Question

given: \\( \angle abc \\) is a right angle, \\( \angle dbc \\) is a straight angle
prove: \\( \angle abc \cong \angle abd \\)

statements	reasons
2. \\( \angle dbc \\) is a straight angle	2. given
3. \\( m\angle abc = 90^{\circ} \\)	3. definition of right angle
4. \\( m\angle dbc = 180^{\circ} \\)	4. definition of straight angle
5. \\( m\angle abd + m\angle abc = m\angle dbc \\)	5. angle addition property
6. \\( m\angle abd + 90^{\circ} = 180^{\circ} \\)	6. substitution property
7. \\( m\angle abd = 90^{\circ} \\)	7. subtraction property
8. \\( 90^{\circ} = m\angle abd \\)	8. symmetric property
9. \\( m\angle abc = m\angle abd \\)	9.
10. \\( \angle abc \cong \angle abd \\)	10. definition of congruent angles

Explanation:

Step1: Recall the given information

We are given that \( \angle ABC\) is a right - angle (\(m\angle ABC = 90^{\circ}\)) and \( \angle DBC\) is a straight - angle (\(m\angle DBC=180^{\circ}\)).

Step2: Use the angle addition property

By the angle addition property \(m\angle ABD + m\angle ABC=m\angle DBC\).

Step3: Substitute the known values

Substitute \(m\angle ABC = 90^{\circ}\) and \(m\angle DBC = 180^{\circ}\) into the equation \(m\angle ABD + m\angle ABC=m\angle DBC\). We get \(m\angle ABD+90^{\circ}=180^{\circ}\).

Step4: Solve for \(m\angle ABD\)

Using the subtraction property of equality, subtract \(90^{\circ}\) from both sides of the equation \(m\angle ABD + 90^{\circ}=180^{\circ}\). So \(m\angle ABD=180^{\circ}-90^{\circ}=90^{\circ}\).

Step5: Use the definition of congruent angles

Since \(m\angle ABC = 90^{\circ}\) and \(m\angle ABD = 90^{\circ}\), by the definition of congruent angles (if \(m\angle X=m\angle Y\), then \(\angle X\cong\angle Y\)), we have \(m\angle ABC=m\angle ABD\) and \(\angle ABC\cong\angle ABD\).

Answer:

The proof is completed as shown in the steps above, and \(\angle ABC\cong\angle ABD\) is proved.