Geometry

Question Analysis:
This question involves compound triangles sharing a common angle and tests the ability to:
Identify SAS congruence in overlapping figures.
Apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to find missing lengths.

Key Concept Explanation:
SAS Congruence: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding parts of another.
CPCTC: Once triangles are proven congruent, all corresponding sides and angles are equal.

Step-by-Step Solution:
Identify Given Information:
AB = AD = 8 cm (Side 1).
AC = AE = 12 cm (Side 2).
∠BAC = ∠DAE = 40° (Included Angle).
Prove ΔABC ≅ ΔADE by SAS:
Both triangles have: Side AB = AD.
Included angle ∠BAC = ∠DAE.
Side AC = AE.
⇒ ΔABC ≅ ΔADE (SAS).
Find DE using CPCTC:
Since ΔABC ≅ ΔADE, corresponding sides BC = DE.
Given BC = 10 cm ⇒ DE = 10 cm.

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