Question

part 1 of 3
prove that quadrilateral abcd is
congruent to quadrilateral
efgh.
complete the proof below.
statement reason
∠a ≅ ∠e,
∠abd ≅ ∠efh, ∠c ≅ ∠g,
bc ≅ fg, cd ≅ gh given
△bcd ≅ △fgh
options for reason: asa, aas, sas, sss

Explanation:

Step1: Identify triangle parts

In $\triangle BCD$ and $\triangle FGH$, we have $\angle C \cong \angle G$ (given), $\overline{BC} \cong \overline{FG}$ (given), $\overline{CD} \cong \overline{GH}$ (given).

Step2: Match congruence criterion

The ASA (Angle - Side - Angle) criterion requires two angles and the included side. Here, we have one angle ($\angle C \cong \angle G$) and two sides ($\overline{BC} \cong \overline{FG}$, $\overline{CD} \cong \overline{GH}$) with the angle between the two sides? Wait, no, wait. Wait, $\angle C$ is between $BC$ and $CD$, and $\angle G$ is between $FG$ and $GH$. So the sides $BC$ and $CD$ with included angle $\angle C$, and sides $FG$ and $GH$ with included angle $\angle G$. So the congruence criterion here is SAS? Wait, no, SAS is Side - Angle - Side (two sides and the included angle). Wait, $BC \cong FG$, $\angle C \cong \angle G$, $CD \cong GH$. So that's two sides and the included angle. So the criterion is SAS? Wait, no, wait the options are ASA, AAS, SAS, SSS. Wait, let's re - check.

Wait, $\angle C$ is the angle between $BC$ and $CD$, and $\angle G$ is the angle between $FG$ and $GH$. So we have $BC \cong FG$, $\angle C \cong \angle G$, $CD \cong GH$. So that's Side - Angle - Side (SAS) because the angle is between the two sides. Wait, but wait the options: ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), SAS (Side - Angle - Side), SSS (Side - Side - Side). So in $\triangle BCD$ and $\triangle FGH$:

• $BC \cong FG$ (side)
• $\angle C \cong \angle G$ (angle)
• $CD \cong GH$ (side)

So this is SAS (Side - Angle - Side) congruence criterion. Wait, no, wait SAS is when the angle is between the two sides. Yes, $\angle C$ is between $BC$ and $CD$, and $\angle G$ is between $FG$ and $GH$. So the correct criterion is SAS? Wait, no, wait the options: let's check again. Wait, maybe I made a mistake. Wait, the given information: $\angle C \cong \angle G$, $BC \cong FG$, $CD \cong GH$. So that's two sides and the included angle, so SAS. But wait, the options include ASA, AAS, SAS, SSS. Wait, maybe I misread. Wait, the triangles are $\triangle BCD$ and $\triangle FGH$. Let's list the corresponding parts:

• $BC \cong FG$ (given)
• $\angle C \cong \angle G$ (given)
• $CD \cong GH$ (given)

So the congruence is by SAS (Side - Angle - Side) because we have two sides and the included angle. Wait, but let's check the options. Wait, maybe the answer is SAS? Wait, no, wait the problem: let's see the statement $\triangle BCD \cong \triangle FGH$. The reasons are the given parts. So $BC \cong FG$, $\angle C \cong \angle G$, $CD \cong GH$. So that's SAS. But wait, maybe I made a mistake. Wait, no, SAS is Side - Angle - Side, which is two sides and the included angle. So here, $BC$ and $CD$ with included angle $\angle C$, and $FG$ and $GH$ with included angle $\angle G$. So yes, SAS. But wait, the options: ASA, AAS, SAS, SSS. So the correct reason is SAS? Wait, no, wait maybe I messed up. Wait, let's check the angle - side - angle. ASA is two angles and the included side. AAS is two angles and a non - included side. SSS is three sides. So in this case, we have two sides and the included angle, so SAS. So the answer should be SAS. Wait, but let's confirm.

Wait, the triangles $\triangle BCD$ and $\triangle FGH$:

• Side: $BC \cong FG$
• Angle: $\angle C \cong \angle G$
• Side: $CD \cong GH$

So this is SAS (Side - Angle - Side) congruence. So the reason is SAS.

Answer:

SAS