Question

question
if ( mangle 2 = 98^{circ}, mangle 3 = 23^{circ} ) and ( mangle 8 = 70^{circ} ), find the measure of each angle.

Explanation:

Step1: Use linear - pair relationship

Since \(\angle1\) and \(\angle2\) form a linear pair, \(m\angle1 + m\angle2=180^{\circ}\). Given \(m\angle2 = 98^{\circ}\), then \(m\angle1=180^{\circ}-98^{\circ}=82^{\circ}\).

Step2: Use alternate - interior angles (assuming parallel lines)

If the lines are parallel, \(\angle1\) and \(\angle6\) are alternate - interior angles, so \(m\angle6 = m\angle1 = 82^{\circ}\). Also, \(\angle2\) and \(\angle7\) are alternate - interior angles, so \(m\angle7 = m\angle2=98^{\circ}\).

Step3: Use triangle - angle sum property

In the triangle, if we consider the triangle with \(\angle7\), \(\angle8\) and the angle adjacent to \(\angle9\). But first, since \(\angle3 = 23^{\circ}\), and if we assume some parallel - line and transversal relationships (for \(\angle4\), if the lines are parallel and using the property of angles formed by a transversal), \(m\angle4\) can be found. But if we consider the fact that \(\angle3\) and \(\angle4\) might be related by some angle - bisector or parallel - line property (assuming the left - hand side lines are cut by a transversal). If we assume the lines are parallel, \(\angle3\) and \(\angle4\) are equal (if it's an angle - bisector or alternate - interior angles in a simple case), so \(m\angle4 = 23^{\circ}\).

Step4: Use linear - pair and parallel - line relationships for \(\angle5\), \(\angle9\) and \(\angle10\)

\(\angle5\) and \(\angle3+\angle4\) form a linear pair. \(m\angle3 + m\angle4=23^{\circ}+23^{\circ}=46^{\circ}\), so \(m\angle5 = 180^{\circ}-46^{\circ}=134^{\circ}\).
For \(\angle9\), if we consider the triangle with \(\angle7\) and \(\angle8\), and using the property of parallel lines (assuming the lower lines are parallel). If we assume the lines are parallel, \(\angle9\) and \(\angle7+\angle8\) are related. \(m\angle7 + m\angle8=98^{\circ}+70^{\circ}=168^{\circ}\), and since \(\angle9\) and \(\angle7+\angle8\) form a linear pair (assuming parallel lines and transversal), \(m\angle9 = 180^{\circ}-(98^{\circ}+70^{\circ}) = 12^{\circ}\).
Since \(\angle10\) and \(\angle9\) form a linear pair, \(m\angle10=180^{\circ}-m\angle9 = 168^{\circ}\).

Answer:

\(m\angle1 = 82^{\circ}\), \(m\angle4 = 23^{\circ}\), \(m\angle5 = 134^{\circ}\), \(m\angle6 = 82^{\circ}\), \(m\angle7 = 98^{\circ}\), \(m\angle9 = 12^{\circ}\), \(m\angle10 = 168^{\circ}\)