Question

mark each diagram with any inherent or given information. if the triangles are congruent by sas, write the congruency statement.
1.
2.
3.
4.
5.
given: \\( \overline { e g } \\) is the angle bisector
of \\( \angle d e f \\).
\\( \triangle d e g \cong \triangle f e g \\)
6.
7.
8.
given: \\( \overline { a b } \cong \overline { r s } , \overline { a c } \cong \overline { r t } , m \angle b = m \angle s \\)
9.
given: \\( \overline { b d } \\) is the perpendicular
bisector of \\( \overline { a c } \\).
\\( \triangle a b d \cong \triangle c b d \\)

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

The SAS congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze each pair of triangles

• For problem 5:
• Given that \(EG\) is the angle bisector of \(\angle DEF\), so \(\angle DEG=\angle FEG\).
• Also, \(DE = FE\) (marked as equal) and \(EG=EG\) (common side).
• By the SAS congruence criterion, \(\triangle DEG\cong\triangle FEG\)
• For problem 9:
• Since \(BD\) is the perpendicular bisector of \(AC\), \(AD = CD\) and \(\angle ADB=\angle CDB = 90^{\circ}\).
• Also, \(BD = BD\) (common side).
• By the SAS congruence criterion, \(\triangle ABD\cong\triangle CBD\)

Answer:

1. \(\triangle DEG\cong\triangle FEG\)
2. \(\triangle ABD\cong\triangle CBD\)