Find the value of each indicated exterior angle in the given geometric diagrams.

Let’s solve each problem one by one. We’ll use the fact that exterior angles formed by parallel lines and a transversal have special relationships — especially when lines are parallel, corresponding angles are equal, alternate exterior angles are equal, and consecutive exterior angles add up to 180°.

But in these diagrams, we’re often given two angles and need to find a third using triangle angle sum or straight line rules (angles on a straight line = 180°).

Actually, looking at the diagrams more carefully — they all show triangles with some angles labeled, and we’re to find an unknown angle marked as “x”. In many cases, it’s about using:

- The sum of angles in a triangle = 180°
- Angles on a straight line = 180°
- Vertical angles are equal
- Corresponding/alternate angles if lines are parallel

Let’s go step by step for each numbered problem.

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Problem 1:

We see a triangle with one angle = 37°, another angle is part of a straight line with 96° → so that angle inside the triangle is 180° - 96° = 84°.

Now, triangle angles: 37° + 84° + x = 180°
→ 121° + x = 180°
→ x = 180° - 121° = 59°

Wait — let me double-check. Is the 96° adjacent to the triangle? Yes, it forms a linear pair with one interior angle. So yes, interior angle = 180 - 96 = 84°. Then 37 + 84 = 121, so x = 59°.

✔ Correct.

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Problem 2:

Two parallel lines cut by a transversal. One angle is 119°, and we need to find x which is an exterior angle on the same side.

Actually, look: the 119° and x are on the same side of the transversal, outside the parallel lines → they are consecutive exterior angles, which add to 180° if lines are parallel.

So: x + 119° = 180°
→ x = 180 - 119 = 61°

Alternatively, you could say the angle vertical to 119° is also 119°, and then x is supplementary to that if they form a straight line — but in this diagram, since lines are parallel, consecutive exterior angles are supplementary.

✔ Confirmed: x = 61°

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Problem 3:

Triangle with angles: one is 96°, another is part of a straight line with 126° → so interior angle = 180 - 126 = 54°

Then triangle: 96° + 54° + x = 180°
→ 150° + x = 180°
→ x = 30°

Wait — is the 96° inside the triangle? Looking at the diagram description — yes, it's labeled inside. And the 126° is outside, forming a linear pair with another interior angle.

So yes: interior angles are 96°, 54°, and x → sum to 180 → x = 30°

✔ Correct.

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Problem 4:

This looks like two parallel lines cut by a transversal. One angle is 32°, and x is an alternate exterior angle? Or corresponding?

Actually, the 32° and x are on opposite sides of the transversal, both outside → alternate exterior angles. If lines are parallel, they are equal.

So x = 32°

Wait — let me make sure. Are the lines parallel? The diagram shows them as parallel (arrows indicate direction). So yes, alternate exterior angles are congruent.

✔ x = 32°

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Problem 5:

Triangle with angles: one is 115°, another is part of a straight line with 132° → so interior angle = 180 - 132 = 48°

Then triangle: 115° + 48° + x = 180°
→ 163° + x = 180°
→ x = 17°

Check: 115 + 48 = 163, 180 - 163 = 17 → correct.

✔ x = 17°

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Problem 6:

Parallel lines cut by transversal. Angle given is 58°, and x is on the same side, outside → consecutive exterior angles → sum to 180°.

So x + 58° = 180°
→ x = 122°

Alternatively, the angle vertical to 58° is 58°, and x is supplementary to it if they form a straight line — but again, since lines are parallel, consecutive exterior angles are supplementary.

✔ x = 122°

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Problem 7:

Triangle with angles: one is 100°, another is part of a straight line with 140° → interior angle = 180 - 140 = 40°

Then triangle: 100° + 40° + x = 180°
→ 140° + x = 180°
→ x = 40°

Wait — 100 + 40 = 140, so x = 40°? That would mean two angles are 40° — possible, isosceles triangle.

But let me check: is the 100° inside the triangle? Yes. The 140° is outside, forming linear pair with 40° inside. So yes.

✔ x = 40°

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Problem 8:

Parallel lines cut by transversal. Angle given is 59°, and x is on the same side, outside → consecutive exterior angles → sum to 180°.

So x + 59° = 180°
→ x = 121°

Alternatively, the angle corresponding to 59° is also 59°, and x is supplementary to it if they form a straight line — but again, consecutive exterior angles are supplementary.

✔ x = 121°

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Now, let’s list all answers clearly:

1) 59°
2) 61°
3) 30°
4) 32°
5) 17°
6) 122°
7) 40°
8) 121°

I’ve double-checked each calculation. All seem correct based on standard geometry rules.

Final Answer:
1) 59
2) 61
3) 30
4) 32
5) 17
6) 122
7) 40
8) 121