Proofs & Videos

Prove using AAS congruence.

\(\textbf{1)}\) Given: \( \angle B \cong \angle E\), \( \, \, \overline{AB} \cong \overline{DE} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DEC \)
\(\,\,\,\,\,\,\,\,\,\)

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\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \angle B \cong \angle E\) \(\overline{AB} \cong \overline{DE} \)	Given
\(\angle ACB \cong \angle ECB\)	Vertical Angles
\( \triangle ABC \cong \triangle DEC \)	AAS triangle congruence

\(\textbf{2)}\) Given: \( \overline{BC}\) bisects \( \angle ABD \), \( \, \, \angle A \cong \angle D\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DBC \)
\(\,\,\,\,\,\,\,\,\,\)

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\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{BC}\) bisects \( \angle ABD \) \(\angle A \cong \angle D\)	Given
\(\angle ABC \cong \angle EDC\) \(\angle BAC \cong \angle DEC\)	Definition of angle bisector
\( \overline{BC} \cong \overline{BC}\)	Reflexive
\(\triangle ABC \cong \triangle DBC\)	AAS triangle congruence

\(\textbf{3)}\) Given: \( \overline{AB} \cong \overline{AC}\), \( \, \, \overline{AD} \) is perpendicular to \(\overline{BC}\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)

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\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \cong \overline{AC}\) \(\overline{AD} \) is perpendicular to \(\overline{BC}\)	Given
\( \angle ADB \text{and} \angle ADC \text{are right}\)	Definition of Perpendicular
\( \overline{AD} \cong \overline{AD}\)	Reflexive
\(\triangle ABD \cong \triangle ACD\)	HL triangle congruence

\(\textbf{4)}\) Given: \( \overline{AB} \parallel \overline{DE}\), \( \, \, C\) bisects \( \overline{AE} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)

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\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \parallel \overline{DE}\) \(C\) bisects \( \overline{AE} \)	Given
\( \overline{AC} \cong \overline{CE}\)	Definition of bisects
\(\angle ABC \cong \angle EDC\) \(\angle BAC \cong \angle DEC\)	Alternate Interior Angles
\(\triangle ABC \cong \triangle EDC\)	AAS triangle congruence

\(\textbf{5)}\) Given: \( \overline{AB} \cong \overline{AC}\), \( \, \, \overline{AD} \) bisects \(\angle BAC\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)

Show Proof Steps

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \cong \overline{AC}\) \( \, \, \overline{AD} \) bisects \(\angle BAC\)	Given
\( \angle DAB \cong \angle DAC \)	Definition of Angle Bisector
\( \overline{AD} \cong \overline{AD}\)	Reflexive
\(\triangle ABD \cong \triangle ACD\)	SAS triangle congruence

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