Question

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

Explanation:

Step1: State given perpendiculars

$\overline{AB}\perp\overline{BC}$ and $\overline{DC}\perp\overline{BC}$ (Given)

Step2: Define right - angle property

$\angle ABC=\angle DCB = 90^{\circ}$ (Definition of perpendicular lines)

Step3: State common side

$\overline{BC}\cong\overline{BC}$ (Reflexive property of congruence)

Step4: Prove right - triangle congruence

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

Step5: State corresponding parts of congruent triangles

$\angle ACB\cong\angle DBC$ (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)

Step6: Consider vertical angles

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

Step7: Use angle - side - angle (ASA)

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

Step8: State corresponding parts of congruent triangles

$\overline{AE}\cong\overline{DE}$ (CPCTC)

Answer:

1. Statements: $\overline{AB}\perp\overline{BC}$, $\overline{DC}\perp\overline{BC}$; Reasons: Given
2. Statements: $\angle ABC = 90^{\circ}$, $\angle DCB=90^{\circ}$; Reasons: Definition of perpendicular lines
3. Statements: $\overline{BC}\cong\overline{BC}$; Reasons: Reflexive property of congruence
4. Statements: $\triangle ABC\cong\triangle DCB$; Reasons: HL congruence criterion
5. Statements: $\angle ACB\cong\angle DBC$; Reasons: CPCTC
6. Statements: $\angle AEB=\angle DEC$; Reasons: Vertical angles are congruent
7. Statements: $\triangle AEB\cong\triangle DEC$; Reasons: ASA congruence criterion
8. Statements: $\overline{AE}\cong\overline{DE}$; Reasons: CPCTC

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