Question

1. 12. (3x)° 49° (7x - 23)° (11y - 1)° (7y - 20)° (3x - 4)° (5x - 38)° m l m © gina wilson (all things algebra®, llc), 2014 - 2015

Explanation:

Step1: Identify angle - relationships in problem 11

Vertical angles are equal. So, \(3x = 7x - 23 - 49\) (since the non - overlapping angles formed by the intersection of lines have a relationship).
\[3x=7x - 72\]

Step2: Solve the equation for \(x\) in problem 11

Subtract \(3x\) from both sides: \(0 = 7x-3x - 72\), which simplifies to \(0 = 4x - 72\). Then add 72 to both sides: \(4x=72\), and divide by 4 to get \(x = 18\).

Step3: Find the value of \(y\) in problem 11

Another pair of vertical angles gives \(11y-1=49 + 3x\). Substitute \(x = 18\) into the equation: \(11y-1=49+3\times18\).
\[11y-1=49 + 54\]
\[11y-1=103\]
Add 1 to both sides: \(11y=104\), so \(y=\frac{104}{11}\approx9.45\)

Step4: Identify angle - relationships in problem 12

The sum of angles in a triangle is \(180^{\circ}\), and we also have parallel lines \(l\) and \(m\). The angle \((3x - 4)\) and \((5x - 38)\) are corresponding angles (because of parallel lines \(l\) and \(m\)), so \(3x-4=5x - 38\).
\[3x-4=5x - 38\]

Step5: Solve the equation for \(x\) in problem 12

Subtract \(3x\) from both sides: \(-4=5x-3x - 38\), which simplifies to \(-4 = 2x-38\). Add 38 to both sides: \(2x=34\), and divide by 2 to get \(x = 17\).

Step6: Find the value of \(y\) in problem 12

In the right - angled triangle, \((3x - 4)+(7y - 20)+90=180\). Substitute \(x = 17\) into the equation: \((3\times17 - 4)+(7y - 20)+90=180\).
\[47+(7y - 20)+90=180\]
\[7y+117 = 180\]
Subtract 117 from both sides: \(7y=63\), and divide by 7 to get \(y = 9\).

Answer:

In problem 11: \(x = 18,y=\frac{104}{11}\); In problem 12: \(x = 17,y = 9\)