Question

statements reasons

1. $overrightarrow{ba}perpoverrightarrow{bc}$ 1. given
2. $angle abc$ is a right - angle. 2. definition of complementary angles
3. $mangle abc = 90^{circ}$ 3. definition of right angle
4. $mangle1 + mangle2 = mangle abc$ 4. angle addition postulate
5. $mangle1 + mangle2 = 90^{circ}$ 5. transitive property of equality
6. $angle1$ and $angle2$ are complementary. 6. definition of perpendicular lines

Explanation:

Step1: Given perpendicular lines

We are given $\overrightarrow{BA}\perp\overrightarrow{BC}$.

Step2: Determine angle type

By the definition of perpendicular lines, $\angle ABC$ is a right - angle.

Step3: Define right - angle measure

By the definition of a right - angle, $m\angle ABC = 90^{\circ}$.

Step4: Apply angle addition postulate

The angle addition postulate states that if $\angle 1$ and $\angle 2$ make up $\angle ABC$, then $m\angle 1 + m\angle 2=m\angle ABC$.

Step5: Use transitive property

Since $m\angle ABC = 90^{\circ}$ and $m\angle 1 + m\angle 2=m\angle ABC$, by the transitive property of equality, $m\angle 1 + m\angle 2 = 90^{\circ}$.

Step6: Define complementary angles

Two angles are complementary if the sum of their measures is $90^{\circ}$. So, $\angle 1$ and $\angle 2$ are complementary.

Answer:

The proof shows that if $\overrightarrow{BA}\perp\overrightarrow{BC}$, then the two non - overlapping angles $\angle 1$ and $\angle 2$ that make up $\angle ABC$ are complementary.