Question

7.) given: $overline{ab}perpoverline{bc}$, $overline{dc}perpoverline{bc}$, and $overline{ac}congoverline{db}$
prove: $overline{ae}congoverline{de}$

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Explanation:

Step1: State right - angle equality

$\angle ABC=\angle DCB = 90^{\circ}$ (Given $\overline{AB}\perp\overline{BC}$ and $\overline{DC}\perp\overline{BC}$)

Step2: Identify common side

$BC = BC$ (Reflexive property of congruence)

Step3: Prove right - triangle congruence

$\triangle ABC\cong\triangle DCB$ (HL congruence criterion since $\overline{AC}\cong\overline{DB}$ and $BC$ is common)

Step4: Get corresponding - angle equality

$\angle ACB=\angle DBC$ (Corresponding parts of congruent triangles are congruent)

Step5: Identify vertical angles

$\angle AEB=\angle DEC$ (Vertical angles are congruent)

Step6: Use angle - side - angle

In $\triangle AEB$ and $\triangle DEC$, $\angle ABE=\angle DCE = 90^{\circ}$, $BC$ is common, and $\angle AEB=\angle DEC$. So, $\triangle AEB\cong\triangle DEC$ (ASA congruence criterion)

Step7: Prove segment equality

$AE\cong DE$ (Corresponding parts of congruent triangles are congruent)

Answer:

$AE\cong DE$