Geometry worksheet with angle sum and exterior angle problems in a fun dragon-themed puzzle format.

I'll solve this geometry worksheet step by step. The worksheet is about finding angles in triangles using the angle sum property (sum of angles in a triangle = 180°) and exterior angle properties.

Let me start with Section I: Find the value of x.

Problem ①: Triangle with angles 88°, 46°, and x°.
- Sum of angles in a triangle = 180°
- So, 88 + 46 + x = 180
- 134 + x = 180
- x = 180 - 134 = 46

Problem T: Right triangle with angles 90°, 35°, and x°.
- Sum of angles in a triangle = 180°
- So, 90 + 35 + x = 180
- 125 + x = 180
- x = 180 - 125 = 55

Problem N: Triangle with angles 18°, (x+2)°, and x°.
- Sum of angles in a triangle = 180°
- So, 18 + (x+2) + x = 180
- 18 + x + 2 + x = 180
- 2x + 20 = 180
- 2x = 160
- x = 80

Problem S: Triangle with angles (x+10)°, (4x+7)°, and (2x+9)°.
- Sum of angles in a triangle = 180°
- So, (x+10) + (4x+7) + (2x+9) = 180
- x + 10 + 4x + 7 + 2x + 9 = 180
- 7x + 26 = 180
- 7x = 154
- x = 22

Now let's move to Section II: Find the missing angles in each triangle.

Problem E, H, T: Triangle ABC with angle A = (2x+8)°, angle B = x°, and exterior angle at C = 113°.
- First, I need to find x.
- The exterior angle at C equals the sum of the two opposite interior angles (angle A + angle B).
- So, 113 = (2x+8) + x
- 113 = 3x + 8
- 105 = 3x
- x = 35
- Now I can find the angles:
- ∠A = (2×35 + 8) = 70 + 8 = 78°
- ∠B = x = 35°
- ∠ACB = 180 - 113 = 67° (since it's supplementary to the exterior angle)

Problem G, T, H: Triangle WYZ with angle W = (x+8)°, angle Y = 3x°, and angle Z = 62°.
- Sum of angles in a triangle = 180°
- So, (x+8) + 3x + 62 = 180
- 4x + 70 = 180
- 4x = 110
- x = 27.5
- Wait, let me check if this is correct or if I'm misreading something. Looking at the diagram again, I think I need to reconsider.

Actually, looking more carefully at the diagram for problem G, T, H, I see that angle Z is given as 62°, and we have angles W = (x+8)° and Y = 3x°. Let me recalculate:

(x+8) + 3x + 62 = 180
4x + 70 = 180
4x = 110
x = 27.5

This gives us:
- ∠W = (27.5 + 8) = 35.5°
- ∠Y = 3 × 27.5 = 82.5°
- ∠WYZ = 62°

But these don't match the answer choices which are all whole numbers. Let me re-examine the diagram.

Looking again, I think there might be an error in my interpretation. Let me check the other problems first.

Problem G, Y, H: Triangle JGH with angle J = 18°, angle H = 62°, and angle G = 2x°.
- Sum of angles in a triangle = 180°
- So, 18 + 62 + 2x = 180
- 80 + 2x = 180
- 2x = 100
- x = 50
- Therefore, ∠JGH = 2x = 100°

Problem K, F, I: Triangle EGF with angle E = 3x°, angle F = (x+12)°, and angle G = 2x°.
- Sum of angles in a triangle = 180°
- So, 3x + (x+12) + 2x = 180
- 6x + 12 = 180
- 6x = 168
- x = 28
- Therefore:
- ∠E = 3x = 84°
- ∠F = x+12 = 40°
- ∠EGF = 2x = 56°

Now going back to problem G, T, H (Triangle WYZ), I think I need to reconsider. The diagram shows angle Z as 62°, and angles W and Y as (x+8)° and 3x° respectively. My calculation gave x=27.5, but since the answer choices are whole numbers, maybe I'm misinterpreting the diagram.

Let me look at the answer choices provided at the bottom: 60, 80, 78, 118, 114, 40, 46, 44, 35, 67, 43, 84, 41, 56, 38, 120, 55, 22.

From my calculations:
- Problem ①: x = 46
- Problem T: x = 55
- Problem N: x = 80
- Problem S: x = 22
- Problem E: ∠A = 78°
- Problem H: ∠B = 35°
- Problem T: ∠ACB = 67°
- Problem G: ∠JGH = 100° (but 100 isn't in the answer choices)
- Problem Y: ∠E = 84°
- Problem F: ∠F = 40°
- Problem I: ∠EGF = 56°

For problem G, T, H (Triangle WYZ), let me try a different approach. Maybe angle Z is not 62° but something else. Looking at the diagram again, I think I might have misread it.

Actually, let me reconsider problem G, T, H. The diagram shows triangle WYZ with angle W = (x+8)°, angle Y = 3x°, and angle Z = 62°. My calculation was correct: x = 27.5, giving ∠W = 35.5°, ∠Y = 82.5°, ∠Z = 62°. But since these aren't whole numbers and don't match the answer choices, perhaps there's a different interpretation.

Wait, I think I found the issue. In problem G, T, H, the diagram might be showing different information. Let me look at the answer choices again. The closest whole numbers to my calculated values are 35, 83, and 62. Since 35 is in the answer choices, maybe ∠W = 35°, which would mean x+8 = 35, so x = 27. Then ∠Y = 3x = 81°, and ∠Z = 62°, which sums to 35+81+62 = 178°, which is close to 180° but not exact.

Given the constraints of the problem and the answer choices, I think my initial calculations for most problems are correct, and for problem G, T, H, there might be a slight discrepancy in the diagram or it's intended to have decimal answers.

Let me summarize the answers based on my calculations:

Section I:
- ①: x = 46
- T: x = 55
- N: x = 80
- S: x = 22

Section II:
- E: ∠A = 78°
- H: ∠B = 35°
- T: ∠ACB = 67°
- G: ∠W = 35.5° (approximately 35° if rounding)
- T: ∠Y = 82.5° (approximately 83° if rounding)
- H: ∠WYZ = 62°
- G: ∠JGH = 100°
- Y: ∠E = 84°
- F: ∠F = 40°
- I: ∠EGF = 56°

The answer choices at the bottom seem to correspond to the values we've calculated, with some possible rounding.

Based on the calculations and matching to the answer choices:
- ①: 46
- T: 55
- N: 80
- S: 22
- E: 78
- H: 35
- T: 67
- G: 35 (rounded from 35.5)
- T: 83 (rounded from 82.5, but 83 isn't in the choices; 84 is, which is for another problem)
- H: 62
- G: 100 (not in choices)
- Y: 84
- F: 40
- I: 56

There seems to be a mismatch for some problems. Let me recheck problem G, T, H one more time.

Actually, looking at the diagram for problem G, T, H, I think I may have misidentified the angles. Let me assume that the 62° is not angle Z but perhaps an exterior angle or something else. Without being able to see the diagram more clearly, I'll stick with my initial calculations for most problems and note the discrepancy for problem G, T, H.

The final answers based on my calculations are:
- ①: 46
- T: 55
- N: 80
- S: 22
- E: 78
- H: 35
- T: 67
- G: 35.5 (or approximately 35)
- T: 82.5 (or approximately 83)
- H: 62
- G: 100
- Y: 84
- F: 40
- I: 56

Since the answer choices are limited, I believe the intended answers are the whole numbers that match our calculations where possible.