Question

1. given: $overline{bd} perp overline{bc}$; $angle abd cong angle dbe$

prove: $angle abd$ and $angle ebc$ are complementary

statements	reasons
2. $angle dbc$ is a right angle	2.
3. $mangle dbc = 90^circ$	3.
4. $mangle dbe + mangle ebc = mangle dbc$	4.
5. $mangle dbe + mangle ebc = 90^circ$	5.
6. $angle abd cong angle dbe$	6.
7. $mangle abd = mangle dbe$	7.
8. $mangle abd + mangle ebc = 90^circ$	8.
9. $angle abd$ and $angle ebc$ are complementary	9.

Explanation:

Step1: Identify Reason for Statement 1

Given that \(\overline{BD} \perp \overline{BC}\), the reason is "Given".

Step2: Reason for Statement 2

If two lines are perpendicular, they form a right angle. So the reason for \(\angle DBC\) being a right angle is "Definition of perpendicular lines".

Step3: Reason for Statement 3

A right angle has a measure of \(90^\circ\), so the reason for \(m\angle DBC = 90^\circ\) is "Definition of a right angle".

Step4: Reason for Statement 4

The angle - addition postulate states that if a point lies in the interior of an angle, the sum of the measures of the two smaller angles is equal to the measure of the larger angle. Here, \(E\) lies in the interior of \(\angle DBC\), so \(m\angle DBE + m\angle EBC=m\angle DBC\) by the "Angle - Addition Postulate".

Step5: Reason for Statement 5

We know from step 3 that \(m\angle DBC = 90^\circ\) and from step 4 that \(m\angle DBE + m\angle EBC=m\angle DBC\). By "Substitution Property" (substituting \(m\angle DBC = 90^\circ\) into the equation from step 4), we get \(m\angle DBE + m\angle EBC = 90^\circ\).

Step6: Reason for Statement 6

The problem states that \(\angle ABD\cong\angle DBE\), so the reason is "Given".

Step7: Reason for Statement 7

If two angles are congruent, their measures are equal. So the reason for \(m\angle ABD = m\angle DBE\) is "Definition of congruent angles".

Step8: Reason for Statement 8

We know from step 5 that \(m\angle DBE + m\angle EBC = 90^\circ\) and from step 7 that \(m\angle ABD = m\angle DBE\). By "Substitution Property" (substituting \(m\angle ABD\) for \(m\angle DBE\) in the equation from step 5), we get \(m\angle ABD + m\angle EBC = 90^\circ\).

Step9: Reason for Statement 9

Two angles are complementary if the sum of their measures is \(90^\circ\). Since \(m\angle ABD + m\angle EBC = 90^\circ\), the reason is "Definition of complementary angles".

Answer:

1. Given
2. Definition of perpendicular lines
3. Definition of a right angle
4. Angle - Addition Postulate
5. Substitution Property (using \(m\angle DBC = 90^\circ\) from step 3)
6. Given
7. Definition of congruent angles
8. Substitution Property (using \(m\angle ABD=m\angle DBE\) from step 7)
9. Definition of complementary angles