Question

3 using the first property of a parallelogram, prove the third property of a parallelogram is as follows. fill the appropriate words in the blanks.
●the third property of a parallelogram
「the two diagonals intersect at the midpoint of each.」
proof
in parallelogram abcd on the right,
let point o be the intersection point of the diagonals.
in △abo and △cdo,
from the first property of a parallelogram,
□ = □ ………①
since ab || dc, ∠bao = ∠dco ………②
∠□ = ∠□ ………③
from ①, ②, and ③,
since the two □ and the included □ are equal respectively,
△□ ≅ △□.
so, ao = co and bo = do.
therefore, the two diagonals of a parallelogram intersect at the midpoint of each.

Explanation:

Step1: Use the property of parallelogram

In a parallelogram \(ABCD\), \(AB = DC\) (the first property of a parallelogram: opposite sides of a parallelogram are equal). So, in \(\triangle ABO\) and \(\triangle CDO\), the first blank is \(AB\) and the second blank is \(DC\).

Step2: Find another pair of equal angles

Since \(AB\parallel DC\), by the alternate - interior angles theorem, \(\angle ABO=\angle CDO\). So, the third blank is \(ABO\) and the fourth blank is \(CDO\).

Step3: Use the ASA (Angle - Side - Angle) congruence criterion

From ① (\(AB = DC\)), ② (\(\angle BAO=\angle DCO\)), and ③ (\(\angle ABO=\angle CDO\)), since the two angles and the included side are equal respectively. The fifth blank is “angles”, the sixth blank is “side”.
For the congruent triangles, \(\triangle ABO\cong\triangle CDO\) (by ASA congruence criterion). So, the seventh blank is \(ABO\) and the eighth blank is \(CDO\).

Answer:

1. \(AB\), \(DC\)
2. \(ABO\), \(CDO\)
3. “angles”, “side”
4. \(ABO\), \(CDO\)