Question

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. given
9. given
10. definition of right angle
11. definition of straight angle
12. angle addition property
13. substitution property
14. subtraction property

what is the format of this proof?
given ∠abc is a right angle, ∠dbc is a straight angle
prove ∠abd is a right angle

Explanation:

Step1: Analyze the given information

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

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

Since \(m\angle ABC = 90^{\circ}\) and \(m\angle DBC = 180^{\circ}\), substituting these values into the equation \(m\angle ABD + m\angle ABC=m\angle DBC\) gives \(m\angle ABD+90^{\circ}=180^{\circ}\).

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

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

Answer:

The proof uses the definitions of right - angle and straight - angle, the angle addition property, and the subtraction property of equality. The steps are:

1. Given \(\angle ABC\) is a right - angle (\(\angle ABC = 90^{\circ}\)) and \(\angle DBC\) is a straight - angle (\(\angle DBC = 180^{\circ}\)).
2. By angle addition property \(m\angle ABD + m\angle ABC=m\angle DBC\).
3. Substitute \(m\angle ABC = 90^{\circ}\) and \(m\angle DBC = 180^{\circ}\) into the equation: \(m\angle ABD+90^{\circ}=180^{\circ}\).
4. By subtraction property of equality, \(m\angle ABD=90^{\circ}\), so \(\angle ABD\) is a right - angle.