Question

1. ( mangle1 = 2x + 6 ) and ( mangle2 = 3x - 2 ). find ( mangle1 ).
2. find ( mangle ust ) if ( mangle2 = 6x - 1 ) and ( mangle ust = 10x + 10 ).

Explanation:

Step1: Use the exterior angle theorem (for problem 12)

Assume \( \angle 2\) is an exterior angle related to \( \angle 1\) (based on the figure structure). If \( \angle 2\) is an exterior angle and \( \angle 1\) is an interior - adjacent angle (common in triangle - related angle problems), and if \( \angle 2\) and \( \angle 1\) are equal (if \( YP = YW\) and \( WP = XP\) type of isosceles - like relations assumed from the figure structure). Then \(2x + 6=3x - 2\).
Solve for \(x\):
\(2x+6 = 3x - 2\)
\(6 + 2=3x - 2x\)
\(x = 8\)

Step2: Calculate \(m\angle1\)

Substitute \(x = 8\) into \(m\angle1=2x + 6\).
\(m\angle1=2\times8 + 6\)
\(m\angle1=16 + 6\)
\(m\angle1 = 22\)

Step3: Use the angle - sum relation (for problem 15)

Assume \(m\angle UST=m\angle1 + m\angle2\) (if \( \angle UST\) is composed of \( \angle1\) and \( \angle2\) as in a non - overlapping angle addition). If \( \angle1=\angle2\) (from figure symmetry assumptions), then \(m\angle UST = 2m\angle2\).
Set \(10x + 10=2(6x - 1)\)
\(10x+10 = 12x-2\)
\(10 + 2=12x - 10x\)
\(2x=12\)
\(x = 6\)

Step4: Calculate \(m\angle UST\)

Substitute \(x = 6\) into \(m\angle UST=10x + 10\)
\(m\angle UST=10\times6+10\)
\(m\angle UST=60 + 10\)
\(m\angle UST = 70\)

Answer:

For problem 12: \(m\angle1 = 22\)
For problem 15: \(m\angle UST = 70\)