Question

iii. materials needed:
worksheet # 1
calculator
iv. instructions:

1. complete the table by filling in the correct data in each column.

\

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

& \

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

\\\\
\hline
\

$$\begin{tabular}{l} 1) find ac \\\\ \\\\ image of triangle abc with angle a = 92 degrees, angle b = 28 degrees, and side bc = 15 yd \\end{tabular}$$

& & \\\\
\hline
\

$$\begin{tabular}{l} 2) find bc \\\\ \\\\ image of triangle abc with angle c = 15 degrees, angle a = 59 degrees, and side ab = 10 yd \\end{tabular}$$

& & \\\\
\hline
\end{tabular}

Explanation:

🆕 New Concept Discovered: Law of Sine Solvability
Determining if a triangle can be solved using the Law of Sines based on given parts.

Step 1: Analyze Triangle 1

Let's look at the given information for the first triangle:

• We are given two angles: \(\angle A = 92^\circ\) and \(\angle B = 28^\circ\).
• We are given one side: \(BC = 15\text{ yd}\), which is opposite to \(\angle A\).

Since the given side is opposite to one of the known angles rather than included between them, this configuration is Angle-Angle-Side (AAS).

To determine if we can use the Law of Sines:

• The Law of Sines requires at least one known angle and its opposite side to set up a ratio:

$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$

• Here, we know angle \(A = 92^\circ\) and its opposite side \(a = 15\text{ yd}\). This gives us a complete ratio \(\frac{15}{\sin 92^\circ}\).
• We can easily find the remaining side \(b\) (which is \(AC\)) using:

$$ \frac{AC}{\sin 28^\circ} = \frac{15}{\sin 92^\circ} $$

• Therefore, this triangle can be solved using the Law of Sines. We write YES.

Step 2: Analyze Triangle 2

Let's look at the given information for the second triangle:

• We are given two angles: \(\angle C = 15^\circ\) and \(\angle A = 59^\circ\).
• We are given one side: \(AB = 10\text{ yd}\), which is opposite to \(\angle C\).

Since the given side is opposite to one of the known angles rather than included between them, this configuration is also Angle-Angle-Side (AAS).

To determine if we can use the Law of Sines:

• We know angle \(C = 15^\circ\) and its opposite side \(c = 10\text{ yd}\). This gives us a complete ratio \(\frac{10}{\sin 15^\circ}\).
• We can find the remaining side \(a\) (which is \(BC\)) using:

$$ \frac{BC}{\sin 59^\circ} = \frac{10}{\sin 15^\circ} $$

• Therefore, this triangle can be solved using the Law of Sines. We write YES.

Answer:

Triangle	Identify the case (AAS, ASA, SSS, ASS, or SAS)	Write YES if the given triangle can be solved using Sine Law
2) Find BC	AAS	YES