Question

1. given: u is the midpoint of \\( \overline { s t } \\), \\( \overline { s u } \cong \overline { t u } \\), \\( \overline { t u } \cong \overline { i o } \\) prove: \\( \angle s i u \cong \angle t i u \\) 10. given: \\( \overline { m n } \parallel \overline { p q } \\), \\( \overline { m p } \parallel \overline { n q } \\) prove: \\( \overline { m p } \cong \overline { q n } \\) 11. given: \\( \overline { d e } \parallel \overline { f g } \\), \\( \overline { d e } \cong \overline { f g } \\), \\( \angle d e f \cong \angle f g h \\) prove: \\( \angle d f e \cong \angle f h g \\)

Explanation:

Step1: Prove for problem 9

• Since \(U\) is the mid - point of \(\overline{ST}\), by the definition of mid - point, \(\overline{SU}\cong\overline{TU}\).
• Given \(\overline{SU}\cong\overline{TW}\) and \(\overline{TU}\cong\overline{WU}\).
• In \(\triangle SUV\) and \(\triangle TWU\), we have \(\overline{SU}\cong\overline{TU}\), \(\overline{SV}\cong\overline{TW}\), \(\overline{VU}\cong\overline{WU}\).
• By SSS (Side - Side - Side) congruence criterion, \(\triangle SUV\cong\triangle TWU\).
• Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle SVU=\angle T WU\).

Step2: Prove for problem 10

• Since \(\overline{MN}\parallel\overline{PO}\) and \(\overline{MP}\parallel\overline{NO}\), by the definition of a parallelogram, \(MPON\) is a parallelogram.
• In a parallelogram, opposite sides are congruent. So, \(\overline{MP}\cong\overline{ON}\).

Step3: Prove for problem 11

• Since \(\overline{DE}\parallel\overline{FG}\), by the alternate interior angles theorem, \(\angle EDF=\angle GFH\).
• Given \(\overline{DE}\cong\overline{FG}\) and \(\angle DEF=\angle FGH\).
• In \(\triangle DEF\) and \(\triangle FGH\), we have \(\overline{DE}\cong\overline{FG}\), \(\angle DEF=\angle FGH\), \(\angle EDF=\angle GFH\).
• By ASA (Angle - Side - Angle) congruence criterion, \(\triangle DEF\cong\triangle FGH\).
• Then, by CPCTC, \(\angle DFE=\angle FHG\).

Answer:

For problem 9: \(\angle SVU=\angle T WU\) is proved.
For problem 10: \(\overline{MP}\cong\overline{ON}\) is proved.
For problem 11: \(\angle DFE=\angle FHG\) is proved.