Question

complete the proof below by filling in the blanks.

1. given: \\(\overline{cd}\perp\overline{ad};\overline{bc}\perp\overline{ab};\overline{ac}\text{ bisects }\angle dcb;\angle1\cong\angle3;\angle2\cong\angle4.\text{ prove: }\angle5\cong\angle6\\

statements reasons

1. \\(\overline{cd}\perp\overline{ad},\overline{bc}\perp\overline{ab}\\) 1.
2. \\(\overline{ac}\text{ bisects }\angle dcb\\) 2.
3. 3. given
4. 4. definition of an angle bisector
5. \\(\angle1\cong\angle4\\) 5. transitive property
6. \\(\angle3\cong\angle4\\) 6.
7. \\(\angle5\text{ is complementary to }\angle3\\) 7.
8. 8. if the exterior sides of a pair of adjacent angles are perpendicular, then the angles are complementary.
9. 9.

Explanation:

Step1: State given perpendiculars

Given

Step2: State given angle - bisector

Given

Step3: State given angle - congruences

$\angle1\cong\angle3;\angle2\cong\angle4$ (Given)

Step4: Use angle - bisector definition

$\angle1=\angle2$ (Definition of an angle bisector)

Step5: Use transitive property for $\angle1\cong\angle4$

Since $\angle1 = \angle2$ and $\angle2\cong\angle4$, by transitive property $\angle1\cong\angle4$

Step6: Use transitive property for $\angle3\cong\angle4$

Since $\angle1\cong\angle3$ and $\angle1\cong\angle4$, by transitive property $\angle3\cong\angle4$

Step7: State complementary - angle relationship for $\angle5$

In right - triangle $ADC$, $\angle5$ and $\angle3$ are complementary (In a right - triangle, the two non - right angles are complementary)

Step8: State complementary - angle relationship for $\angle6$

In right - triangle $ABC$, $\angle6$ and $\angle4$ are complementary (In a right - triangle, the two non - right angles are complementary)

Step9: Prove $\angle5\cong\angle6$

Since $\angle3\cong\angle4$, and $\angle5$ is complementary to $\angle3$ and $\angle6$ is complementary to $\angle4$, angles complementary to congruent angles are congruent.

Answer:

1. Given
2. Given
3. $\angle1\cong\angle3;\angle2\cong\angle4$
4. $\angle1 = \angle2$
5. Given (used for transitive property)
6. Transitive Property (from $\angle1\cong\angle3$ and $\angle1\cong\angle4$)
7. In right - triangle $ADC$, the non - right angles are complementary
8. $\angle6$ is complementary to $\angle4$
9. Angles complementary to congruent angles are congruent.