Prove using SAS congruence.
\(\textbf{1)}\) Given: \(C\) bisects \( \overline{AE} \) and \( \overline{BD} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\(C\) bisects \( \overline{AE} \) and \( \overline{BD} \) Given
\( \overline{AC} \cong \overline{CE}\)
\( \overline{BC} \cong \overline{CD}\) Definition of bisects
\(\angle ACB \cong \angle ECD\) Vertical Angles
\(\triangle ACB \cong \triangle ECD\) SAS triangle congruence
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\( \overline{BC} \cong \overline{CD}\) |
|
\(\textbf{2)}\) Given: \( \angle B \cong \angle C \), \( \, \, D\) bisects \( \overline{BC} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \angle B \cong \angle C \)
Point \(D\) bisects \( \overline{BC} \) Given
\(\overline{BD} \cong \overline{CD} \) Definition of bisects
\(\overline{AB} \cong \overline{AC} \) Base Angles Theorem
\(\triangle ABD \cong \triangle ACD\) SAS triangle congruence
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
Point \(D\) bisects \( \overline{BC} \) |
|
\(\textbf{3)}\) Given: \( \overline{AD} \cong \overline{AF}\), \( \, \, \overline{BD} \cong \overline{CF} \), \( \, \, E\) bisects \( \overline{DF} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle DBE \cong \triangle FCE \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \overline{AD} \cong \overline{AF}\)
\(\overline{BD} \cong \overline{CF} \)
\(E\) bisects \( \overline{DF} \) Given
\(\angle D \cong \angle F\) Isosceles Triangles
Base Angles Theorem
\( \overline{DE} \cong \overline{FE}\) Definition of bisects
\( \triangle DBE \cong \triangle FCE \) SAS triangle congruence
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overline{BD} \cong \overline{CF} \) \(E\) bisects \( \overline{DF} \) |
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Base Angles Theorem |
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Challenge Problem
\(\textbf{4)}\) Given: \(ABDC\) is a parallelogram
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DCB \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\(ABDC\) is a parallelogram Given
\(\overline{AC} \cong \overline{BD} \)
\(\overline{AB} \cong \overline{CD} \) Opposite sides congruent in a parallelogram
\(\angle A \cong \angle D\) Opposite angles congruent in a parallelogram
\(\triangle ABC \cong \triangle DCB\) SAS triangle congruence
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overline{AB} \cong \overline{CD} \) |
|
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