Question

use the image to answer the question

drag and drop the statements and reasons into their correct locations in the two - column proof that parallelogram ( a b c d ) with congruent diagonals ( overline{d b} ) and ( overline{a c} ) is a rectangle.
(2 points)

two - column proof

1 ( a b c d ) is a parallelogram. given
2 ( overline{d b} cong overline{a c} ) given
3 ( overline{d a} cong overline{c b} ) opposite sides of a parallelogram are congruent
4 ( overline{a b} cong overline{b a} ) reflexive property of congruence
5 ( \triangle b a d cong \triangle a b c ) sss congruence theorem
6 ( angle b a d cong angle a b c )
7 ( m angle b a d=m angle a b c ) definition of congruence
8 ( m angle b a d+m angle a b c = 180^{circ} ) consecutive angles of a parallelogram are supplementary
9 ( m angle b a d+m angle b a d = 180^{circ} )
10 ( 2 m angle b a d = 180^{circ} ) combine like terms
11
12 ( m angle a b c = 90^{circ} ) transitive property of equality
13 ( m angle a d c=m angle d a b = 90^{circ} ) opposite angles of a parallelogram are congruent
14 ( angle b a d, angle d c b, angle a b c ), and ( angle a d c ) are right angles definition of a right angle
15 ( a b c d ) is a rectangle definition of a rectangle

Explanation:

Step1: Find reason for statement 6

Since \(\triangle BAD\cong\triangle ABC\) (from statement 5), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle BAD\cong\angle ABC\).

Step2: Find reason for statement 9

Because \(m\angle BAD = m\angle ABC\) (from statement 7) and \(m\angle BAD+m\angle ABC = 180^{\circ}\) (from statement 8), we substitute \(m\angle ABC\) with \(m\angle BAD\) in the equation \(m\angle BAD+m\angle ABC = 180^{\circ}\). So the reason is Substitution Property of Equality.

Step3: Find statement 11

From \(2m\angle BAD=180^{\circ}\) (statement 10), using the Division Property of Equality (\(a = b\Rightarrow\frac{a}{c}=\frac{b}{c},c
eq0\)), we divide both sides of the equation \(2m\angle BAD = 180^{\circ}\) by 2. So \(m\angle BAD=90^{\circ}\).

Answer:

1. Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
2. Reason: Substitution Property of Equality
3. Statement: \(m\angle BAD = 90^{\circ}\)