Question

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

1. definition of a

rectangle

1. all right angles are

congruent.
9.

1. \\( \triangle bla \cong \triangle cmd \\)

Explanation:

Step1: Use the property of rectangle

Since \(ABCD\) is a rectangle, \(AB = CD\) (opposite sides of a rectangle are equal).

Step2: Use the given congruent segments

Given \(BP\cong CP\), and from the rectangle \( \angle BAL=\angle CDM = 90^{\circ}\) (definition of a rectangle).

Step3: Prove triangles congruent

In \(\triangle BLA\) and \(\triangle CMD\), \(BP = CP\) (given), \(AB=CD\) (property of rectangle), \(\angle BAL=\angle CDM\) (all right angles are congruent). By Hypotenuse - Leg (HL) criterion (if we consider right - angled triangles \(\triangle BLA\) and \(\triangle CMD\) where \(BP\) and \(CP\) are hypotenuses in a sense related to the right - angled triangles formed), \(\triangle BLA\cong\triangle CMD\).

Step4: Conclude the segment congruence

Since \(\triangle BLA\cong\triangle CMD\), then \(AL\cong DM\) (corresponding parts of congruent triangles are congruent).

Answer:

1. \(AB = CD\)
2. \(\angle BAL=\angle CDM\)
3. \(BP = CP\)
4. Hypotenuse - Leg (HL) criterion