Question

the figure shows parallelogram abcd with diagonal $overline{ac}$. complete the two column proof to prove that opposite angles $angle abc$ and $angle cda$ are congruent.

statements	reasons
	opposite sides of a parallelogram are congruent
$overline{ab} \parallel \overline{cd}$
	alternate interior angles are congruent
$overline{ac} \cong \overline{ac}$	reflexive property
$\triangle bac \cong \triangle dca$
	corresponding parts of congruent triangles are congruent

$\angle bca \cong \angle dac$ $\angle bac \cong \angle dca$ $\angle abc \cong \angle cda$ $\overline{ab} \cong \overline{cd}$ $\overline{bc} \cong \overline{da}$ sas sss aas definition of a parallelogram definition of a quadrilateral

Explanation:

To solve the two - column proof for showing that \(\angle ABC\cong\angle CDA\) in parallelogram \(ABCD\) with diagonal \(\overline{AC}\), we follow these steps:

Step 1: Fill in the statement for "Opposite sides of a parallelogram are congruent"

In a parallelogram, opposite sides are congruent. So, for parallelogram \(ABCD\), we have \(\overline{AB}\cong\overline{CD}\) and \(\overline{BC}\cong\overline{DA}\). We can use \(\overline{AB}\cong\overline{CD}\) (or \(\overline{BC}\cong\overline{DA}\)) here. Let's take \(\overline{AB}\cong\overline{CD}\) (and we can also note \(\overline{BC}\cong\overline{DA}\) later).

Step 2: Fill in the reason for \(\overline{AB}\parallel\overline{CD}\)

By the definition of a parallelogram, opposite sides of a parallelogram are parallel. So the reason for \(\overline{AB}\parallel\overline{CD}\) is "Definition of a parallelogram".

Step 3: Find the alternate interior angles

Since \(\overline{AB}\parallel\overline{CD}\) and \(\overline{AC}\) is a transversal, the alternate interior angles are \(\angle BAC\cong\angle DCA\).

Step 4: Determine the congruence criterion for \(\triangle BAC\cong\triangle DCA\)

We know that \(\overline{AB}\cong\overline{CD}\) (from step 1), \(\angle BAC\cong\angle DCA\) (from step 3), and \(\overline{AC}\cong\overline{AC}\) (by the reflexive property). So, by the Side - Angle - Side (SAS) congruence criterion, \(\triangle BAC\cong\triangle DCA\). The reason for \(\triangle BAC\cong\triangle DCA\) is "SAS".

Step 5: Conclude the congruence of \(\angle ABC\) and \(\angle CDA\)

Since \(\triangle BAC\cong\triangle DCA\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we have \(\angle ABC\cong\angle CDA\).

Filling the table row - by - row:

Statements	Reasons
\(\overline{AB}\cong\overline{CD}\) (and \(\overline{BC}\cong\overline{DA}\))	Opposite sides of a parallelogram are congruent
\(\overline{AB}\parallel\overline{CD}\)	Definition of a parallelogram
\(\angle BAC\cong\angle DCA\)	Alternate interior angles are congruent
\(\overline{AC}\cong\overline{AC}\)	Reflexive Property
\(\triangle BAC\cong\triangle DCA\)	SAS
\(\angle ABC\cong\angle CDA\)	Corresponding parts of congruent triangles are congruent

The key steps are using the properties of parallelograms (parallel and congruent opposite sides), the alternate interior angles theorem, the reflexive property of congruence, the SAS congruence criterion for triangles, and then CPCTC to show that the opposite angles \(\angle ABC\) and \(\angle CDA\) are congruent.

So, to complete the proof:

• The statement for "Opposite sides of a parallelogram are congruent" is \(\boldsymbol{\overline{AB}\cong\overline{CD}}\) (or \(\overline{BC}\cong\overline{DA}\))
• The reason for \(\overline{AB}\parallel\overline{CD}\) is \(\boldsymbol{\text{Definition of a parallelogram}}\)
• The alternate interior angles are \(\boldsymbol{\angle BAC\cong\angle DCA}\)
• The congruence criterion for \(\triangle BAC\cong\triangle DCA\) is \(\boldsymbol{\text{SAS}}\)
• The final statement to show \(\angle ABC\cong\angle CDA\) is \(\boldsymbol{\angle ABC\cong\angle CDA}\) (by CPCTC)

Answer:

To solve the two - column proof for showing that \(\angle ABC\cong\angle CDA\) in parallelogram \(ABCD\) with diagonal \(\overline{AC}\), we follow these steps:

Step 1: Fill in the statement for "Opposite sides of a parallelogram are congruent"

In a parallelogram, opposite sides are congruent. So, for parallelogram \(ABCD\), we have \(\overline{AB}\cong\overline{CD}\) and \(\overline{BC}\cong\overline{DA}\). We can use \(\overline{AB}\cong\overline{CD}\) (or \(\overline{BC}\cong\overline{DA}\)) here. Let's take \(\overline{AB}\cong\overline{CD}\) (and we can also note \(\overline{BC}\cong\overline{DA}\) later).

Step 2: Fill in the reason for \(\overline{AB}\parallel\overline{CD}\)

By the definition of a parallelogram, opposite sides of a parallelogram are parallel. So the reason for \(\overline{AB}\parallel\overline{CD}\) is "Definition of a parallelogram".

Step 3: Find the alternate interior angles

Since \(\overline{AB}\parallel\overline{CD}\) and \(\overline{AC}\) is a transversal, the alternate interior angles are \(\angle BAC\cong\angle DCA\).

Step 4: Determine the congruence criterion for \(\triangle BAC\cong\triangle DCA\)

We know that \(\overline{AB}\cong\overline{CD}\) (from step 1), \(\angle BAC\cong\angle DCA\) (from step 3), and \(\overline{AC}\cong\overline{AC}\) (by the reflexive property). So, by the Side - Angle - Side (SAS) congruence criterion, \(\triangle BAC\cong\triangle DCA\). The reason for \(\triangle BAC\cong\triangle DCA\) is "SAS".

Step 5: Conclude the congruence of \(\angle ABC\) and \(\angle CDA\)

Since \(\triangle BAC\cong\triangle DCA\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we have \(\angle ABC\cong\angle CDA\).

Filling the table row - by - row:

Statements	Reasons
\(\overline{AB}\cong\overline{CD}\) (and \(\overline{BC}\cong\overline{DA}\))	Opposite sides of a parallelogram are congruent
\(\overline{AB}\parallel\overline{CD}\)	Definition of a parallelogram
\(\angle BAC\cong\angle DCA\)	Alternate interior angles are congruent
\(\overline{AC}\cong\overline{AC}\)	Reflexive Property
\(\triangle BAC\cong\triangle DCA\)	SAS
\(\angle ABC\cong\angle CDA\)	Corresponding parts of congruent triangles are congruent

The key steps are using the properties of parallelograms (parallel and congruent opposite sides), the alternate interior angles theorem, the reflexive property of congruence, the SAS congruence criterion for triangles, and then CPCTC to show that the opposite angles \(\angle ABC\) and \(\angle CDA\) are congruent.

So, to complete the proof:

• The statement for "Opposite sides of a parallelogram are congruent" is \(\boldsymbol{\overline{AB}\cong\overline{CD}}\) (or \(\overline{BC}\cong\overline{DA}\))
• The reason for \(\overline{AB}\parallel\overline{CD}\) is \(\boldsymbol{\text{Definition of a parallelogram}}\)
• The alternate interior angles are \(\boldsymbol{\angle BAC\cong\angle DCA}\)
• The congruence criterion for \(\triangle BAC\cong\triangle DCA\) is \(\boldsymbol{\text{SAS}}\)
• The final statement to show \(\angle ABC\cong\angle CDA\) is \(\boldsymbol{\angle ABC\cong\angle CDA}\) (by CPCTC)