Question

4 in parallelogram abcd on the right, given m and n are the midpoints of sides ad and bc respectively, prove an = cm using the properties of a parallelogram, as follows. fill the appropriate words in the blanks.
proof
in △abn and △
,
since the opposite sides of a parallelogram are equal,
(the first property of a parallelogram)
= …………①
since the opposite angles of a parallelogram are equal,
(the second property of a parallelogram)
∠= ∠ …………②
bn = 1/2 and dm = 1/2.
from bc = ad, = …………③
from ①, ②, and ③, since the two and the included are equal respectively,
△≅△.
therefore, an = cm.
★ like 3 and 4, the property of a parallelogram can be derived from its other properties.

Explanation:

Step1: Identify the triangles

We want to prove \(AN = CM\). Consider \(\triangle ABN\) and \(\triangle CDM\).

Step2: Use the property of parallelogram sides

In parallelogram \(ABCD\), \(AB = CD\) (opposite sides of a parallelogram are equal).

Step3: Use the property of parallelogram angles

\(\angle ABN=\angle CDM\) (opposite angles of a parallelogram are equal).

Step4: Analyze the lengths of \(BN\) and \(DM\)

Since \(N\) is the mid - point of \(BC\) and \(M\) is the mid - point of \(AD\), and \(BC = AD\) (opposite sides of a parallelogram), \(BN=\frac{1}{2}BC\) and \(DM = \frac{1}{2}AD\), so \(BN=DM\).

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

In \(\triangle ABN\) and \(\triangle CDM\), two sides (\(AB = CD\) and \(BN = DM\)) and the included angle (\(\angle ABN=\angle CDM\)) are equal. So, \(\triangle ABN\cong\triangle CDM\) (by SAS congruence).

Answer:

In \(\triangle ABN\) and \(\triangle CDM\), \(AB = CD\), \(\angle ABN=\angle CDM\), \(BN=\frac{1}{2}BC\), \(DM=\frac{1}{2}AD\), \(BN = DM\), two sides and the included angle, \(\triangle ABN\cong\triangle CDM\)