Practice Proofs

Prove using ASA congruence.

\(\textbf{1)}\) Given: \(C\) bisects \( \overline{AE} \), \( \, \, \) \( \angle A \cong \angle E \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)

Show Proof

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\(C\) bisects \( \overline{AE} \) \(\angle A \cong \angle E \)	Given
\( \overline{AC} \cong \overline{CE}\)	Definition of bisects
\(\angle ACB \cong \angle ECD\)	Vertical angles
\(\triangle ABC \cong \triangle EDC\)	ASA triangle congruence

\(\textbf{2)}\) Given: \( \overline{AB} \parallel \overline{CD}\), \( \, \, \overline{AC} \parallel \overline{BD} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DCB \)
\(\,\,\,\,\,\,\,\,\,\)

Show Proof

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \parallel \overline{CD}\) \(\overline{AC} \parallel \overline{BD} \)	Given
\(\angle ABC \cong \angle DCB\)	Alternate Interior Angles
\(\angle ACB \cong \angle DBC\)	Alternate Interior Angles
\( \overline{BC} \cong \overline{CB}\)	Reflexive
\(\triangle ACB \cong \triangle DCB\)	ASA triangle congruence

\(\textbf{3)}\) Given: \( \overline{AD} \perp \overline{BC}\), \( \, \, \overline{AD}\) bisects \( \angle BAC \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)

Show Proof

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AD} \perp \overline{BC}\) \( \overline{AD}\) bisects \( \angle BAC \)	Given
\(\angle ADB \text{ and } \angle ADC\) are right angles	Definition of perpendicular
\(\angle ADB \cong \angle ADC\)	All right angles congruent
\(\angle DAB \cong \angle DAC\)	Definition of angle bisector
\( \overline{AD} \cong \overline{AD}\)	Reflexive
\(\triangle ABD \cong \triangle ACD\)	ASA triangle congruence

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

Show Proof (First Method)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \parallel \overline{DE}\) \(\overline{AB} \cong \overline{DE} \)	Given
\(\angle ABC \cong \angle EDC\) \(\angle BAC \cong \angle DEC\)	Alternate Interior Angles
\(\triangle ABC \cong \triangle EDC\)	ASA triangle congruence

Show Proof (alternate method)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
\( \overline{AB} \parallel \overline{DE}\) \(\overline{AB} \cong \overline{DE} \)	Given
\(\angle ABC \cong \angle EDC\)	Alternate Interior Angles
\(\angle ACB \cong \angle ECD\)	Vertical Angles
\(\triangle ABC \cong \triangle EDC\)	AAS triangle congruence

See Related Pages\(\)

\(\bullet\text{ Geometry Homepage}\)
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\(\bullet\text{ Proving Triangle Congruence SAS}\)
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\(\bullet\text{ Proving Triangle Congruence SSS}\)
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\(\bullet\text{ Proving Triangle Congruence ASA}\)
\(\,\,\,\,\,\,\,\,\) \(…\)

\(\bullet\text{ Proving Triangle Congruence AAS}\)
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\(\bullet\text{ Proving Triangle Congruence CPCTC}\)
\(\,\,\,\,\,\,\,\,\text{Corresponding Parts of Congruent Triangles are}\)\(…\)