Question

given: \\( \overline { j g } \cong \overline { h i }, \angle j g i \cong \angle h i g \\) prove: \\( \angle j \cong \angle h \\) given: \\( \angle p \cong \angle t, r \\) is the midpoint of \\( \overline { q s } \\) prove: \\( \triangle p r q \cong \triangle t r s \\) given: \\( \angle b \cong \angle e, \angle b c a \cong \angle e c d, c \\) is the midpoint of \\( \overline { a d } \\) prove: \\( \overline { a b } \cong \overline { d e } \\)

Explanation:

First Proof:

Step1: Reflexive Property

By the reflexive property of congruence, \(GI\cong IG\).

Step2: SAS Congruence

Since \(JG\cong HI\), \(\angle JGI\cong\angle HIG\), and \(GI\cong IG\), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle JGI\cong\triangle HIG\).

Step3: CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC), so \(\angle J\cong\angle H\).

Step1: Mid - point Definition

Since \(R\) is the mid - point of \(QS\), by the definition of a mid - point, \(QR\cong SR\).

Step2: Vertical Angles

\(\angle PRQ\) and \(\angle TRS\) are vertical angles. Vertical angles are congruent, so \(\angle PRQ\cong\angle TRS\).

Step3: AAS Congruence

Given \(\angle P\cong\angle T\), \(QR\cong SR\), and \(\angle PRQ\cong\angle TRS\), by the Angle - Angle - Side (AAS) congruence criterion, \(\triangle PRQ\cong\triangle TRS\).

Step1: Mid - point Definition

Since \(C\) is the mid - point of \(AD\), by the definition of a mid - point, \(AC\cong DC\).

Step2: AAS Congruence

Given \(\angle B\cong\angle E\), \(\angle BCA\cong\angle ECD\), and \(AC\cong DC\), by the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ABC\cong\triangle DEC\).

Step3: CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC), so \(AB\cong DE\).

Answer:

1. Reflexive Property of Congruence; 4. SAS (Side - Angle - Side); 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Second Proof: