Question

given: $overline{ab} cong overline{ac}$, $overline{ad} perp overline{bc}$
prove: $angle b cong angle c$
proof

statements	reasons
2. $mangle adb = mangle adc = 90^circ$	2.
3. $overline{ad} cong overline{ad}$	3.
4. $\triangle adb cong \triangle adc$	4.
5. $angle b cong angle c$	5.

what should the corresponding reasons be?

1. given; 2. definition of congruence; 3. definition of congruency; 4. sas theorem; 5. cpctc
2. given; 2. definition of congruence; 3. reflexive property; 4. hl theorem; 5. cpctc
3. given; 2. definition of perpendicular lines; 3. definition of congruency; 4. sas theorem; 5. cpctc
4. given; 2. definition of perpendicular lines; 3. reflexive property; 4. hl theorem; 5. cpctc

Explanation:

1. The first statement is exactly the given information, so the reason is "Given".
2. $\overline{AD} \perp \overline{BC}$ means the angles formed are right angles (90°), which is the definition of perpendicular lines.
3. A segment is congruent to itself, which is the reflexive property of congruence.
4. $\triangle ADB$ and $\triangle ADC$ are right triangles, we have a pair of congruent hypotenuses ($\overline{AB} \cong \overline{AC}$) and a shared congruent leg ($\overline{AD} \cong \overline{AD}$), so we use the HL (Hypotenuse-Leg) theorem for right triangle congruence.
5. Once the triangles are proven congruent, corresponding parts are congruent, which is CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Given; 2. Definition of perpendicular lines; 3. Reflexive property; 4. HL theorem; 5. CPCTC