internal angle of 58°. The line continues to the right along the top parallel line, and angle d° is the angle between the triangle’s side and the extension of the top line to the right.

So yes — at point Q, the internal angle of the triangle is 58°, and the external angle d° is adjacent to it along the straight top line. Therefore:

> 58° + d° = 180°
> d° = 122°

That makes sense.

But then what about the parallel lines? How does that affect d°?

Actually, angle d° is not necessarily related directly to a° via correspondence unless we consider the direction.

Wait — perhaps I made a mistake earlier with a°.

Let’s recast everything clearly.

We have triangle with angles:

- Top left: 48°
- Top right: 58°
- Bottom: c° = 74° (as calculated)

Now, at the bottom vertex (let’s call it R), the triangle’s angle is 74°. The bottom line is straight, so the angles on either side of the triangle’s sides at point R must add to 180° with the triangle’s angle.

Specifically:

- To the left of the triangle at R: angle b° — this is between the left side of the triangle and the bottom line extending left.
- To the right of the triangle at R: angle a° — between the right side of the triangle and the bottom line extending right.

Since the bottom line is straight, the total around point R on the bottom side is 180°.

But the triangle’s angle at R is 74°, which is *between* the two sides going up. So the angles b° and a° are on the outside.

Actually, at point R, the three angles around the point on the plane sum to 360°, but we’re only concerned with the straight line.

The key is: the bottom line is straight, so the angle between the left extension and the right extension is 180°.

The triangle occupies 74° of that space at point R, so the remaining 180° - 74° = 106° is split into b° and a°? No — that’s not right.

Actually, at point R, the two rays going up form the 74° angle of the triangle. The bottom line is horizontal. So the angle between the left ray and the bottom line to the left is b°, and between the right ray and the bottom line to the right is a°.

And since the bottom line is straight, the sum of b° + 74° + a° should be 180°? No — that would be if they were all on one side, but they are not.

Let me draw it mentally:

Imagine point R on the bottom line. From R, one line goes up-left to P, making angle b° with the bottom line to the left. Another line goes up-right to Q, making angle a° with the bottom line to the right. The angle between UP-LEFT and UP-RIGHT is the triangle’s angle at R, which is 74°.

So, the total angle from left-bottom to right-bottom passing through the triangle is b° + 74° + a° = 180°, because it's a straight line.

Yes! That’s correct.

So:

> b° + 74° + a° = 180°

But we already have from earlier:

From the left side: at point P (top-left), the triangle’s angle is 48°, and it’s adjacent to b° along the bottom line? No — at point P, the 48° is between the top line and the left side of the triangle.

I think I mixed up the locations.

Let me define:

- At the top-left vertex (call it A): the triangle has angle 48°. This angle is between the top parallel line and the left side of the triangle.
- At the top-right vertex (B): triangle angle 58°, between the top parallel line and the right side of the triangle.
- At the bottom vertex (C): triangle angle c° = 74°, between the two sides of the triangle.

Now, at vertex C (bottom), the two sides go up to A and B. The bottom line is straight. So, the angle between the left side (CA) and the bottom line to the left is b°. The angle between the right side (CB) and the bottom line to the right is a°.

And since the bottom line is straight, the sum of b° + angle ACB + a° = 180°, because they are adjacent angles on a straight line.

Angle ACB is c° = 74°, so:

> b° + 74° + a° = 180°
> b° + a° = 106° ...(equation 1)

Now, at vertex A (top-left): the triangle’s angle is 48°, between the top line and side AC. The bottom line is parallel to the top line. The side AC is a transversal.

The angle between side AC and the bottom line at C is b°. Since the lines are parallel, the alternate interior angles should be equal.

Specifically, the angle at A between the top line and AC is 48°, and the alternate interior angle at C between the bottom line and AC should be equal to that — but is it b°?

Let’s see: at A, the 48° is measured from the top line down to AC. At C, the angle b° is measured from the bottom line up to CA. Since the lines are parallel, and AC is the transversal, these are alternate interior angles — so they should be equal.

Yes! So:

> b° = 48°

But earlier I thought b° was 132° — that was wrong.

Where did I go wrong before? I assumed that b° and 48° are on a straight line, but they are not — they are at different vertices.

At vertex A, the 48° is inside the triangle, between the top line and side AC. At vertex C, b° is between the bottom line and side CA. Since the top and bottom lines are parallel, and AC is the transversal, then the alternate interior angles are equal: so b° = 48°.

Similarly, at vertex B, the triangle’s angle is 58°, between the top line and side BC. At vertex C, a° is between the bottom line and side CB. Again, alternate interior angles — so a° = 58°.

Then, from equation 1: b° + a° = 48° + 58° = 106°, and 106° + 74° = 180°, which checks out.

Now, what about d°? At vertex B, the triangle’s angle is 58°, between the top line and side BC. The angle d° is the angle between side BC and the extension of the top line to the right. So, at vertex B, on the top line, the angle between the left part of the top line and side BC is 58° (inside the triangle), and the angle between side BC and the right part of the top line is d°. Since the top line is straight, these two angles add to 180°.

So:

> 58° + d° = 180°
> d° = 122°

Also, we can check with parallel lines: the angle d° and the angle at C on the other side — but anyway, this seems consistent.

So to summarize:

- c° = 180° - 48° - 58° = 74°
- b° = 48° (alternate interior angle with the 48° at top-left)
- a° = 58° (alternate interior angle with the 58° at top-right)
- d° = 180° - 58° = 122° (adjacent on straight line at top-right)

But let me confirm b° and a°.

If b° is the angle at bottom-left between the bottom line and the left side of the triangle, and since the top and bottom lines are parallel, and the left side is a transversal, then the alternate interior angle to the 48° at top-left is indeed the angle at bottom-left between the bottom line and the transversal — which is b°. So yes, b° = 48°.

Similarly for a° = 58°.

And for d°, at top-right, the 58° is inside the triangle, so the adjacent angle on the straight line is d° = 180° - 58° = 122°.

Perfect.

So final values:

a = 58
b = 48
c = 74
d = 122

But let me write them in order as per the question — the problem doesn't specify which to find, but typically we need all.

In the image, it's labeled a°, b°, c°, d°, so probably we need to find all four.

So:

a = 58
b = 48
c = 74
d = 122

Now, to box the answer.

Final Answer:
a = 58, b = 48, c = 74, d = 122