Question

1. if m∠8 = 23°, find each measure. give your reasoning.
2. if m∠9 = 97° and m∠12 = 114°, find each measure.
3. if m∠2 = 98°, m∠3 = 23° and m∠8 = 70°, find each measure.
4. if m∠3 = 54°, find each measure.

Explanation:

Step1: Identify angle - relationships

Use properties like vertical - angles are equal, linear - pair angles sum to 180°, and corresponding, alternate - interior, and alternate - exterior angles for parallel lines.

Step2: Solve for each angle

For example, if two angles are a linear pair, and one angle is \(x\), the other is \(180 - x\). If two angles are vertical angles, they are equal.

Since the figures are not clearly labeled with parallel - line information and specific relationships are not given in a detailed way for each sub - problem, we'll assume some general angle - relationship rules.

Let's take the first part where \(m\angle8 = 23^{\circ}\):

• \(m\angle1\) and \(m\angle8\) are a linear pair. So \(m\angle1=180 - 23=157^{\circ}\) (linear - pair property: sum of angles in a linear pair is 180°).
• \(m\angle2\) and \(m\angle8\) are vertical angles. So \(m\angle2 = 23^{\circ}\) (vertical angles are equal).
• \(m\angle3\) and \(m\angle8\) are a linear pair. So \(m\angle3=180 - 23 = 157^{\circ}\).
• \(m\angle4\) and \(m\angle8\) are corresponding angles (assuming parallel lines). So \(m\angle4 = 23^{\circ}\).
• \(m\angle5\) and \(m\angle1\) are vertical angles. So \(m\angle5=157^{\circ}\).
• \(m\angle6\) and \(m\angle2\) are vertical angles. So \(m\angle6 = 23^{\circ}\).
• \(m\angle7\) and \(m\angle3\) are vertical angles. So \(m\angle7=157^{\circ}\).

For the other parts, we would follow similar logic based on angle - relationships such as:

• Vertical angles: \(\angle a=\angle b\) if \(\angle a\) and \(\angle b\) are vertical angles.
• Linear pairs: If \(\angle x\) and \(\angle y\) form a linear pair, then \(m\angle x + m\angle y=180^{\circ}\).
• Corresponding angles: If two parallel lines are cut by a transversal, corresponding angles are equal.
• Alternate - interior angles: If two parallel lines are cut by a transversal, alternate - interior angles are equal.
• Alternate - exterior angles: If two parallel lines are cut by a transversal, alternate - exterior angles are equal.

However, without specific information about parallel lines and detailed labels for each problem, a complete step - by - step for all sub - problems cannot be fully provided.

Since we don't have all the necessary details for a full solution of all parts, we'll just note the general approach.

Answer:

The general approach to find angle measures is to use angle - relationship properties such as vertical angles, linear pairs, and angle relationships for parallel lines (corresponding, alternate - interior, alternate - exterior). Without full information about the figures (like which lines are parallel), we cannot give exact measures for all the remaining parts. For the part where \(m\angle8 = 23^{\circ}\), \(m\angle1 = 157^{\circ}\), \(m\angle2 = 23^{\circ}\), \(m\angle3 = 157^{\circ}\), \(m\angle4 = 23^{\circ}\), \(m\angle5 = 157^{\circ}\), \(m\angle6 = 23^{\circ}\), \(m\angle7 = 157^{\circ}\) based on angle - relationship properties.