Question

The following is an incomplete paragraph proving that the opposite sides of parallelogram ABCD are congruent:

Answer 1

Answer:

The sentence which accurately completes the proof is: "Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem." ⇒ 2nd answer

Step-by-step explanation:

Let us revise the cases of congruence

• SSS ⇒ 3 sides in the 1st Δ ≅ 3 sides in the 2nd Δ
• SAS ⇒ 2 sides and including angle in the 1st Δ ≅ 2 sides and including angle in the 2nd Δ
• ASA ⇒ 2 angles and the side whose joining them in the 1st Δ ≅ 2 angles and the side whose joining them in the 2nd Δ
• AAS ⇒ 2 angles and one side in the 1st Δ ≅ 2 angles and one side in the 2nd Δ
• HL ⇒ hypotenuse leg of the 1st right Δ ≅ hypotenuse leg of the 2nd right Δ  

In Parallelogram ABCD

∵ Segment AB is parallel to segment DC

∵ Segment BC is parallel to segment AD

- Construct diagonal A C with a straightedge

In Δs BCA and DAC

∵ AC is congruent to itself ⇒ Reflexive Property of Equality

∵ ∠BAC and ∠DCA are congruent ⇒ Alternate Interior Angles

∵ ∠BCA and ∠DAC are congruent ⇒ Alternate Interior Angles

- AC is joining the congruent angles

∴ Δ BCA is congruent to Δ DAC by ASA Theorem of congruence

By CPCTC

∴ AB is congruent to CD

∴ BC is congruent to DA

The sentence which accurately completes the proof is: "Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem."