Question

statements1. $angle abc$ is a right angle2. $angle dbc$ is a straight angle3. $mangle abc=90^circ$4. $mangle dbc=180^circ$5. $mangle abd + mangle abc = mangle dbc$6. $mangle abd + 90^circ = 180^circ$7. $mangle abd=90^circ$8. $90^circ=mangle abd$9. $mangle abc = mangle abd$10. $angle abc cong angle abd$reasons1. given2. given3. definition of right angle4. definition of straight angle5. angle addition property6. substitution property7. subtraction property8. symmetric property9.10. definition of congruent angles$\bigcirc$ definition of angle bisector$\bigcirc$ segment addition property$\bigcirc$ definition of congruent angles$\bigcirc$ transitive property

Explanation:

We need to identify the reason for step 9: $m\angle ABC = m\angle ABD$. From steps 3 and 7, we know $m\angle ABC=90^\circ$ and $m\angle ABD=90^\circ$. The transitive property states that if $a=b$ and $b=c$, then $a=c$; here, since both angles equal $90^\circ$, they equal each other via this property. The other options do not fit: angle bisector relates to splitting an angle, segment addition applies to lengths, and congruent angles definition is used in step 10, not step 9.

Answer:

transitive property