Question

solve for x and find all the angle measures:

Explanation:

Step1: Identify angle - sum relationships

We assume the first set of intersecting lines. Since the sum of angles around a point is 360° and vertical angles are equal. Let's consider the non - vertical - angle case. If we assume a linear pair (adjacent angles on a straight line sum to 180°), we can't directly solve for \(x\) from the given 55° and 85° angles without more information.

Let's consider the right - angled triangle. In a right - angled triangle, if one non - right angle is 20°, and we want to find the other non - right angle \(F\).
The sum of the interior angles of a triangle is 180°.

Step2: Calculate angle \(F\) in the triangle

Using the formula \(A + B+C=180^{\circ}\) for a triangle, where \(A = 90^{\circ}\), \(B = 20^{\circ}\), and \(C=F\).
\[F=180^{\circ}-90^{\circ}-20^{\circ}=70^{\circ}\]

For the third set of angles \((x + 1)^{\circ}\), \((x-1)^{\circ}\), and \((x + 3)^{\circ}\) which form a right - angle (90°).
We set up the equation \((x + 1)+(x - 1)+(x + 3)=90\).

Step3: Simplify the equation

Combine like terms: \((x+x+x)+(1 - 1+3)=90\), which gives \(3x+3 = 90\).
Subtract 3 from both sides: \(3x=90 - 3=87\).
Divide both sides by 3: \(x=\frac{87}{3}=29\)

For the fourth set of angles \(5x+4\), \(3x + 7\), and \(x-2\) which form a straight - line (180°).
Set up the equation \((5x + 4)+(3x+7)+(x - 2)=180\).

Step4: Combine like terms in the new equation

\((5x+3x+x)+(4 + 7-2)=180\), so \(9x + 9=180\).
Subtract 9 from both sides: \(9x=180 - 9 = 171\).
Divide both sides by 9: \(x=\frac{171}{9}=19\)

Answer:

In the triangle, \(F = 70^{\circ}\). For the angles forming a right - angle, \(x = 29\). For the angles forming a straight - line, \(x = 19\)