Question

in the triangle below, \\( \angle c \\) is a right angle. suppose that \\( m\angle a = (5x + 15)^\circ \\) and \\( m\angle b = (3x + 11)^\circ \\).\
(a) write an equation to find \\( x \\). use your answer from part (a) to solve for \\( x \\).\
equation: \\( \square \\)\
(b) find the degree measure of each angle.\
\\( m\angle a = \square^\circ \\)\
\\( m\angle b = \square^\circ \\)\
\\( m\angle c = \square^\circ \\)

Explanation:

Part (a)

Step1: Recall triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). Since \(\angle C\) is a right angle, \(m\angle C = 90^\circ\). So, \(m\angle A + m\angle B + m\angle C = 180^\circ\).

Step2: Substitute angle expressions

Substitute \(m\angle A=(5x + 15)^\circ\), \(m\angle B=(3x + 11)^\circ\), and \(m\angle C = 90^\circ\) into the angle - sum formula:
\[
(5x + 15)+(3x + 11)+90 = 180
\]
Simplify the left - hand side:
\[
5x+3x + 15 + 11+90=180\\
8x+116 = 180
\]

Part (b)

Step1: Solve for \(x\)

From the equation \(8x+116 = 180\), subtract 116 from both sides:
\[
8x=180 - 116\\
8x = 64
\]
Divide both sides by 8:
\[
x=\frac{64}{8}=8
\]

Step2: Find \(m\angle A\)

Substitute \(x = 8\) into \(m\angle A=(5x + 15)^\circ\):
\[
m\angle A=5\times8 + 15=40 + 15 = 55^\circ
\]

Step3: Find \(m\angle B\)

Substitute \(x = 8\) into \(m\angle B=(3x + 11)^\circ\):
\[
m\angle B=3\times8+11 = 24 + 11=35^\circ
\]

Step4: Recall \(m\angle C\)

Since \(\angle C\) is a right angle, \(m\angle C = 90^\circ\)

Answer:

(a) The equation is \(8x + 116=180\) (or the expanded form \((5x + 15)+(3x + 11)+90 = 180\))
(b) \(m\angle A=\boldsymbol{55^\circ}\), \(m\angle B=\boldsymbol{35^\circ}\), \(m\angle C=\boldsymbol{90^\circ}\)