Look at the quadrilateral shown below: Image Description: A quadrilateral ABCD is shown with diagonals AC and BD intersecting in point O. Angle AOB is labeled as 1, angle BOC is labeled as 4, angle COD is labeled as 2, and angle AOD is labeled as 3.

Melissa writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram:

Melissa's proof

1. For triangles AOB and COD, angle 1 is equal to angle 2, as they are vertical angles.
2. AO = OC and BO = OD because it is given that diagonals bisect each other.
3. The _ are congruent by SAS postulate.
4. Similarly, triangles AOD and COB are congruent.
5. By CPCTC, AB is equal to DC.
6. By CPCTC, AD is equal to BC.
7. As the opposite sides are congruent, the quadrilateral ABCD is a parallelogram.

Which is the missing phrase in Melissa's proof?

Options:
A. angles ADB and CBD
B. angles AOB and COD
C. triangles ADB and CBD
D. triangles AOB and COD

A

Step-by-step explanation: it was right on math nation

The correct option is;

D. Triangles AOB and COD

Step-by-step explanation:

The given parameters are;

The diagonals of the quadrilateral ABCD = AC and BD

The angles at the point of intersection are ∠AOB = 1, ∠BOC = 4, ∠COD = 2, and ∠AOD = 3

Given that the diagonals bisect each other, we have;

Segments AO = OC and BO = OD

Also, we have;

∠1 = ∠2, and ∠3 = ∠4, given that  vertically opposite angles are equal

∴ ΔAOB ≅ ΔCOD, and ΔAOD ≅ ΔCOB, by Side-Angle-Side (SAS) rule of congruency

Therefore;

Segments AD ≅ CB, and AB ≅ DC

∠BAO ≅∠DCO and ∠DAO ≅ ∠BCO

Therefore;

∠DCB ≅ ∠DAB by angle addition

Similarly ∠CBA ≅ ∠ADC by angle addition

∴ ∠DCB and ∠CBA are supplementary and ∠ADC and ∠DAB are also supplementary

Therefore, quadrilateral ABCD is a parallelogram.

The Triangles  AOB and COD are congruent by SAS Postulate

y = - 6

step-by-step explanation:

given the point (- 2, y) lies on the graph then it is a solution of it's equation.

substitute x = - 2 into the equation for value of y

y = 3(- 2) = 3 × - 2 = - 6

the point on the graph is (- 2, 6)