Question

iv. instructions:
i. complete the table by filling in the correct data in each column.

\

$$\begin{tabular}{|l|l|l|} \\hline triangle & \\begin{tabular}{l} identify the cases of the given \\\\ in the triangle. write \\\\ aas, asa, sss, \\\\ ass, or sas. \\end{tabular}$$

& \

$$\begin{tabular}{l} write yes if the given \\\\ triangle can be solved \\\\ using sine law. \\end{tabular}$$

\\\\
\hline

1. find ac & & \\\\

\hline

1. find bc & & \\\\

\hline

1. find ac & & \\\\

\hline

1. find m\angle a & & \\\\

\hline

1. find m\angle b & & \\\\

\hline

1. find m\angle c & & \\\\

\hline
\end{tabular}

ii. solve the unknown in each triangle.
1.
4.

Explanation:

🆕 New Concept Discovered: Law of Sines and Cosines Cases
*Matching given triangle parts to solve for unknowns*

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Step 1: Analyze Triangle 1

* **Given information:**
* Angle \( A = 60^\circ \)
* Angle \( B = 28^\circ \)
* Side \( BC = 15\text{ yd} \) (opposite to Angle \( A \))
* **Case classification:** We are given two angles and a non-included side. This is the **AAS** (Angle-Angle-Side) case.
* **Can it be solved using the Law of Sines?** Yes. We have a known angle-side opposite pair (\( A \) and \( BC \)), which allows us to set up the Law of Sines ratio:
\[ \frac{\sin(60^\circ)}{15} = \frac{\sin(28^\circ)}{AC} \]
* **Answer:** **YES**

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Step 2: Analyze Triangle 2

* **Given information:**
* Angle \( C = 15^\circ \)
* Angle \( A = 50^\circ \)
* Side \( AB = 10\text{ yd} \) (opposite to Angle \( C \))
* **Case classification:** We are given two angles and a non-included side. This is the **AAS** (Angle-Angle-Side) case.
* **Can it be solved using the Law of Sines?** Yes. We have a known angle-side opposite pair (\( C \) and \( AB \)):
\[ \frac{\sin(15^\circ)}{10} = \frac{\sin(50^\circ)}{BC} \]
* **Answer:** **YES**

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Step 3: Analyze Triangle 3

* **Given information:**
* Angle \( B = 62^\circ \)
* Angle \( A = 38^\circ \)
* Side \( AB = 26\text{ m} \) (the side between the two angles)
* **Case classification:** We are given two angles and the included side. This is the **ASA** (Angle-Side-Angle) case.
* **Can it be solved using the Law of Sines?** Yes. We can find the third angle \( C = 180^\circ - (62^\circ + 38^\circ) = 80^\circ \), which gives us a known angle-side opposite pair to set up the Law of Sines:
\[ \frac{\sin(80^\circ)}{26} = \frac{\sin(38^\circ)}{BC} \]
* **Answer:** **YES**

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Step 4: Analyze Triangle 4

* **Given information:**
* Side \( BC = 7\text{ yd} \)
* Side \( AC = 21\text{ yd} \)
* Angle \( B = 72^\circ \) (non-included angle opposite to side \( AC \))
* **Case classification:** We are given two sides and a non-included angle. This is the **ASS** (Angle-Side-Side) case.
* **Can it be solved using the Law of Sines?** Yes. We have a known angle-side opposite pair (\( B \) and \( AC \)):
\[ \frac{\sin(72^\circ)}{21} = \frac{\sin(A)}{7} \]
* **Answer:** **YES**

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Step 5: Analyze Triangle 5

* **Given information:**
* Side \( AB = 32\text{ mi} \)
* Side \( AC = 21\text{ mi} \)
* Angle \( C = 28^\circ \) (non-included angle opposite to side \( AB \))
* **Case classification:** We are given two sides and a non-included angle. This is the **ASS** (Angle-Side-Side) case.
* **Can it be solved using the Law of Sines?** Yes. We have a known angle-side opposite pair (\( C \) and \( AB \)):
\[ \frac{\sin(28^\circ)}{32} = \frac{\sin(B)}{21} \]
* **Answer:** **YES**

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Step 6: Analyze Triangle 6

* **Given information:**
* Side \( BC = 19\text{ ft} \)
* Side \( AB = 11\text{ ft} \)
* Angle \( A = 98^\circ \) (included angle between sides \( AB \) and \( AC \))
* **Case classification:** We are given two sides and their included angle. This is the **SAS** (Side-Angle-Side) case.
* **Can it be solved using the Law of Sines?** No. We do not have any complete angle-side opposite pair to set up a Law of Sines ratio. This case must be solved first using the Law of Cosines.
* **Answer:** **NO**

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Step 7: Solve Part II, Triangle 1

* **Given:**
* Angle \( A = 35^\circ \)
* Angle \( B = 88^\circ \)
* Side \( AC = 44\text{ mm} \) (opposite to Angle \( B \))
* Unknown side \( AB = x \) (opposite to Angle \( C \))
* **Find Angle \( C \):**
\[ C = 180^\circ - (35^\circ + 88^\circ) = 57^\circ \]
* **Apply Law of Sines to find \( x \):**
\[ \frac{\sin(88^\circ)}{44} = \frac{\sin(57^\circ)}{x} \]
\[ x = \frac{44 \cdot \sin(57^\circ)}{\sin(88^\circ)} \approx \frac{44 \cdot 0.8387}{0.9994} \approx 36.93\text{ mm} \]

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Step 8: Solve Part II, Triangle 4

* **Given:**
* Angle \( G = 48^\circ \)
* Angle \( M = 61^\circ \)
* Side \( GM = 21\text{ cm} \) (opposite to Angle \( A \))
* Unknown side \( GA = x \) (opposite to Angle \( M \))
* **Find Angle \( A \):**
\[ A = 180^\circ - (48^\circ + 61^\circ) = 71^\circ \]
* **Apply Law of Sines to find \( x \):**
\[ \frac{\sin(71^\circ)}{21} = \frac{\sin(61^\circ)}{x} \]
\[ x = \frac{21 \cdot \sin(61^\circ)}{\sin(71^\circ)} \approx \frac{21 \cdot 0.8746}{0.9455} \approx 19.43\text{ cm} \]

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Answer:

Part I: Table Completion

Triangle	Case (AAS, ASA, SSS, ASS, SAS)	Can be solved using Law of Sines? (YES/NO)
2) Find BC	AAS	YES
3) Find AC	ASA	YES
4) Find \(m\angle A\)	ASS	YES
5) Find \(m\angle B\)	ASS	YES
6) Find \(m\angle C\)	SAS	NO

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Part II: Solving the Unknowns

1. \( x \approx 36.93\text{ mm} \)
2. \( x \approx 19.43\text{ cm} \)