Question

given: ∠abc is a right angle, ∠dbc is a straight angle
prove: ∠abc ≅ ∠abd
statements

1. ∠abc is a right angle
2. ∠dbc is a straight angle
3. m∠abc = 90°
4. m∠dbc = 180°
5. m∠abd + m∠abc = m∠dbc
6. m∠abd + 90° = 180°
7. m∠abd = 90°
8. 90° = m∠abd
9. m∠abc = m∠abd
10. ∠abc ≅ ∠abd

reasons

1. given
2. given
3. definition of right angle
4. definition of straight angle
5. angle addition property
6. substitution property
7. subtraction property
8. symmetric property

9.

1. definition of congruent angles

Explanation:

Step1: Recall given information

We know $\angle ABC$ is a right - angle and $\angle DBC$ is a straight - angle.

Step2: Define angle measures

By definition, $m\angle ABC = 90^{\circ}$ (right - angle) and $m\angle DBC=180^{\circ}$ (straight - angle).

Step3: Apply angle addition property

Since $\angle ABD+\angle ABC=\angle DBC$, we substitute the known measures: $m\angle ABD + 90^{\circ}=180^{\circ}$.

Step4: Solve for $m\angle ABD$

Using the subtraction property, we get $m\angle ABD=180^{\circ}-90^{\circ}=90^{\circ}$.

Step5: Prove congruence

Since $m\angle ABC = 90^{\circ}$ and $m\angle ABD = 90^{\circ}$, by the definition of congruent angles (angles with equal measures are congruent), $\angle ABC\cong\angle ABD$.

Answer:

The proof is completed as shown in the statements and reasons in the table. $\angle ABC\cong\angle ABD$ because both angles have a measure of $90^{\circ}$.