Question

1. polygon carbon is a regular hexagon. \\( \triangle acn \cong \triangle \underline{?} \\) 20. \\( \triangle \underline{?} \cong \triangle \underline{?} \\)

Explanation:

17.

Step1: Analyze the properties of a regular hexagon

In a regular hexagon \(CARBON\), all sides are equal (\(CA = AR=RB = BO=ON = NC\)) and all interior angles are equal.

Step2: Use the Side - Side - Side (SSS) congruence criterion

Since \(CA = AR = RB = BO=ON = NC\), and in \(\triangle ACN\) and \(\triangle OAR\)

• \(CA=OA\) (sides of the hexagon)
• \(CN = OR\) (sides of the hexagon)
• \(AN = AR\) (sides of the hexagon)

By SSS congruence, \(\triangle ACN\cong\triangle OAR\)

20.

Step1: Identify the given information

We know that \(IG = BN\) (marked as equal), \(IB = BE\) (marked as equal), and \(\angle IBG+\angle GBN=\angle NBE+\angle GBN\) (since \(\angle IBG=\angle NBE\) as they are marked equal). So, \(\angle IBG=\angle NBE\)

Step2: Use the Side - Angle - Side (SAS) congruence criterion

In \(\triangle IBG\) and \(\triangle EBN\)

• \(IG = BN\) (given)
• \(\angle IBG=\angle EBN\) (proved above)
• \(IB = EB\) (given)

By SAS congruence, \(\triangle IBG\cong\triangle EBN\)

Answer:

1. \(\triangle ACN\cong\triangle OAR\)
2. \(\triangle IBG\cong\triangle EBN\)