Question

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

Explanation:

Step1: State given perpendiculars

$\overline{CD}\perp\overline{AD},\overline{BC}\perp\overline{AB}$ are given.

Step2: State given angle - bisector

$\overline{AC}$ bisects $\angle DCB$ is given.

Step3: State given congruent angles

$\angle1\cong\angle3;\angle2\cong\angle4$ are given.

Step4: Define angle - bisector result

Since $\overline{AC}$ bisects $\angle DCB$, we have $\angle DCA=\angle BCA$.

Step5: Identify right - triangles

Because of the perpendiculars, $\triangle ADC$ and $\triangle ABC$ are right - triangles.

Step6: State reflexive property

The side $\overline{AC}$ is common to both triangles, so $\overline{AC}=\overline{AC}$.

Step7: Prove triangle congruence

Using the Angle - Angle - Side (AAS) congruence criterion ($\angle ADC=\angle ABC = 90^{\circ},\angle DCA=\angle BCA,\overline{AC}=\overline{AC}$), we can say $\triangle ADC\cong\triangle ABC$.

Step8: Prove required angle congruence

Since the triangles are congruent, corresponding parts are congruent, so $\angle5\cong\angle6$.

Answer:

1. Given
2. Given
3. $\angle1\cong\angle3;\angle2\cong\angle4$
4. $\angle DCA=\angle BCA$ (Definition of angle - bisector)
5. $\triangle ADC$ and $\triangle ABC$ are right - triangles ($\overline{CD}\perp\overline{AD},\overline{BC}\perp\overline{AB}$)
6. $\overline{AC}=\overline{AC}$ (Reflexive property)
7. $\triangle ADC\cong\triangle ABC$ (AAS congruence criterion since $\angle ADC=\angle ABC = 90^{\circ},\angle DCA=\angle BCA,\overline{AC}=\overline{AC}$)
8. $\angle5\cong\angle6$ (Corresponding parts of congruent triangles are congruent)