10. given: $\overline{ax} \cong \overline{cx}$ and $\overline{bx} \cong \overline{dx}$ prove: (abcd) is a parallelogram. $\triangle axd \cong \triangle cxb$ and $\triangle axb \cong \triangle cxd$ vertical angles theorem $\angle cad \cong \angle bca$ and $\angle axb \cong \angle cxd$ aas congruence theorem sas congruence theorem $\angle axd \cong \angle cxb$ and $\angle axb \cong \angle cxd$ alternate interior angles theorem parallelogram opposite sides converse parallelogram diagonals converse

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# Answer:
To solve the proof that $ABCD$ is a parallelogram given $\overline{AX} \cong \overline{CX}$ and $\overline{BX} \cong \overline{DX}$, we follow these steps:

## Step 1: Identify Vertical Angles
$\angle AXD \cong \angle CXB$ and $\angle AXB \cong \angle CXD$ (Vertical Angles Theorem: Vertical angles are congruent).

## Step 2: Prove Triangles Congruent (SAS)
- For $\triangle AXD$ and $\triangle CXB$:
  - $\overline{AX} \cong \overline{CX}$ (Given)
  - $\angle AXD \cong \angle CXB$ (Vertical Angles Theorem)
  - $\overline{DX} \cong \overline{BX}$ (Given)
  - By the **SAS Congruence Theorem**, $\triangle AXD \cong \triangle CXB$.

- For $\triangle AXB$ and $\triangle CXD$:
  - $\overline{AX} \cong \overline{CX}$ (Given)
  - $\angle AXB \cong \angle CXD$ (Vertical Angles Theorem)
  - $\overline{BX} \cong \overline{DX}$ (Given)
  - By the **SAS Congruence Theorem**, $\triangle AXB \cong \triangle CXD$.

So, $\triangle AXD \cong \triangle CXB$ and $\triangle AXB \cong \triangle CXD$ (SAS Congruence Theorem).

## Step 3: Use CPCTC
From the congruent triangles ($\triangle AXD \cong \triangle CXB$ and $\triangle AXB \cong \triangle CXD$), we use **CPCTC** (Corresponding Parts of Congruent Triangles are Congruent) to conclude:
- $\overline{AD} \cong \overline{BC}$ (from $\triangle AXD \cong \triangle CXB$)
- $\overline{AB} \cong \overline{CD}$ (from $\triangle AXB \cong \triangle CXD$)

## Step 4: Conclude Parallelogram
A quadrilateral with both pairs of opposite sides congruent is a parallelogram (**Parallelogram Opposite Sides Converse**). Thus, $ABCD$ is a parallelogram.

Filling in the proof table:

| Statement | Reasons |
| --- | --- |
| $\overline{AX} \cong \overline{CX}$ and $\overline{BX} \cong \overline{DX}$ | Given |
| $\angle AXD \cong \angle CXB$ and $\angle AXB \cong \angle CXD$ | Vertical Angles Theorem |
| $\triangle AXD \cong \triangle CXB$ and $\triangle AXB \cong \triangle CXD$ | SAS Congruence Theorem |
| $\overline{AD} \cong \overline{BC}$ and $\overline{AB} \cong \overline{CD}$ | CPCTC |
| $ABCD$ is a parallelogram | Parallelogram Opposite Sides Converse |

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QUESTION IMAGE

Question

Explanation:

Answer:

Step 1: Identify Vertical Angles

Step 2: Prove Triangles Congruent (SAS)

Step 3: Use CPCTC

Step 4: Conclude Parallelogram

Statement	Reasons
\(\angle AXD \cong \angle CXB\) and \(\angle AXB \cong \angle CXD\)	Vertical Angles Theorem
\(\triangle AXD \cong \triangle CXB\) and \(\triangle AXB \cong \triangle CXD\)	SAS Congruence Theorem
\(\overline{AD} \cong \overline{BC}\) and \(\overline{AB} \cong \overline{CD}\)	CPCTC
\(ABCD\) is a parallelogram	Parallelogram Opposite Sides Converse