Question

here is quadrilateral abcd. prove abcd is a parallelogram. the diagram shows ad =, de =, and ea =. therefore triangle is congruent to triangle bce by the triangle congruence theorem. so m∠ade = m∠cbe because they are corresponding parts of triangles. therefore ad is parallel to bc by the angles theorem. the diagram shows ab is parallel to. both pairs of opposite sides are parallel so quadrilateral abcd must be a.

Explanation:

Step1: Identificar lados congruentes

El diagrama muestra que $\overline{AD}\cong\overline{CB}$, $\overline{DE}\cong\overline{EB}$ y $\overline{EA}\cong\overline{EC}$. Entonces, en los triángulos $\triangle ADE$ y $\triangle CBE$, tenemos tres lados congruentes respectivamente.

Step2: Aplicar teorema de congruencia de triángulos

Por el teorema Side - Side - Side (SSS), $\triangle ADE\cong\triangle CBE$.

Step3: Encontrar ángulos congruentes

Como $\triangle ADE\cong\triangle CBE$, entonces $m\angle ADE = m\angle CBE$ ya que son partes correspondientes de triángulos congruentes.

Step4: Demostrar lados paralelos

Debido a que $m\angle ADE = m\angle CBE$, entonces $\overline{AD}\parallel\overline{BC}$ por el teorema de ángulos alternos internos. Además, el diagrama muestra que $\overline{DC}\parallel\overline{AB}$.

Step5: Concluir tipo de cuadrilátero

Como ambos pares de lados opuestos son paralelos, el cuadrilátero $ABCD$ es un paralelogramo.

Answer:

El diagrama muestra $AD = CB$, $DE = EB$, y $EA = EC$. Therefore triangle $ADE$ is congruent to triangle $BCE$ by the Side - Side - Side Triangle Congruence Theorem. So $m\angle ADE = m\angle CBE$ because they are corresponding parts of congruent triangles. Therefore $AD$ is parallel to $BC$ by the Alternate Interior Angles Theorem. The diagram shows $AB$ is parallel to $DC$. Both pairs of opposite sides are parallel so quadrilateral $ABCD$ must be a parallelogram.