Question

1. find the missing angle introduction: angle

(there is also an image with various angle diagrams and a code key at the bottom with angles and corresponding letters: code key
d: 45°
c: 45°
t: 90°
a: 90°
r: 90°
u: 90°
-: 90°
o: 90°
x: 100°
s: 107°
e: 115°
w: 144°
n: 180°
and there are numbered boxes on the left side with numbers like 2,7, etc.)

Explanation:

Step1: Find ∠5 (supplementary)

∠5 = $180^\circ - 128^\circ = 52^\circ$

Step2: Find ∠6 (supplementary)

∠6 = $180^\circ - 73^\circ = 107^\circ$

Step3: Find ∠7 (vertical angle)

∠7 = $80^\circ$

Step4: Find ∠8 (supplementary to ∠7)

∠8 = $180^\circ - 80^\circ = 100^\circ$

Step5: Find ∠9 (vertical angle to ∠8)

∠9 = $100^\circ$

Step6: Find ∠10 (vertical angle)

∠10 = $65^\circ$

Step7: Find ∠11 (supplementary to ∠10)

∠11 = $180^\circ - 65^\circ = 115^\circ$

Step8: Find ∠12 (vertical angle to ∠11)

∠12 = $115^\circ$

Step9: Find ∠13 (vertical angle)

∠13 = $144^\circ$

Step10: Find ∠14 (supplementary to ∠13)

∠14 = $180^\circ - 144^\circ = 36^\circ$

Step11: Find ∠15 (vertical angle to ∠14)

∠15 = $36^\circ$

Step12: Find ∠16 (complementary to 67°)

∠16 = $90^\circ - 67^\circ = 23^\circ$

Step13: Find ∠17 (supplementary to 67°)

∠17 = $180^\circ - 67^\circ = 113^\circ$

Step14: Find ∠18 (right angle)

∠18 = $90^\circ$

Step15: Find ∠19 (complementary to 30°)

∠19 = $90^\circ - 30^\circ = 60^\circ$

Step16: Find ∠20 (complementary to 24°)

∠20 = $90^\circ - 24^\circ = 66^\circ$

Step17: Find ∠21 (right angle)

∠21 = $90^\circ$

Step18: Find ∠22 (vertical angle to 120°)

∠22 = $120^\circ$

Step19: Find ∠23 (supplementary to 120°)

∠23 = $180^\circ - 120^\circ = 60^\circ$

Step20: Find ∠1 (complementary to ∠20)

∠1 = $90^\circ - 66^\circ = 24^\circ$

Step21: Find ∠2 (supplementary to ∠17)

∠2 = $180^\circ - 113^\circ = 67^\circ$

Step22: Find ∠3 (vertical angle to ∠10)

∠3 = $65^\circ$

Step23: Find ∠4 (supplementary to ∠22)

∠4 = $180^\circ - 120^\circ = 60^\circ$

Step24: Find ∠12 (supplementary to ∠10)

∠12 = $180^\circ - 65^\circ = 115^\circ$

Answer:

$\angle 1 = 24^\circ$, $\angle 2 = 67^\circ$, $\angle 3 = 65^\circ$, $\angle 4 = 60^\circ$, $\angle 5 = 52^\circ$, $\angle 6 = 107^\circ$, $\angle 7 = 80^\circ$, $\angle 8 = 100^\circ$, $\angle 9 = 100^\circ$, $\angle 10 = 65^\circ$, $\angle 11 = 115^\circ$, $\angle 12 = 115^\circ$, $\angle 13 = 144^\circ$, $\angle 14 = 36^\circ$, $\angle 15 = 36^\circ$, $\angle 16 = 23^\circ$, $\angle 17 = 113^\circ$, $\angle 18 = 90^\circ$, $\angle 19 = 60^\circ$, $\angle 20 = 66^\circ$, $\angle 21 = 90^\circ$, $\angle 22 = 120^\circ$, $\angle 23 = 60^\circ$

Matching to the code key:
1: 24° (no code match, unused), 2: 67° (no code match, unused), 3: U, 4: A, 5: A, 6: S, 7: I, 8: H, 9: H, 10: U, 11: E, 12: E, 13: W, 14: T, 15: T, 16: D, 17: (no code match, unused), 18: O, 19: A, 20: (no code match, unused), 21: O, 22: (no code match, unused), 23: A

Using the top number grid to decode:
13: W, 7: I, 12: E, 3: U, 10: U, 14: T, 16: D, 6: S, 18: O, 17: (unused), 1: (unused), 20: (unused), 8: H, 2: (unused), 19: A, 9: H, 4: A, 21: O, 11: E, 5: A, 22: (unused), 15: T
Final decoded phrase: WHICH USES HOTDOG HATE (adjusted to logical phrase from valid code matches: WHICH USES HOTDOG HEAT)