Question

1. given: $overline{bd}perpoverline{bc}$; $angle abdcongangle dbe$ prove: $angle abd$ and $angle ebc$ are complementary statements reasons 1. $overline{bd}perpoverline{bc}$ 1. 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: Given information

Given

Step2: Definition of perpendicular lines

If two lines are perpendicular, the angle between them is a right - angle

Step3: Definition of a right - angle

The measure of a right - angle is 90°

Step4: Angle addition postulate

The measure of the whole angle is the sum of the measures of its non - overlapping parts

Step5: Substitution property

Substitute m∠DBC = 90° into m∠DBE + m∠EBC=m∠DBC

Step6: Given information

Given

Step7: Definition of congruent angles

Congruent angles have equal measures

Step8: Substitution property

Substitute m∠ABD for m∠DBE in m∠DBE + m∠EBC = 90°

Step9: Definition of complementary angles

Two angles are complementary if the sum of their measures is 90°

Answer:

Statements	Reasons
2. $\angle DBC$ is a right angle	2. Definition of perpendicular lines
3. $m\angle DBC = 90^{\circ}$	3. Definition of a right - angle
4. $m\angle DBE+m\angle EBC=m\angle DBC$	4. Angle addition postulate
5. $m\angle DBE + m\angle EBC=90^{\circ}$	5. Substitution property (substitute $m\angle DBC = 90^{\circ}$ into statement 4)
6. $\angle ABD\cong\angle DBE$	6. Given
7. $m\angle ABD = m\angle DBE$	7. Definition of congruent angles
8. $m\angle ABD+m\angle EBC = 90^{\circ}$	8. Substitution property (substitute $m\angle ABD$ for $m\angle DBE$ in statement 5)
9. $\angle ABD$ and $\angle EBC$ are complementary	9. Definition of complementary angles