Which two triangles are congruent by the ASA Theorem? Complete the congruence statement.
Understand the Problem
The question asks to identify which two triangles are congruent based on the Angle-Side-Angle (ASA) theorem, and to complete the congruence statement. The ASA theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
Answer
$\triangle BCD \cong \triangle QRS$
Answer for screen readers
$\triangle BCD \cong \triangle QRS$
Steps to Solve
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Identify the triangles with two angles and the included side marked We need to find two triangles where two angles and the side between those angles are marked congruent to the corresponding parts of the other triangle.
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Compare triangle EFG with triangle QRS Triangle EFG has $\angle F$ and $\angle G$ marked, with side $FG$ included between them. Triangle QRS has $\angle Q$ and $\angle R$ marked, with side $QR$ included. The markings on the angles and included sides of triangle $EFG$ and triangle $QRS$ indicate that $\angle F \cong \angle Q$, $\angle G \cong \angle R$, and $FG \cong QR$. Therefore, $\triangle EFG \cong \triangle QRS$ by ASA.
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Compare triangle EFG with triangle BCD Triangle $EFG$ and triangle $BCD$ have two angles and the included side marked. The markings indicate that $\angle F \cong \angle B$, $\angle G \cong \angle D$, and $FG \cong BD$. Therefore, $\triangle EFG \cong \triangle BCD$ by ASA.
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Write the congruence statement Since we have $\angle F \cong \angle Q$, $\angle G \cong \angle R$, and $FG \cong QR$, then $\triangle EFG \cong \triangle QRS$. Since we have $\angle F \cong \angle B$, $\angle G \cong \angle D$, and $FG \cong BD$, then $\triangle EFG \cong \triangle BCD$. Therefore, $\triangle BCD \cong \triangle EFG \cong \triangle QRS$.
$\triangle BCD \cong \triangle QRS$
More Information
The Angle-Side-Angle (ASA) theorem is a fundamental concept in geometry that helps determine when two triangles are congruent. Congruent triangles have the same size and shape, meaning all corresponding sides and angles are equal.
Tips
A common mistake is to choose the wrong order of vertices in the congruence statement. The order matters because it indicates which angles and sides correspond. For example, $\triangle BCD \cong \triangle QRS$ means that $\angle B \cong \angle Q$, $\angle C \cong \angle R$, $\angle D \cong \angle S$, $BC \cong QR$, $CD \cong RS$, and $BD \cong QS$. Another common mistake is assuming that the side has to be between the angles. If it is not included, it is not ASA.
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