In the diagram below, $\overline{A C} \cong \overline{D B}$.
Complete the proof that a parallelogram whose diagonals are congruent has a right angle.

This proof uses this theorem: Opposite sides in a parallelogram are congruent.
\begin{tabular}{lll}
& Statement & Reason \
\hline 1 & $\overline{A B} \| \overline{C D}$ and & Given \
& $\overline{A D} \| \overline{B C}$ & \
2 & $\overline{A B} \cong \overline{D C}$ & Opposite sides in a parallelogram are congruent. (1) \
3 & $\overline{A D} \cong \overline{D A}$ & Line segments are congruent to themselves. \
4 & $\overline{A C} \cong \overline{D B}$ & Given \
5 & Pick statement $\vee$ & Side-side-side congruence (2,3,4) \
6 & Pick statement $\vee$ & Pick reason $\vee$ \
7 & $m \angle 1+m \angle 4=180^{\circ}$ & When a transversal crosses parallel lines, same-side \
interior angles sum up to $180^{\circ}$. (1)
\end{tabular}

Question

In the diagram below, $\overline{A C} \cong \overline{D B}$.
Complete the proof that a parallelogram whose diagonals are congruent has a right angle.

This proof uses this theorem: Opposite sides in a parallelogram are congruent.
\begin{tabular}{lll} 
& Statement & Reason \
\hline 1 & $\overline{A B} \| \overline{C D}$ and & Given \
& $\overline{A D} \| \overline{B C}$ & \
2 & $\overline{A B} \cong \overline{D C}$ & Opposite sides in a parallelogram are congruent. (1) \
3 & $\overline{A D} \cong \overline{D A}$ & Line segments are congruent to themselves. \
4 & $\overline{A C} \cong \overline{D B}$ & Given \
5 & Pick statement $\vee$ & Side-side-side congruence (2,3,4) \
6 & Pick statement $\vee$ & Pick reason $\vee$ \
7 & $m \angle 1+m \angle 4=180^{\circ}$ & When a transversal crosses parallel lines, same-side \
interior angles sum up to $180^{\circ}$. (1)
\end{tabular}

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To prove that a parallelogram with congruent diagonals has a right angle, we need to show that the triangles formed by the diagonals are congruent. ‖ ‖ step 2 ⋮ By the given information, we know that $\overline{AC} \cong \overline{DB}$ and $\overline{AB} \cong \overline{DC}$ (opposite sides of a parallelogram are congruent). ‖ ‖ step 3 ⋮ Since $\overline{AD}$ is congruent to itself (reflexive property), we have all three sides of $ riangle ABD$ congruent to the corresponding sides of $ riangle BAC$. ‖ ‖ step 4 ⋮ By the Side-Side-Side (SSS) congruence postulate, $ riangle ABD \cong riangle BAC$. ‖ ‖ step 5 ⋮ From the congruence of the triangles, we know that corresponding angles are congruent, so $\angle A \cong \angle C$. ‖ ‖ step 6 ⋮ Since $\overline{AB} \| \overline{CD}$ and $\overline{AD}$ is a transversal, $\angle A$ and $\angle D$ are same-side interior angles. ‖ ‖ step 7 ⋮ By the same-side interior angles theorem, $m\angle A + m\angle D = 180^\circ$. Since $\angle A \cong \angle C$, we can substitute to get $m\angle C + m\angle D = 180^\circ$. ‖ ‖ step 8 ⋮ If two angles sum to $180^\circ$ and are congruent, each must be $90^\circ$. Therefore, $m\angle A = m\angle C = m\angle D = 90^\circ$. ‖ ∻Answer∻ ⚹ The parallelogram has a right angle at each vertex, proving that a parallelogram with congruent diagonals is a rectangle. ⚹ ∻Key Concept∻ ⚹Side-Side-Side (SSS) Congruence Postulate⚹ ∻Explanation∻ ⚹The SSS Congruence Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. This leads to the conclusion that corresponding angles are congruent, and in the case of a parallelogram with congruent diagonals, it results in right angles at each vertex.◊What is the relationship between the angles in a parallelogram when the diagonals are congruent?⍭ Generate me a similar question◊⚹

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