Question

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

1. \\( \angle b l a \cong \angle p b c \\),

\\( \angle c m d \cong \angle p c b \\)
5.

1. \\( \angle b l a \cong \angle c m d \\)

Explanation:

Step1: Use the property of rectangle

In rectangle \(ABCD\), \(AB = CD\) and \(AB\parallel CD\), so \(\angle ABP=\angle DCP\) (alternate - interior angles).

Step2: Use the given congruent angles

Since \(\angle BLA\cong\angle PBC\) and \(\angle CMD\cong\angle PCB\), and \(\angle PBC=\angle PCB\) (because \(BP = CP\), isosceles - triangle property).

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

In \(\triangle ABL\) and \(\triangle DCM\), \(\angle BLA=\angle CMD\) (from step 2), \(\angle A=\angle D = 90^{\circ}\) (rectangle property), \(AB = CD\) (rectangle property). So \(\triangle ABL\cong\triangle DCM\) (AAS).

Step4: Use the property of congruent triangles

If \(\triangle ABL\cong\triangle DCM\), then \(AL\cong DM\) (corresponding parts of congruent triangles are congruent).

Answer:

1. \(BP = CP\) (given), \(\angle ABP=\angle DCP\) (alternate - interior angles of \(AB\parallel CD\) in rectangle \(ABCD\)), \(\angle PBC=\angle PCB\) (isosceles - triangle property as \(BP = CP\)); 5. \(\triangle ABL\cong\triangle DCM\); 6. Corresponding parts of congruent triangles are congruent.