Proofs & Videos
Prove using AAS congruence.
\(\textbf{1)}\) Given: \( \angle B \cong \angle E\), \( \, \, \overline{AB} \cong \overline{DE} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DEC \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \angle B \cong \angle E\)
\(\overline{AB} \cong \overline{DE} \) Given
\(\angle ACB \cong \angle ECB\) Vertical Angles
\( \triangle ABC \cong \triangle DEC \) AAS triangle congruence
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overline{AB} \cong \overline{DE} \) |
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\(\textbf{2)}\) Given: \( \overline{BC}\) bisects \( \angle ABD \), \( \, \, \angle A \cong \angle D\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle DBC \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \overline{BC}\) 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
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\angle A \cong \angle D\) |
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\(\angle BAC \cong \angle DEC\) |
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\(\textbf{3)}\) Given: \( \overline{AB} \cong \overline{AC}\), \( \, \, \overline{AD} \) is perpendicular to \(\overline{BC}\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \overline{AB} \cong \overline{AC}\)
\(\overline{AD} \) 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
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overline{AD} \) is perpendicular to \(\overline{BC}\) |
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\(\textbf{4)}\) Given: \( \overline{AB} \parallel \overline{DE}\), \( \, \, C\) bisects \( \overline{AE} \)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABC \cong \triangle EDC \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \overline{AB} \parallel \overline{DE}\)
\(C\) 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
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(C\) bisects \( \overline{AE} \) |
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\(\angle BAC \cong \angle DEC\) |
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\(\textbf{5)}\) Given: \( \overline{AB} \cong \overline{AC}\), \( \, \, \overline{AD} \) bisects \(\angle BAC\)
\(\,\,\,\,\,\,\,\,\,\)Prove: \( \triangle ABD \cong \triangle ACD \)
\(\,\,\,\,\,\,\,\,\,\)
\(\underline{\text{Statement}}\)
\(\underline{\text{Reason}}\)
\( \overline{AB} \cong \overline{AC}\)
\( \, \, \overline{AD} \) 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
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\( \, \, \overline{AD} \) bisects \(\angle BAC\) |
|
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