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brianna4455
08.11.2020 •
Mathematics
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
Solved
Show answers
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Ответ:
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.
Ответ:
The Triangles AOB and COD are congruent by SAS Postulate
Ответ:
A
Step-by-step explanation: it was right on math nation
Ответ:
None of them have perpendicular lines except for the blue one if I am seeing right.