3. For triangles AOB and COD, angle 1 is equal to angle 2 as they are vertical angles. Therefore, the missing phrase in Terra's proof is: D. vertical angles.
In Mathematics and Euclidean Geometry, the vertical angles theorem states that two (2) opposite vertical angles that are formed whenever two lines intersect each other are always congruent, which means being equal to each other.
By applying the vertical angles theorem to triangle AOB and triangle COD, we have the following congruent angles:
m∠1 ≅ m∠2
m∠3 ≅ m∠4
In this context, we can reasonably infer and logically deduce that the missing phrase in Terra's proof is vertical angles.
Complete Question:
Look at the quadrilateral shown below.
Terra writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram:
Terra's proof
1. AO = OC because it is given that diagonals bisect each other.
2. BO = OD because it is given that diagonals bisect each other.
3. For triangles AOB and COD, angle 1 is equal to angle 2 as they are ____
4. Therefore, the triangles AOB and COD are congruent by SAS postulate.
5. Similarly, triangles AOD and COB are congruent.
6. By CPCTC, angle ABD is equal to angle BDC and angle ADB is equal to angle DBC.
7. As the alternate interior angles are congruent the opposite sides of quadrilateral ABCD are parallel
8. Therefore, ABCD is a parallelogram.
Which is the missing phrase in Terra's proof?
alternate interior angles
corresponding angles
same-side interior angles
vertical angles