Question

vertical angles
∠1 and ∠4 are vertical angles. measure angle 1: m∠1 = 35°.
now use your knowledge of vertical and corresponding angles to find the other missing angles.
assume ∠2 and ∠3 are vertical angles.
m∠2 = __° m∠3 = ° m∠4 = __°
∠6 and ∠7 are vertical angles. measure angle 6: m∠6 = 100°.
now use your knowledge of vertical and corresponding angles to find the other missing angles.
assume ∠8 and ∠9 are vertical angles.
m∠7 = __° m∠8 = ° m∠9 = __°
what is the relationship between the measures of vertical angles?

Explanation:

Step1: Vertical angles property

Vertical angles are equal. Since \( \angle1\) and \( \angle4\) are vertical angles, \(m\angle4 = m\angle1=35^{\circ}\).

Step2: Linear - pair property

\(\angle1\) and \( \angle2\) form a linear pair (\(m\angle1 + m\angle2=180^{\circ}\)). So \(m\angle2=180 - 35=145^{\circ}\).

Step3: Vertical angles property again

Since \( \angle2\) and \( \angle3\) are vertical angles, \(m\angle3 = m\angle2 = 145^{\circ}\).

For the second part:

Step4: Vertical angles property

Since \( \angle6\) and \( \angle7\) are vertical angles, \(m\angle7 = m\angle6 = 100^{\circ}\).

Step5: Linear - pair property

\(\angle6\) and \( \angle8\) form a linear pair (\(m\angle6 + m\angle8=180^{\circ}\)). So \(m\angle8=180 - 100 = 80^{\circ}\).

Step6: Vertical angles property

Since \( \angle8\) and \( \angle9\) are vertical angles, \(m\angle9 = m\angle8=80^{\circ}\).

The relationship between the measures of vertical angles: Vertical angles are congruent (i.e., their measures are equal).

Answer:

\(m\angle2 = 145^{\circ}\), \(m\angle3 = 145^{\circ}\), \(m\angle4 = 35^{\circ}\), \(m\angle7 = 100^{\circ}\), \(m\angle8 = 80^{\circ}\), \(m\angle9 = 80^{\circ}\). Vertical angles are congruent (have equal measures).