Question

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

1.

(03.04 MC)

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


Parallelogram ABCD is shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD.


According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD. Construct diagonal A C with a straightedge. It is congruent to itself by the Reflexive Property of Equality. Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem. Angles BCA and DAC are congruent by the same theorem. __________. By CPCTC, opposite sides AB and CD, as well as sides BC and DA, are congruent.


Which sentence accurately completes the proof? (5 points)



Triangles BCA and DAC are congruent according to the Angle-Angle-Side (AAS) Theorem.


Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.


Angles ABC and CDA are congruent according to a property of parallelograms (opposite angles congruent).


Angles BAD and ADC, as well as angles DCB and CBA, are supplementary by the Same-Side Interior Angles Theorem.

Answer 1

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

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."