Answer:
1. Given
2. Diagonals of a parallelogram bisect each other.
3. Vertical angles are equal.
4. Definition of parallelogram.
5. If lines parallel, then alternate interior angles are equal.
6. ASA
7. CPCTE
Step-by-step explanation:
Statement 1:
The first statement is a parallelogram ABCD, which is already given in the question. So, reason 1 is: Given.
Statement 2:
BT and TD are equal because for a parallelogram, its diagonal bisect each other. Here, BD and AC are the diagonals of parallelogram ABCD. So, the diagonals bisect each other at T. Hence, [tex]BT = TD[/tex]
Statement 3:
Angles 1 and 2 is a pair of vertical angles. A pair of vertical angles are always equal to each other.
Statement 4:
A parallelogram is a quadrilateral whose opposite sides are parallel and equal. Hence, [tex]BC ||AD[/tex] is because of the definition of a parallelogram.
Statement 5:
Angles 3 and 4 is a pair of alternate interior angles. If two lines are parallel, then the alternate interior angles are always equal.
Statement 6:
The triangles BET and DFT are now congruent because:
i.Angle- [tex]\angle1=\angle2[/tex]
ii. Side - [tex]BT = TD[/tex]
iii. Angle - [tex]\angle3=\angle4[/tex]
Therefore, by ASA postulate the two triangles are congruent.
Statement 7:
As the two triangles are congruent, then their corresponding parts are also equal.
So, by CPCTE, [tex]ET=FT[/tex]
