Diagrams showing triangles with interior and exterior angles, using the exterior angle theorem to find unknown angles.

Let’s solve each triangle problem one by one. We’ll use two key rules:

1. The angles inside a triangle always add up to 180°.
2. An exterior angle (like x in these problems) is equal to the sum of the two opposite interior angles.
OR — you can find the third interior angle first, then subtract from 180° to get the exterior angle (since they form a straight line).

We’ll go step by step for each figure.

---

(I)
Interior angles: 50° and 70°
Third interior angle = 180° - (50° + 70°) = 180° - 120° = 60°
Exterior angle x is next to this 60° angle → so x = 180° - 60° = 120°
*(Or directly: x = 50° + 70° = 120° — same answer!)*

✔ x = 120°

---

(II)
Interior angles: 65° and 45°
Third interior angle = 180° - (65° + 45°) = 180° - 110° = 70°
Exterior angle x is next to this 70° angle → x = 180° - 70° = 110°
*(Or: x = 65° + 45° = 110°)*

✔ x = 110°

---

(III)
This one looks different — it’s not a full triangle with an exterior angle on the base. Let’s look carefully.

We have a triangle with angles 30° and 40°. The third angle inside the triangle is:
180° - (30° + 40°) = 110°

Now, angle x is shown as the angle between the extended side and the other side — actually, looking at the diagram, x is the *exterior* angle at the top vertex? Wait — no.

Actually, in diagram (iii), the angle marked x is formed by extending one side, and it's adjacent to the 110° angle we just found? No — let me re-express.

Wait — better way: In any triangle, if you extend one side, the exterior angle equals the sum of the two non-adjacent interior angles.

In (iii), the two given interior angles are 30° and 40°, and x is the exterior angle at the third vertex — so yes, x = 30° + 40° = 70°

But wait — that doesn’t match the drawing? Let me double-check.

Actually, looking again: In diagram (iii), the angle labeled x is NOT the exterior angle of the triangle — it’s the angle between the extension and the side — but actually, it IS the exterior angle corresponding to the vertex where the 30° and 40° are NOT located.

Standard rule: Exterior angle = sum of two remote interior angles.

So if the triangle has angles A=30°, B=40°, C=?, then exterior at C = A + B = 70°.

Yes — so x = 70°.

But wait — in the diagram, the angle x appears to be obtuse? Hmm — maybe I misread.

Alternative approach: Maybe the 30° and 40° are not both interior? Let me think differently.

Actually, upon closer inspection of typical textbook diagrams like this, in (iii), the angle x is the exterior angle at the top, and the two given angles (30° and 40°) are the two remote interior angles — so x = 30° + 40° = 70°.

But visually, if you draw it, extending the left side, the exterior angle should be large... unless...

Wait — perhaps the 30° and 40° are on the bottom, and x is at the top right? Actually, standard interpretation: in such diagrams, when two angles are given inside the triangle and x is outside at the third vertex, x = sum of the two given.

But let’s calculate the third interior angle: 180 - 30 - 40 = 110°. Then the exterior angle adjacent to it would be 180 - 110 = 70°. So yes, x = 70°.

Even though it might look small in the sketch, mathematically it’s correct.

✔ x = 70°

---

(IV)
Triangle with two 60° angles → equilateral? Third angle = 180 - 60 - 60 = 60°
Exterior angle x is next to one of the 60° angles → x = 180 - 60 = 120°
*(Or: x = 60° + 60° = 120° — since the two remote interior angles are both 60°)*

✔ x = 120°

---

(V)
Interior angles: 50° and 50°
Third interior angle = 180 - 50 - 50 = 80°
Exterior angle x is next to this 80° angle → x = 180 - 80 = 100°
*(Or: x = 50° + 50° = 100°)*

✔ x = 100°

---

(VI)
This one is tricky. It shows a triangle with angles 30° and 60°, and x is an exterior angle — but which one?

Looking at the diagram: there’s a vertical line, and the triangle is attached. The angle marked x is outside, adjacent to the 60° angle? Or is it related to the 30°?

Actually, let’s identify the triangle: it has angles 30° and 60°, so third angle = 180 - 30 - 60 = 90°.

Now, angle x is shown as the exterior angle at the vertex with the 60° angle? But in the diagram, it seems x is formed by extending the side opposite the 30° angle.

Better: Use the exterior angle theorem. If x is the exterior angle at the vertex where the 60° is, then x = 30° + 90° = 120°? But we don’t know the 90° yet.

Wait — let’s label:

Assume the triangle has:
- Bottom-left angle: 30°
- Bottom-right angle: 60°
- Top angle: 90° (since 30+60+90=180)

Now, angle x is drawn at the bottom-right vertex, outside the triangle, formed by extending the bottom side to the right. So x is adjacent to the 60° angle → so x = 180° - 60° = 120°.

Alternatively, using exterior angle theorem: x = sum of the two remote interior angles = 30° + 90° = 120°.

Same answer.

✔ x = 120°

---

Final check for all:

(I) 50+70=120 → x=120
(II) 65+45=110 → x=110
(III) 30+40=70 → x=70
(IV) 60+60=120 → x=120
(V) 50+50=100 → x=100
(VI) 30+90=120 or 180-60=120 → x=120

All consistent.

Final Answer:
(I) 120°
(II) 110°
(III) 70°
(IV) 120°
(V) 100°
(VI) 120°