Question

given: rectangle abcd,
\\( \overline { bp } \cong \overline { cp } \\)
prove: \\( \overline { al } \cong \overline { dm } \\)

1. \\( \overline { bl } \cong \overline { cm } \\)
2. \\( \triangle bla \cong \triangle cmd \\)
3. \\( \overline { al } \cong \overline { dm } \\)
4. if a quadrilateral is

a rectangle/parallel-
ogram, then both
pairs of opposite
sides are congruent.

1. hypotenuse - leg
2. click here to insert

Explanation:

Step1: Properties of rectangle

In rectangle \(ABCD\), \(AB = CD\) (opposite sides of a rectangle are equal) and \(\angle BAL=\angle CDM = 90^{\circ}\) (angles of a rectangle are right - angles).

Step2: Given information

Given \(BP = CP\), and since \(ABCD\) is a rectangle, \(AB = CD\). Also, from \(\triangle BLA\) and \(\triangle CMD\), we know that \(\triangle BLA\cong\triangle CMD\) (by Hypotenuse - Leg congruence criterion as \(BP = CP\) (hypotenuse), \(AB = CD\) (leg) and \(\angle BAL=\angle CDM = 90^{\circ}\)).

Step3: Congruent parts of congruent triangles

If \(\triangle BLA\cong\triangle CMD\), then by the property of congruent triangles (corresponding parts of congruent triangles are congruent), \(AL = DM\).

Answer:

\(\overline{AL}\cong\overline{DM}\) because \(\triangle BLA\cong\triangle CMD\) (by Hypotenuse - Leg) and corresponding parts of congruent triangles are congruent.