Question

3x3 systems applications situation c. the interior angles of a triangle sum to 180°. the measure of ∠a is four degrees more than the sum of ∠b and ∠c. if ∠a measures 49° more than ∠c, what is the measure of each angle? 1 declare your variables a = b = c = 2 write the system of equations eq.1: eq.2: eq.3: 3 solve the system m∠a = m∠b = m∠c = 4 explain your process which method did you use to solve the problem? explain your process.

Explanation:

Step1: Declare variables

Let \(a = m\angle A\), \(b = m\angle B\), \(c=m\angle C\)

Step2: Write the system of equations

• EQ.1: Since the sum of interior - angles of a triangle is \(180^{\circ}\), we have \(a + b + c=180\).
• EQ.2: The measure of \(\angle A\) is four degrees more than the sum of \(\angle B\) and \(\angle C\), so \(a=b + c+4\).
• EQ.3: \(\angle A\) measures \(49^{\circ}\) more than \(\angle C\), so \(a=c + 49\).

Step3: Substitute and solve

• Substitute \(a=b + c+4\) into \(a + b + c=180\). We get \((b + c+4)+b + c=180\), which simplifies to \(2b + 2c=176\), then \(b + c = 88\), so \(b=88 - c\).
• Substitute \(a=c + 49\) into \(a + b + c=180\), we have \((c + 49)+b + c=180\), which simplifies to \(b+2c=131\).
• Substitute \(b = 88 - c\) into \(b+2c=131\), we get \((88 - c)+2c=131\), \(88 + c=131\), so \(c = 43\).
• Since \(a=c + 49\), then \(a=43 + 49=92\).
• Since \(b=88 - c\), then \(b=88 - 43 = 45\).

of process:
We used the substitution - method to solve the system of equations. First, we expressed one variable in terms of the others using the given relationships. Then we substituted these expressions into the sum - of - angles equation for the triangle. By doing so, we were able to solve for one variable (\(c\)) first, and then find the values of the other two variables (\(a\) and \(b\)) using the previously established relationships.

Answer:

\(m\angle A = 92^{\circ}\), \(m\angle B = 45^{\circ}\), \(m\angle C = 43^{\circ}\)