Question

complete the proof below by filling in the blanks. this will become our theorem 8.

1. given: $overline{ac}perpoverline{bc}$, $angle1$ and $angle2$ are adjacent.

prove: $mangle1 + mangle2=90$
statements reasons

1. $overline{ac}perpoverline{bc}$ 1. given
2. $angle1$ and $angle2$ are adjacent. 2.
3. $angle acb$ is a right - angle. 3. perpendicular lines intersect to form right angles.
4. 4. definition of a right angle
5. 5. betweenness of rays
6. $mangle1 + mangle2 = 90$ 6.

theorem 8 if the exterior sides of a pair of adjacent angles are perpendicular, then the angles are complementary.

Explanation:

Step1: State given facts

1. $\overline{AC}\perp\overline{BC}$ (Given); 2. $\angle1$ and $\angle2$ are adjacent (Given)

Step2: Define right - angle

Since $\overline{AC}\perp\overline{BC}$, $\angle ACB$ is a right - angle (Perpendicular lines intersect to form right angles)

Step3: Recall right - angle measure

$m\angle ACB = 90^{\circ}$ (Definition of a right angle)

Step4: Use angle addition

Since $\angle ACB=\angle1+\angle2$ (Betweenness of Rays)

Step5: Substitute and prove

$m\angle1 + m\angle2=90^{\circ}$ (Substitution)

Answer:

The completed proof is as follows:

Statements	Reasons
2. $\angle1$ and $\angle2$ are adjacent	2. Given
3. $\angle ACB$ is a right angle	3. Perpendicular lines intersect to form right angles
4. $m\angle ACB = 90^{\circ}$	4. Definition of a right angle
5. $\angle ACB=\angle1+\angle2$	5. Betweenness of Rays
6. $m\angle1 + m\angle2 = 90^{\circ}$	6. Substitution