Question

in parallelogram abcd on the right, draw the two diagonal lines and one straight line passing through intersection point o of the diagonal lines in the direction of sides ad and bc. then, let the intersection points of sides ad and bc be e and f respectively. in this case, prove that eo = fo as follows.
(1) draw a line segment on the right figure according to the problem.
(2) fill in the blanks and complete the proof.
proof
in △aoe and △
,
since the diagonals intersect at the midpoint of each,
( the third property of a parallelogram )
= …………①
from ad || cb, ∠ = ∠ …………②
since the opposite angles are equal,
∠ = ∠ …………③
from ①, ②, and ③, since the two and the included are equal respectively,
△ ≅ △.
therefore, =.

Explanation:

Step1: Identify the congruent triangles

In $\triangle AOE$ and $\triangle COF$, because of the properties of parallelograms.

Step2: Use the property of parallelogram diagonals

Since the diagonals of a parallelogram bisect each other, $AO = CO$ (①).

Step3: Use the property of parallel lines

From $AD\parallel CB$, $\angle OAE=\angle OCF$ (②) (alternate - interior angles).

Step4: Use the property of vertical angles

Since the opposite angles (vertical angles) are equal, $\angle AOE=\angle COF$ (③).

Step5: Apply the ASA (Angle - Side - Angle) congruence criterion

From ①, ②, and ③, since the two angles and the included side are equal respectively, $\triangle AOE\cong\triangle COF$.

Step6: Conclude the result

Therefore, $EO = FO$.

Answer:

In $\triangle AOE$ and $\triangle COF$, $AO = CO$ (①), $\angle OAE=\angle OCF$ (②), $\angle AOE=\angle COF$ (③). From ①, ②, and ③, since the two angles and the included side are equal respectively, $\triangle AOE\cong\triangle COF$. Therefore, $EO = FO$.

So the blanks are filled as follows:
In $\triangle AOE$ and $\triangle\boldsymbol{COF}$, $AO = CO$ (①), $\angle\boldsymbol{OAE}=\angle\boldsymbol{OCF}$ (②), $\angle\boldsymbol{AOE}=\angle\boldsymbol{COF}$ (③). From ①, ②, and ③, since the two angles and the included side are equal respectively, $\triangle\boldsymbol{AOE}\cong\triangle\boldsymbol{COF}$. Therefore, $\boldsymbol{EO}=\boldsymbol{FO}$.