Parallelogram ABCD is represented with diagonal AC.
Quadrilateral ABCD is a parallelogram. Complete the statements to prove that
and
.
Statement Reason
Quadrilateral ABCD is a parallelogram. given
and
definition of a parallelogram
Draw
.
is a transversal that intersects two pairs of parallel lines,
and
and
and
. drawing a line segment
∠CAB ≅ ∠ACD and ∠BCA ≅ ∠CAD
m∠CAB =m∠ACD and m∠BCA = m∠CAD
Reflexive Property of Equality
ASA criterion for congruence
and
Corresponding sides of congruent triangles are congruent.
Explanation:
- The statement "Quadrilateral ABCD is a parallelogram" is given in the problem.
- The definition of a parallelogram states that opposite sides of a parallelogram are parallel.
- By drawing diagonal AC, line segment AD forms a transversal that intersects two pairs of parallel lines, AB and DC.
- By definition, corresponding angles formed by a transversal and two parallel lines are congruent. Therefore, ∠CAB ≅ ∠ACD and ∠BCA ≅ ∠CAD.
- The statement "m∠CAB = m∠ACD and m∠BCA = m∠CAD" is using the notation for angle measures.
- The reflexive property of equality states that a quantity is equal to itself, so m∠CAB = m∠ACD and m∠BCA = m∠CAD are true.
- The ASA criterion for congruence states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Since ∠CAB ≅ ∠ACD and ∠BCA ≅ ∠CAD, and AC is the included side, triangle ABC ≅ triangle ACD by ASA.
- Finally, corresponding sides of congruent triangles are congruent. Therefore, AB ≅ AD and BC ≅ CD.