Question

if two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
given: $overrightarrow{ba}perpoverrightarrow{bc}$
prove: $angle1$ and $angle2$ are complementary.

statements reasons

1. 1. given
2. $angle abc$ is a right - angle 2.
3. 3. definition of right angle
4. $mangle1 + mangle2=mangle abc$ 4.
5. 5. transitive property of equality
6. $angle1$ and $angle2$ are complementary 6.

Explanation:

Step1: State given information

Given that two adjacent acute - angles are perpendicular (i.e., $\overrightarrow{BA}\perp\overrightarrow{BC}$).

Step2: Define right - angle

If two lines are perpendicular, the angle formed between them is a right - angle. So, $\angle ABC$ is a right - angle and $m\angle ABC = 90^{\circ}$ by the definition of perpendicular lines.

Step3: Apply angle addition postulate

We know that $\angle ABC=\angle 1+\angle 2$ by the angle addition postulate.

Step4: Substitute the value of $\angle ABC$

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

Step5: Define complementary angles

By the definition of complementary angles (two angles whose sum is $90^{\circ}$ are complementary), $\angle 1$ and $\angle 2$ are complementary.

Answer:

1. $\overrightarrow{BA}\perp\overrightarrow{BC}$ (Given)
2. $\angle ABC$ is a right - angle (Definition of perpendicular lines)
3. $m\angle ABC = 90^{\circ}$ (Definition of right - angle)
4. $\angle ABC=\angle 1+\angle 2$ (Angle Addition Postulate)
5. $m\angle 1 + m\angle 2=90^{\circ}$ (Substitution Property of Equality)
6. $\angle 1$ and $\angle 2$ are complementary (Definition of complementary angles)