To prove that triangles \( \text{BIG} \) and \( \text{COW} \) are congruent using the AAS (Angle-Angle-Side) congruence theorem, we need to establish two angles and the side included between them in one triangle and the corresponding two angles and the corresponding side in the other triangle.
From the information provided, we have:
- \( \angle I = \angle O \) (one pair of corresponding angles)
- We need another angle from the triangles to relate, using given sides:
- \( GB = WC \) (these sides can help relate the other angles)
- \( BG = CW \) (these sides can help relate the other angles)
- \( BI = CO \) (one pair of corresponding sides)
Steps to Prove Triangles BIG and COW are Congruent:
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Step 1: Identify Angles
- Given \( \angle I = \angle O \), we've identified one pair of congruent angles.
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Step 2: Identify the Sides
- Given \( BI = CO \), we have a pair of corresponding sides.
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Step 3: Use the properties of triangles
- We need to show another pair of angles is congruent.
- Knowing that triangles share the same sides, we can verify that the side \( GB \) is equal to \( WC \) and that \( BG \) is equal to \( CW \). The congruence relationships of the sides will allow us to establish that angles \( \angle BGI \) and \( \angle CWO \) are equal due to the Alternate Interior Angles Theorem if lines are parallel or will relate due to the two triangles being created.
Conclusion
Using the AAS theorem, we can conclude that \( \triangle BIG \cong \triangle COW \) by proving:
- \( \angle I = \angle O \)
- \( BI = CO \)
- \( \angle BGI = \angle CWO \) (established from relationships of triangles or through supplementary angles).
Final Congruence Statement
Thus, by confirming two angles and the included side are congruent, we can conclude \( \triangle BIG \cong \triangle COW \) through AAS congruence.
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