Question

problem 24
given: $overline{cd}perpoverline{ad}$; $overline{bc}perpoverline{ab}$; $overline{ac}$ bisects $angle dcb$; $angle1congangle3$; $angle2congangle4$.
prove: $angle5congangle6$

1. $angle1congangle4$ 5. transitive property
2. $angle3congangle4$ 6. click here to insert
3. $angle5$ is complementary to $angle3$. 7. click here to insert
4. click here to insert 8. if the exterior sides of a pair of adjacent

Explanation:

Step1: Justify $\angle3\cong\angle4$

$\overline{AC}$ bisects $\angle DCB$. By the definition of an angle - bisector, it divides an angle into two congruent angles. So $\angle3\cong\angle4$.

Step2: Justify $\angle5$ is complementary to $\angle3$

Given $\overline{CD}\perp\overline{AD}$, $\triangle ADC$ is a right - triangle. In a right - triangle, the two non - right angles are complementary. So $\angle5+\angle3 = 90^{\circ}$, which means $\angle5$ is complementary to $\angle3$.

Step3: Prove $\angle5\cong\angle6$

We know $\angle6$ is complementary to $\angle4$ (from right - triangle $ABC$ as $\overline{BC}\perp\overline{AB}$). Since $\angle3\cong\angle4$, we can substitute $\angle4$ with $\angle3$. So $\angle6$ is complementary to $\angle3$. If two angles ($\angle5$ and $\angle6$) are complementary to the same angle ($\angle3$), then $\angle5\cong\angle6$ (angles complementary to the same angle are congruent).

Answer:

1. Definition of angle - bisector.
2. Since $\overline{CD}\perp\overline{AD}$, in right - triangle $ADC$, $\angle5+\angle3 = 90^{\circ}$ (the two non - right angles in a right - triangle are complementary).
3. $\angle6$ is complementary to $\angle4$. Since $\angle3\cong\angle4$ and $\angle5$ is complementary to $\angle3$, $\angle6$ is complementary to $\angle3$ (substitution as $\angle4$ and $\angle3$ are congruent). Then, if two angles are complementary to the same angle, they are congruent, so $\angle5\cong\angle6$.