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

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?

a) triangles bca and dac are congruent according to the angle-angle-side (aas) theorem.

b) triangles bca and dac are congruent according to the angle-side-angle (asa) theorem.

c) angles abc and cda are congruent according to a property of parallelograms (opposite angles congruent).

d) angles bad and adc, as well as angles dcb and cba, are supplementary by the same-side interior angles theorem.

The correct option is B.

Step-by-step explanation:

Given information: and  .

Draw a diagonal AC.

In triangle BCA and DAC,

                  (Reflexive Property of Equality)

                   ( Alternate Interior Angles Theorem)

                   ( Alternate Interior Angles Theorem)

The ASA (Angle-Side-Angle) postulate states that two triangles are congruent if two corresponding angles and the included side of are congruent.

By ASA postulate,

Therefore option B is correct.

(B)

Step-by-step explanation:

We have to prove that the opposite sides of parallelogram ABCD are congruent.

Proof: 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.

From ΔABC and ΔACD, we have

∠BAC≅∠DCA (Alternate interior angles theorem)

AC≅AC (Reflexive property)

∠BCA≅∠DAC(Alternate interior angles theorem)

Therefore,  ΔABC ≅ ΔACD by ASA rule of congruency.

Thus, option B is correct that is Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

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d

step-by-step explanation:

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

  region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

  region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true