Question

the figure below shows a parallelogram abcd. side ab is parallel to side dc, and side ad is parallel to side bc.
a student wrote the following sentences to prove that parallelogram abcd has two pairs of opposite sides equal.
for triangles abd and cdb, alternate interior angle abd is congruent to angle cdb because ab and dc are parallel lines. similarly, alternate interior angle adb is equal to angle cbd because ad and bc are parallel lines. db is equal to db by the reflexive property. therefore, triangles abd and cdb are congruent by ____. therefore, ab is congruent to dc and ad is congruent to bc by cpctc.
which phrase best completes the students proof?
asa postulate
hl postulate
sas postulate
sss postulate

Explanation:

Step1: Analizar los triángulos ABD y CDB

En el paralelogramo ABCD, \(AB\parallel DC\) y \(AD\parallel BC\). Entonces, \(\angle ABD=\angle CDB\) (ángulos internos alternos) y \(\angle ADB = \angle CBD\) (ángulos internos alternos). Además, \(BD = BD\) (lado común).

Step2: Aplicar el postulado ASA

El postulado ASA (Ángulo - Lado - Ángulo) establece que si dos ángulos y el lado comprendido de un triángulo son congruentes a dos ángulos y el lado comprendido de otro triángulo, entonces los triángulos son congruentes. Aquí, en \(\triangle ABD\) y \(\triangle CDB\), \(\angle ABD=\angle CDB\), \(BD = BD\) y \(\angle ADB=\angle CBD\).

Answer:

ASA Postulate