Question

in quadrilateral abcd, segment ad is congruent to segment bc, and line ad is parallel to line bc.
prove that abcd is a parallelogram.
since line ad is parallel to line, alternate interior angles dac and are congruent.
segment ac is congruent to segment since segments are congruent to themselves. along with the given information that segment ad is congruent to segment, triangle adc is congruent to cba by (sas, sss, or asa) triangle congruence. since the triangles are congruent, all pairs of corresponding angles are congruent, so angle dca is congruent to. since those alternate interior angles are congruent, line ab must be parallel to line. since we define a parallelogram of opposite sides parallel, abcd is a parallelogram.

Explanation:

Step1: Identificar ángulos internos alternos

Dado que \(AD\parallel BC\), los ángulos internos alternos \(\angle DAC\) y \(\angle BCA\) son congruentes.

Step2: Segmento común

El segmento \(AC\) es congruente a sí mismo (\(AC = AC\)) por la propiedad reflexiva.

Step3: Triángulos congruentes

Dado que \(AD = BC\) (por hipótesis), \(\angle DAC=\angle BCA\) (ángulos internos alternos) y \(AC = AC\) (propiedad reflexiva), entonces \(\triangle ADC\cong\triangle CBA\) por el criterio \(SAS\) (lado - ángulo - lado).

Step4: Ángulos correspondientes

Como \(\triangle ADC\cong\triangle CBA\), los ángulos correspondientes \(\angle DCA\) y \(\angle BAC\) son congruentes.

Step5: Paralelismo de \(AB\) y \(DC\)

Como \(\angle DCA=\angle BAC\) (ángulos internos alternos), entonces \(AB\parallel DC\).

Answer:

Since line \(AD\) is parallel to line \(BC\), alternate interior angles \(DAC\) and \(BCA\) are congruent. Segment \(AC\) is congruent to segment \(AC\) since segments are congruent to themselves. Along with the given information that segment \(AD\) is congruent to segment \(BC\), triangle \(ADC\) is congruent to \(CBA\) by \(SAS\) triangle congruence. Since the triangles are congruent, all pairs of corresponding angles are congruent, so angle \(DCA\) is congruent to \(BAC\). Since those alternate interior angles are congruent, line \(AB\) must be parallel to line \(DC\). Since we define a parallelogram of opposite sides parallel, \(ABCD\) is a parallelogram.