Question

2 using the diagram in conjunction with postulates and theorems, determine the measure of all unknown angles. given: ∠2 ≅ ∠7 ≅ ∠19 m∠2 = 125°

Explanation:

Step1: Vertical Angles & Linear Pairs

Vertical angles are equal, linear pairs sum to \(180^\circ\).

• \(\angle1\) and \(\angle3\) (vertical), \(\angle2\) and \(\angle4\) (vertical).
• \(\angle1 + \angle2 = 180^\circ\), so \(m\angle1 = 180^\circ - 125^\circ = 55^\circ\).
• \(m\angle3 = m\angle1 = 55^\circ\), \(m\angle4 = m\angle2 = 125^\circ\).

Step2: \(\angle7 \cong \angle2\)

Given \(\angle7 \cong \angle2\), so \(m\angle7 = 125^\circ\).

• \(\angle6\) (linear pair with \(\angle7\)): \(m\angle6 = 180^\circ - 125^\circ = 55^\circ\).
• \(\angle5\) (vertical with \(\angle7\)): \(m\angle5 = 125^\circ\).
• \(\angle8\) (vertical with \(\angle6\)): \(m\angle8 = 55^\circ\).

Step3: \(\angle19 \cong \angle2\)

Given \(\angle19 \cong \angle2\), so \(m\angle19 = 125^\circ\).

• \(\angle18\) (linear pair with \(\angle19\)): \(m\angle18 = 180^\circ - 125^\circ = 55^\circ\).
• \(\angle20\) (vertical with \(\angle19\)): \(m\angle20 = 125^\circ\).
• \(\angle17\) (vertical with \(\angle18\)): \(m\angle17 = 55^\circ\).

Step4: Corresponding/Alternate Angles (Parallel Lines Implied)

Assume lines \(L \parallel M\) (from angle congruences).

• \(\angle15 \cong \angle19\) (corresponding), so \(m\angle15 = 125^\circ\).
• \(\angle14 \cong \angle20\) (corresponding), so \(m\angle14 = 125^\circ\).
• \(\angle6 \cong \angle13\) (corresponding), so \(m\angle13 = 55^\circ\).
• \(\angle5 \cong \angle16\) (corresponding), so \(m\angle16 = 125^\circ\).

Step5: Intersecting Lines \(N\) & \(P\)

Vertical angles and linear pairs:

• \(\angle9\) and \(\angle11\) (vertical), \(\angle10\) and \(\angle12\) (vertical).
• \(\angle9 + \angle10 = 180^\circ\) (linear pair). From \(\angle10 \cong \angle8\) (corresponding, \(m\angle8 = 55^\circ\)), so \(m\angle10 = 55^\circ\).
• \(m\angle9 = 180^\circ - 55^\circ = 125^\circ\), \(m\angle11 = m\angle9 = 125^\circ\), \(m\angle12 = m\angle10 = 55^\circ\).

Answer:

• \(m\angle1 = 55^\circ\), \(m\angle3 = 55^\circ\), \(m\angle4 = 125^\circ\)
• \(m\angle5 = 125^\circ\), \(m\angle6 = 55^\circ\), \(m\angle8 = 55^\circ\)
• \(m\angle13 = 55^\circ\), \(m\angle14 = 125^\circ\), \(m\angle15 = 125^\circ\), \(m\angle16 = 125^\circ\)
• \(m\angle17 = 55^\circ\), \(m\angle18 = 55^\circ\), \(m\angle20 = 125^\circ\)
• \(m\angle9 = 125^\circ\), \(m\angle10 = 55^\circ\), \(m\angle11 = 125^\circ\), \(m\angle12 = 55^\circ\)

(All unknown angles calculated using vertical angles, linear pairs, and congruence/postulates.)