Question

1. complete the flowchart proof. then write a two - column proof. given $overline{ab}perpoverline{bc},overline{dc}perpoverline{bc}$ prove $angle bcongangle c$

statements
reasons
theorems
2.4 congruent supplements theorem
if two angles are supplementary to the same angle (or to congruent angles), then they are congruent. if $angle1$ and $angle2$ are supplementary and $angle3$ and $angle2$ are supplementary, then $angle1congangle3$. prove this theorem exercise 20 (case 2), page 179
2.5 congruent complements theorem
if two angles are complementary to the same angle (or to congruent angles), then they are congruent. if $angle4$ and $angle5$ are complementary and $angle6$ and $angle5$ are complementary, then $angle4congangle6$. prove this theorem exercise 19 (case 1), page 179
study tip
to prove the congruent supplements theorem, you must prove two cases: one with angles supplementary to the same angle and one with angles supplementary to congruent angles. the proof of the congruent complements theorem also requires

Explanation:

Step1: Recall definition of perpendicular lines

If two lines are perpendicular, the angles formed between them are right - angles. Since $\overline{AB}\perp\overline{BC}$ and $\overline{DC}\perp\overline{BC}$, $\angle B = 90^{\circ}$ and $\angle C=90^{\circ}$.

Step2: Use the definition of congruent angles

Angles that have the same measure are congruent. Since $\angle B = 90^{\circ}$ and $\angle C = 90^{\circ}$, then $\angle B\cong\angle C$.

For the flowchart:
The middle box should be $\angle B = 90^{\circ},\angle C = 90^{\circ}$

For the two - column proof:

STATEMENTS	REASONS
2. $\angle B = 90^{\circ},\angle C = 90^{\circ}$	Definition of $\perp$ lines
3. $\angle B\cong\angle C$	Definition of congruent angles (angles with equal measures are congruent)

Answer:

For the flowchart, the middle box: $\angle B = 90^{\circ},\angle C = 90^{\circ}$
For the two - column proof:

STATEMENTS	REASONS
2. $\angle B = 90^{\circ},\angle C = 90^{\circ}$	Definition of $\perp$ lines
3. $\angle B\cong\angle C$	Definition of congruent angles