Question

1. in △rst, if m∠r is five more than twice x, m∠s is one more than x, and m∠t is sixteen less than seven times x, find x and the measure of each angle.
2. in △abc, if m∠a is thirteen less than m∠c and m∠b is eleven less than four times m∠c, find the measure of each angle.
3. in △jkl, if m∠k is nine more than m∠j and m∠l is 21 less than twice m∠j, find the measure of each angle.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180 degrees. For \(\triangle RST\), \(m\angle R + m\angle S+m\angle T=180\). Given \(m\angle R = 2x + 5\), \(m\angle S=x + 1\), and \(m\angle T=7x-16\).
So, \((2x + 5)+(x + 1)+(7x-16)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(2x+x + 7x+5 + 1-16=180\), which gives \(10x-10 = 180\).

Step3: Solve for \(x\)

Add 10 to both sides of the equation: \(10x=180 + 10=190\). Then divide both sides by 10: \(x=\frac{190}{10}=19\).

Step4: Find the measure of each angle

\(m\angle R=2x + 5=2\times19+5=38 + 5=43\) degrees.
\(m\angle S=x + 1=19+1=20\) degrees.
\(m\angle T=7x-16=7\times19-16=133-16 = 117\) degrees.

For \(\triangle ABC\):
Let \(m\angle C=x\). Then \(m\angle A=x - 13\) and \(m\angle B=4x-11\).
Since \(m\angle A + m\angle B+m\angle C=180\), we have \((x - 13)+(4x-11)+x=180\).
Combine like terms: \(x+4x+x-13-11=180\), \(6x-24 = 180\).
Add 24 to both sides: \(6x=180 + 24=204\).
Divide both sides by 6: \(x=\frac{204}{6}=34\).
\(m\angle A=x - 13=34-13=21\) degrees.
\(m\angle B=4x-11=4\times34-11=136-11 = 125\) degrees.
\(m\angle C=34\) degrees.

For \(\triangle JKL\):
Let \(m\angle J=x\). Then \(m\angle K=x + 9\) and \(m\angle L=2x-21\).
Since \(m\angle J + m\angle K+m\angle L=180\), we have \(x+(x + 9)+(2x-21)=180\).
Combine like terms: \(x+x+2x+9-21=180\), \(4x-12 = 180\).
Add 12 to both sides: \(4x=180 + 12=192\).
Divide both sides by 4: \(x=\frac{192}{4}=48\).
\(m\angle J=48\) degrees.
\(m\angle K=x + 9=48+9=57\) degrees.
\(m\angle L=2x-21=2\times48-21=96-21 = 75\) degrees.

Answer:

For \(\triangle RST\):
\(x = 19\)
\(m\angle R=43\)
\(m\angle S=20\)
\(m\angle T=117\)

For \(\triangle ABC\):
\(m\angle A=21\)
\(m\angle B=125\)
\(m\angle C=34\)

For \(\triangle JKL\):
\(m\angle J=48\)
\(m\angle K=57\)
\(m\angle L=75\)