Find the measure of each indicated angle in the given geometric figures.

Let's solve each of these angle problems step by step. We'll use geometric principles such as:

- Triangle Angle Sum Theorem: The sum of the interior angles in a triangle is always 180°.
- Straight Angle: A straight line forms 180°.
- Supplementary Angles: Two angles that form a straight line add up to 180°.
- Vertical Angles: Opposite angles formed by intersecting lines are equal.
- Parallel Lines and Transversals: Corresponding, alternate interior, and same-side interior angles have specific relationships.

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1) Triangle with angles 65° and 57°

We need the third angle.

$$
\text{Third angle} = 180^\circ - 65^\circ - 57^\circ = 58^\circ
$$

✔ Answer: 58°

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2) Right triangle with one angle 40°

One angle is 90° (right angle), another is 40°.

$$
\text{Third angle} = 180^\circ - 90^\circ - 40^\circ = 50^\circ
$$

✔ Answer: 50°

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3) Triangle with angles 20° and 130°

Wait — this looks like a triangle with two given angles: 20° and 130°?

But 20° + 130° = 150° → so third angle is:

$$
180^\circ - 150^\circ = 30^\circ
$$

But wait — is 130° an interior angle? Let’s check the diagram: it shows a triangle with one angle labeled 20°, and another labeled 130°, which seems large for a triangle.

Wait — actually, looking at the shape, it appears that 130° is an exterior angle, or maybe it's adjacent to an interior angle?

But let’s look carefully: the figure shows a triangle with one vertex marked 20°, and another side extending with an angle labeled 130°. That suggests 130° is an exterior angle.

But if we assume it's just a triangle with two angles: 20° and 130°, then:

$$
180 - 20 - 130 = 30^\circ
$$

So the missing angle is 30°.

But wait — 130° is already very large; possible, but let’s verify.

Alternatively, perhaps 130° is an exterior angle adjacent to one of the triangle’s interior angles.

But in this case, since it's drawn inside the triangle, and the triangle has three angles, and only two are labeled: 20° and 130°, we can proceed:

$$
180 - 20 - 130 = 30^\circ
$$

✔ Answer: 30°

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4) Triangle with angles 85° and 50°

$$
\text{Third angle} = 180^\circ - 85^\circ - 50^\circ = 45^\circ
$$

✔ Answer: 45°

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5) Triangle with angles 137° and 102°, and a question mark

This one is tricky. There are two external angles shown: 137° and 102°, and we're to find the third angle (likely interior).

But note: the angles shown may be exterior angles.

Let’s suppose the triangle has three vertices, and two of the exterior angles are given: 137° and 102°.

Then the interior angles at those vertices are:

- At first vertex: $ 180^\circ - 137^\circ = 43^\circ $
- At second vertex: $ 180^\circ - 102^\circ = 78^\circ $

Now, the third interior angle is:

$$
180^\circ - 43^\circ - 78^\circ = 59^\circ
$$

So the missing angle (interior) is 59°.

✔ Answer: 59°

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6) Diagram with angles 35°, 100°, and a question mark

We see two lines intersecting, forming angles. One angle is 100°, another is 35°, and we’re to find the missing angle.

It appears that there is a triangle formed with two known angles: 35° and 100°?

Wait — no. Let’s interpret:

There is a triangle with one angle labeled 100°, and an adjacent angle outside labeled 35°.

But likely, the 35° is an exterior angle to the triangle.

Alternatively, consider the triangle: two lines intersect, and a triangle is formed.

Let’s suppose the triangle has angles: one is 100°, and another angle is formed by a 35° angle.

Wait — better idea: the 35° and 100° are not both interior angles of the same triangle.

But observe: there is a triangle where one angle is 100°, and another angle is formed by a line making 35° with a side.

Actually, this looks like a triangle with one angle being 100°, and the other two angles are related via the exterior angle.

But here’s a better interpretation: the 35° is an exterior angle to the triangle, and the triangle has an interior angle of 100°.

Wait — no. Look again.

Actually, this appears to be a triangle with one angle of 100°, and an exterior angle of 35° at another vertex.

But exterior angle = sum of opposite interior angles.

Alternatively, the triangle has two known angles: 100° and 35°?

Wait — 100° + 35° = 135° → third angle = 45°?

But that might not fit.

Wait — perhaps the 35° is not an interior angle.

Looking closely: there is a triangle with one angle labeled 100°, and a line extending from one side making a 35° angle with the extension.

That means the exterior angle is 35°, so the interior angle is $ 180^\circ - 35^\circ = 145^\circ $? But that can’t be, because triangle already has 100°, and 145° would make total over 180°.

No.

Alternative: the 35° is an interior angle, and 100° is another interior angle.

Then:

$$
\text{Third angle} = 180^\circ - 35^\circ - 100^\circ = 45^\circ
$$

But the question mark is pointing to the third angle, so yes.

But wait — the diagram shows a triangle with two sides extended, and angles labeled.

Wait — perhaps the 35° is an exterior angle, and the 100° is an interior angle.

Then, using exterior angle theorem: the exterior angle equals the sum of the two non-adjacent interior angles.

So if 35° is an exterior angle, then:

$$
35^\circ = \text{sum of two remote interior angles}
$$

But one of them is 100°? No, that’s impossible — 100° > 35°.

So 35° cannot be an exterior angle if one interior angle is 100°.

So 35° must be an interior angle, and 100° is another.

Then:

$$
\text{Third angle} = 180 - 35 - 100 = 45^\circ
$$

✔ Answer: 45°

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7) Two parallel lines cut by a transversal, with angles 30° and 20°

We see two lines: one horizontal, one slanted, intersecting at a point.

There is a triangle formed with angles 30° and 20°, and we need to find the missing angle.

Wait — the diagram shows:

- A horizontal line.
- A diagonal line crossing it.
- An angle of 30° on the left, and 20° on the right side of the triangle.

Wait — perhaps it's a triangle with two angles: 30° and 20°?

Then:

$$
\text{Third angle} = 180^\circ - 30^\circ - 20^\circ = 130^\circ
$$

But the question mark is on the top angle.

Wait — but the figure shows a triangle with base angles 30° and 20°, so apex angle is:

$$
180 - 30 - 20 = 130^\circ
$$

But 130° is large — possible.

Alternatively, is it a straight line?

Wait — the diagram shows two lines intersecting, forming a triangle.

Yes — it's a triangle with two angles: 30° and 20°, so third is:

$$
180 - 30 - 20 = 130^\circ
$$

✔ Answer: 130°

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8) Triangle with angles 155° and 60°, find missing angle

Wait — 155° and 60° — sum is 215° > 180° → impossible for interior angles.

So 155° must be an exterior angle.

Let’s suppose the triangle has an exterior angle of 155°, and an interior angle of 60°.

Then the interior angle adjacent to 155° is:

$$
180^\circ - 155^\circ = 25^\circ
$$

Now, the triangle has two interior angles: 25° and 60°.

Then the third interior angle is:

$$
180^\circ - 25^\circ - 60^\circ = 95^\circ
$$

But the question mark is likely asking for the angle opposite or the third angle.

But the diagram shows a triangle with one angle labeled 60°, and an exterior angle of 155°.

So the interior angle adjacent to 155° is 25°, and the third angle is 95°.

But the question mark is pointing to the third angle, so:

✔ Answer: 95°

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## ✔ Final Answers:

1) 58°
2) 50°
3) 30°
4) 45°
5) 59°
6) 45°
7) 130°
8) 95°

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Let me know if you'd like diagrams or further explanation!