Practice Proofs
Prove using ASA congruence.
\(\textbf{1)}\) Given: \(C\) bisects \( \overline{AE} \), \( \, \, \) \( \angle A \cong \angle E \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) |
\(\underline{\text{Reason}}\) |
\(C\) 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: \( \triangle ABC \cong \triangle DCB \)
\(\,\,\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) |
\(\underline{\text{Reason}}\) |
\( \overline{AB} \parallel \overline{CD}\)
\(\overline{AC} \parallel \overline{BD} \) |
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: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) |
\(\underline{\text{Reason}}\) |
\( \overline{AD} \perp \overline{BC}\)
\( \overline{AD}\) 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: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) |
\(\underline{\text{Reason}}\) |
\( \overline{AB} \parallel \overline{DE}\)
\(\overline{AB} \cong \overline{DE} \) |
Given |
\(\angle ABC \cong \angle EDC\)
\(\angle BAC \cong \angle DEC\) |
Alternate Interior Angles |
| \(\triangle ABC \cong \triangle EDC\) |
ASA triangle congruence |
| \(\underline{\text{Statement}}\) |
\(\underline{\text{Reason}}\) |
\( \overline{AB} \parallel \overline{DE}\)
\(\overline{AB} \cong \overline{DE} \) |
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\(\)
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