Question

1. given: $overline{bd}perpoverline{bc}$; $angle abdcongangle dbe$ 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. $angle abdcongangle dbe$	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: Definition of perpendicular lines

If two lines are perpendicular, the angle between them is a right - angle. So, if $\overline{BD}\perp\overline{BC}$, then $\angle DBC$ is a right - angle.

Step2: Definition of a right - angle

A right - angle has a measure of $90^{\circ}$. So, if $\angle DBC$ is a right - angle, then $m\angle DBC = 90^{\circ}$.

Step3: Angle addition postulate

The measure of the whole angle ($\angle DBC$) is equal to the sum of the measures of its non - overlapping parts ($\angle DBE$ and $\angle EBC$), i.e., $m\angle DBE + m\angle EBC=m\angle DBC$.

Step4: Substitution property

Since $m\angle DBC = 90^{\circ}$ (from step 2) and $m\angle DBE + m\angle EBC=m\angle DBC$ (from step 3), we substitute $m\angle DBC$ with $90^{\circ}$ to get $m\angle DBE + m\angle EBC = 90^{\circ}$.

Step5: Given

We are given that $\angle ABD\cong\angle DBE$.

Step6: Definition of congruent angles

Congruent angles have equal measures. So, if $\angle ABD\cong\angle DBE$, then $m\angle ABD=m\angle DBE$.

Step7: Substitution property

Since $m\angle DBE + m\angle EBC = 90^{\circ}$ (from step 4) and $m\angle ABD=m\angle DBE$ (from step 6), we substitute $m\angle DBE$ with $m\angle ABD$ to get $m\angle ABD + m\angle EBC = 90^{\circ}$.

Step8: Definition of complementary angles

Two angles are complementary if the sum of their measures is $90^{\circ}$. Since $m\angle ABD + m\angle EBC = 90^{\circ}$, $\angle ABD$ and $\angle EBC$ are complementary.

Reasons:

1. Given
2. Definition of perpendicular lines
3. Definition of a right - angle
4. Angle addition postulate
5. Substitution property
6. Given
7. Definition of congruent angles
8. Substitution property
9. Definition of complementary angles

Answer:

The reasons for each step are as follows:

1. Given
2. Definition of perpendicular lines
3. Definition of a right - angle
4. Angle addition postulate
5. Substitution property
6. Given
7. Definition of congruent angles
8. Substitution property
9. Definition of complementary angles