Question

mp3 write a 2 - column proof for each exercise.

1. given: the figure

prove: ( \triangle abdcong\triangle cdb )

Explanation:

Step1: Identify given information

• \(AB\parallel CD\) (given by the arrow marks indicating parallel lines)
• \(AB = CD\) (given by the tick marks)
• \(\angle A=\angle C = 90^{\circ}\) (right - angle marks)

Step2: Use SAS (Side - Angle - Side) congruence criterion

In \(\triangle ABD\) and \(\triangle CDB\):

• \(AB = CD\) (side)
• \(\angle A=\angle C\) (angle)
• \(AD = CB\) (since \(AB\parallel CD\) and \(AB = CD\), the figure is a parallelogram, and in a parallelogram \(AD = CB\))

Answer:

Statements	Reasons
\(AB = CD\)	Given (from the figure's tick marks)
\(\angle A=\angle C = 90^{\circ}\)	Given (from the right - angle marks)
\(AD = CB\)	If \(AB\parallel CD\) and \(AB = CD\), the quadrilateral \(ABCD\) is a parallelogram (by the definition of a parallelogram: a quadrilateral with one pair of opposite sides parallel and equal is a parallelogram), and in a parallelogram opposite sides are equal
\(\triangle ABD\cong\triangle CDB\)	SAS (Side - Angle - Side) congruence criterion (\(AB = CD\), \(\angle A=\angle C\), \(AD = CB\))