Answer: ∠A = 55° (acute)
∠B = 105° (obtuse)
∠C = 86° (acute)
∠D = 114° (obtuse)
Step-by-step explanation:
Use Law of Cosines: a² = b² + c² - 2bc · cosA for each of the angles.
Note that ∠B from each triangle will have to be added to solve for ∠B.
Similarly for ∠D.
For ΔDAB: a = 9.1, b = 7.3, d = 11
9.1² = 7.3² + 11² - 2(7.3)(11) · cosA
82.81 = 53.29 + 121 - 160.6 · cosA
-91.48 = -160.6 cosA
0.5696 = cos A
55° = A
b² = a² + c² - 2ac · cosB
7.3² = 9.1² + 11² - 2(9.1)(11) · cosB
53.29 = 82.81 + 121 - 200.2 · cosB
-150.52 = -200.2 cosB
0.7518 = cosB
41° = B
ΔDAB: ∠A + ∠B + ∠D = 180°
55° + 41° + ∠D = 180°
96° + ∠D = 180°
∠D = 84°
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For ΔBCD: b = 8.2, c = 9.1, d = 4.6
b² = c² + d² - 2cd · cosB
8.2² = 9.1² + 4.6² - 2(9.1)(4.6) · cosB
67.24 = 82.81 + 21.16 - 83.72 · cosB
-36.73 = -83.72 cosB
0.4387 = cosB
64° = B ∠B in ABCD = 41° + 64° = 105°
c² = b² + d² - 2bd · cosC
9.1² = 8.2² + 4.6² - 2(8.2)(4.6) · cosC
82.81 = 67.24 + 21.16 - 75.44 · cosC
-5.59 = -75.44 cosC
0.074 = cosC
86° = C
d² = b² + c² - 2bc · cosD
4.6² = 8.2² + 9.1² - 2(8.2)(9.1) · cosD
21.16 = 67.24 + 82.81 - 149.24 · cosD
-128.89 = -149.24 cosD
0.8636 = cosD
30° = D ∠D in ABCD = 84° + 30° = 114°
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Acute angles are those that are less than 90° --> ∠A & ∠C
Obtuse angles are those that are greater than 90° --> ∠B & ∠D
