Triangle with angles 48° and 58°, and parallel lines forming angles a°, b°, c°, and d°.

Let's solve the problem step by step using geometry principles.

We are given a triangle with two known angles:

- One angle is 48°
- Another angle is 58°

We need to find the values of angles a°, b°, c°, and d° as labeled in the diagram.

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Step 1: Use the Triangle Angle Sum Theorem

In any triangle, the sum of the interior angles is 180°.

So, for the triangle:
$$
\text{Third angle} = 180^\circ - 48^\circ - 58^\circ = 74^\circ
$$

So, the third angle inside the triangle (at the bottom vertex) is 74°.

This angle is split into two parts: b° and c°, but wait — looking at the diagram:

Actually, angle c° is the interior angle of the triangle at the bottom vertex. So:

> c° = 74°

Now we can proceed.

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Step 2: Find angle d°

Angle d° is an exterior angle to the triangle at the top-right vertex.

It forms a linear pair with the 58° angle, meaning they add up to 180°.

So:
$$
d^\circ = 180^\circ - 58^\circ = 122^\circ
$$

✔ So, d = 122°

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Step 3: Analyze the bottom line

At the bottom vertex, there is a straight line, so the angles on that line must add up to 180°.

We already found that the interior angle of the triangle is 74° (this is c°).

But this angle c° is adjacent to a° and b° — actually, from the diagram:

- c° is the interior angle of the triangle.
- a° and b° are adjacent angles forming a straight line with c°.

Wait — let's clarify.

Looking at the diagram:

- The triangle has three vertices.
- At the bottom vertex, the triangle’s interior angle is c° = 74°.
- But the line extends beyond the triangle, forming two exterior angles: a° and b°, which together with c° form a straight line?

No — actually, a° and b° are on the same side of the triangle, forming a straight line.

Wait — more carefully:

The bottom side of the triangle is part of a straight line. The triangle’s interior angle at the bottom is c° = 74°, and this angle is between two sides of the triangle.

Then, a° and b° are exterior angles formed when the sides extend outward.

But looking at the diagram:

- The side from the bottom-left to the top is extended to the left.
- The side from the bottom-right to the top is extended to the right.
- At the bottom vertex, the interior angle is c° = 74°.
- Then, a° and b° are the exterior angles on either side of the triangle.

But since the bottom line is straight, the total angle around that point is 180°.

So:
$$
a^\circ + c^\circ + b^\circ = 180^\circ
$$

But wait — no. Actually, a° and b° are on opposite sides of the triangle, and c° is the interior angle between them.

So yes, the three angles a°, c°, b° meet at a point on a straight line.

But actually, a° and b° are not both adjacent to c° unless the figure shows otherwise.

Let’s re-express based on standard interpretation.

From the diagram:

- The triangle has vertices: top, bottom-left, bottom-right.
- The top angle is between 48° and 58°, so the third angle at the bottom vertex is 74° → that's c°.
- Now, a° is the angle between the bottom side and the extension of the right side (from the triangle).
- b° is the angle between the bottom side and the extension of the left side.

So, at the bottom vertex, the full angle on the straight line is 180°.

We have:
- The interior angle of the triangle: c° = 74°
- The two exterior angles: a° and b°, which are adjacent to the extensions.

But actually, a° and b° are on the same side? No.

Wait — looking at the diagram:

There are two lines extending from the triangle:
- From the top-right side, it extends to the right → forms d°
- From the bottom-left side, it extends to the left → forms b°
- From the bottom-right side, it extends to the right → forms a°

But at the bottom vertex, the triangle's interior angle is c° = 74°, and the two exterior angles are a° and b°, which are on the opposite sides of the triangle.

But since the bottom edge is a straight line, the angles on one side must add up to 180°.

So, a° and b° are on opposite sides of the triangle, so they are not adjacent.

Wait — perhaps a° and b° are linear pairs with c°?

Let’s think differently.

At the bottom vertex, the interior angle is c° = 74°.

Then, the exterior angles on each side would be supplementary to the adjacent interior angles.

But here, a° and b° are exterior angles at the bottom vertex, formed by extending the two sides.

But the key is: the sum of the exterior angles at a vertex is not directly helpful.

Wait — better idea:

At the bottom vertex, the interior angle is c° = 74°.

Then, the exterior angles on each side (left and right) must each be supplementary to the adjacent interior angles.

But in this case, a° and b° are not the exterior angles at the bottom vertex — they are angles formed by extending the sides.

Let’s consider the straight line at the bottom.

The bottom side of the triangle is part of a straight line.

At the bottom-left corner of the triangle, the angle between the bottom side and the left side is b°, and this b° is an exterior angle.

Similarly, at the bottom-right corner, the angle between the bottom side and the right side is a°, and this is also an exterior angle.

But a° and b° are not at the same vertex — they are at different vertices.

Wait — now I see: a°, b°, c° are all at the same vertex — the bottom vertex.

Yes — because the triangle has three vertices, and c° is the interior angle at the bottom vertex.

Then, a° and b° are the two exterior angles formed by extending the two sides from that vertex.

But at a single vertex, you can only have one interior angle and two exterior angles — but the exterior angles are on opposite sides.

However, in this diagram, a° and b° appear to be on the same side of the triangle.

Wait — looking again:

The diagram shows:

- A triangle with angles 48°, 58°, and unknown (we found 74°)
- The bottom side is extended to the left and right.
- The right side of the triangle is extended to the right, forming angle a° with the bottom extension.
- The left side of the triangle is extended to the left, forming angle b° with the bottom extension.
- And c° is the interior angle at the bottom vertex.

So, at the bottom vertex, we have:
- The interior angle: c° = 74°
- The exterior angles on the left and right are b° and a°, respectively.

But a°, c°, and b° are all at the same vertex, and they lie along a straight line.

Wait — no. The bottom side is a straight line. The triangle sits above it.

So, at the bottom vertex, the interior angle c° = 74° is between the two sides of the triangle.

Then, the extensions of the two sides go off to the left and right.

So, the exterior angles at that vertex are:
- On the left: b°, between the extension of the left side and the bottom line.
- On the right: a°, between the extension of the right side and the bottom line.

But since the bottom line is straight, the total angle around the point is 180°.

So:
$$
a^\circ + c^\circ + b^\circ = 180^\circ
$$

Because a°, c°, and b° are adjacent angles that together make a straight line.

Wait — is that correct?

Actually, a° and b° are on opposite sides of the triangle, so they are not adjacent to each other.

But a° and b° are both adjacent to c°?

No — if the triangle is sitting on the bottom line, then:
- The interior angle c° is above the bottom line.
- The exterior angles a° and b° are below the bottom line, on the left and right.

But the bottom line is straight, so the angles below it must add up to 180°.

But a° and b° are not adjacent — they are separated by the triangle.

Wait — actually, a° and b° are not at the same vertex.

Ah! That's the confusion.

Let me clarify the labels:

- c° is the interior angle at the bottom vertex of the triangle.
- a° is the exterior angle at the bottom-right vertex, formed by extending the right side of the triangle.
- b° is the exterior angle at the bottom-left vertex, formed by extending the left side of the triangle.

But a° and b° are not at the same vertex — they are at different vertices.

So let's assign:

- Vertex A: top
- Vertex B: bottom-left
- Vertex C: bottom-right

Then:
- Angle at A: 48°
- Angle at C: 58°
- Angle at B: 180 - 48 - 58 = 74° → this is c°

Now, at vertex B (bottom-left), the interior angle is 74°, but the side is extended to the left, forming an exterior angle b°.

Similarly, at vertex C (bottom-right), the interior angle is 58°, and the side is extended to the right, forming a°.

Wait — but a° is shown at the bottom-right, between the bottom side and the extension of the right side.

So, a° is the exterior angle at vertex C.

And b° is the exterior angle at vertex B.

And c° is the interior angle at vertex B? Wait — no.

Wait — earlier we said the third angle is 74°, and it's at the bottom vertex.

But which one? Is it at B or C?

Let’s look:

- The triangle has:
- Top angle: 48°
- Right angle: 58°
- So the bottom angle (between left and right sides) is 74° → this is c°

So c° is the interior angle at the bottom vertex, which is the vertex where the two bottom sides meet — but there is only one bottom vertex.

Wait — no, the triangle has three vertices.

Let me label:

- Let’s say:
- Top vertex: T
- Bottom-left vertex: L
- Bottom-right vertex: R

Then:
- Angle at T: 48°
- Angle at R: 58°
- Angle at L: 180 - 48 - 58 = 74° → this is c°

So c° = 74° is at L, the bottom-left vertex.

Then:
- b° is the exterior angle at vertex L, formed by extending the left side (from L to T) to the left.
- Since the interior angle at L is 74°, and the side is extended, the exterior angle b° is supplementary to the interior angle.

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

Similarly, at vertex R, the interior angle is 58°, and the side is extended to the right, forming a°.

So:
$$
a^\circ = 180^\circ - 58^\circ = 122^\circ
$$

Wait — but earlier we had d° = 122°.

Is a° = d°?

Let’s check:

- d° is at the top-right vertex, formed by extending the top side to the right.
- The interior angle at the top-right vertex is 58°, so the exterior angle d° is:
$$
d^\circ = 180^\circ - 58^\circ = 122^\circ
$$

So d° = 122°

Now, a° is at the bottom-right vertex, and its interior angle is 58°, so:
$$
a^\circ = 180^\circ - 58^\circ = 122^\circ
$$

So a° = 122°

But d° = 122°, so a° = d° = 122°

And b° = 180° - 74° = 106°

And c° = 74°

So final answers:

- a° = 122°
- b° = 106°
- c° = 74°
- d° = 122°

Wait — but c° is the interior angle at the bottom-left vertex, which is 74°, yes.

But earlier I thought c° was at the bottom vertex, but there are two bottom vertices.

But in the diagram, c° is labeled at the bottom vertex, and it's the angle between the two bottom sides — but there is only one such vertex.

Wait — no, the triangle has only one vertex at the bottom? No, it has two: bottom-left and bottom-right.

But c° is labeled at the bottom-left vertex, between the bottom side and the left side.

Yes.

So summary:

- c° = interior angle at bottom-left = 74°
- b° = exterior angle at bottom-left = 180° - 74° = 106°
- a° = exterior angle at bottom-right = 180° - 58° = 122°
- d° = exterior angle at top-right = 180° - 58° = 122°

So:
- a = 122
- b = 106
- c = 74
- d = 122

But wait — is a° really the exterior angle at bottom-right?

Let’s double-check the diagram.

The diagram shows:

- A triangle with:
- Left angle: 48°
- Right angle: 58°
- So bottom angle: 74° → this is c°

- The bottom side is horizontal.
- The left side is extended to the left, forming b° with the bottom side.
- The right side is extended to the right, forming a° with the bottom side.
- The top side is extended to the right, forming d° with the top side.

So:
- At bottom-left vertex: interior angle = 74° = c°
- The side is extended to the left → the exterior angle b° is between the extension and the bottom side.
- So b° = 180° - 74° = 106°
- At bottom-right vertex: interior angle = 58°
- The side is extended to the right → the exterior angle a° is between the extension and the bottom side.
- So a° = 180° - 58° = 122°
- At top-right vertex: interior angle = 58°
- The top side is extended to the right → the exterior angle d° is between the extension and the right side.
- So d° = 180° - 58° = 122°

So all consistent.

Therefore:

Final Answers:

- a° = 122°
- b° = 106°
- c° = 74°
- d° = 122°

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✔ Solution:

$$
\boxed{
\begin{aligned}
a^\circ &= 122^\circ \\
b^\circ &= 106^\circ \\
c^\circ &= 74^\circ \\
d^\circ &= 122^\circ
\end{aligned}
}
$$

Explanation:

- The third interior angle of the triangle is $180^\circ - 48^\circ - 58^\circ = 74^\circ$, which is c°.
- b° is the exterior angle at the bottom-left vertex: $180^\circ - 74^\circ = 106^\circ$.
- a° is the exterior angle at the bottom-right vertex: $180^\circ - 58^\circ = 122^\circ$.
- d° is the exterior angle at the top-right vertex: $180^\circ - 58^\circ = 122^\circ$.

All exterior angles are supplementary to their adjacent interior angles.