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Answer :
Certainly! The question is about the SAS Congruence Theorem, which is a rule in geometry regarding the congruence of triangles. Let's go through the question step-by-step to understand the correct answer.
1. Understanding the SAS Congruence Theorem:
- SAS stands for "Side-Angle-Side."
- The SAS Congruence Theorem states that if two triangles have two pairs of corresponding sides that are congruent (equal in length), and the angle between those pairs of sides is also congruent, then the triangles are congruent. This means the triangles are identical in shape and size, though they may be oriented differently.
2. Exploring the given options:
- Option 1: If two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent.
- Option 2: If a triangle has an angle between two congruent sides, then it is isosceles.
- Option 3: If two triangles have one pair of congruent angles and two pairs of congruent sides, then the triangles are congruent.
- Option 4: If two triangles have one pair of congruent angles and at least one pair of congruent sides, then the triangles are congruent.
3. Analyzing each option:
- Option 1 correctly summarizes the SAS Congruence Theorem. It specifies that the congruent angle must be between the two pairs of congruent sides, which is essential for the theorem to apply.
- Option 2 is not related to triangle congruence. It describes a property of isosceles triangles, which is different from the SAS condition.
- Option 3 seems similar, but it is not specific about the position of the congruent angle. The angle must be between the two pairs of congruent sides, which this option does not specifically mention.
- Option 4 is incorrect for SAS as it requires the angle to be specifically between the two pairs of congruent sides rather than being just any angle.
4. Conclusion:
- Clearly, the correct statement of the SAS Congruence Theorem is encapsulated in Option 1. Hence, according to the theorem, if two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent.
Therefore, the correct answer is Option 1: "If two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent."
1. Understanding the SAS Congruence Theorem:
- SAS stands for "Side-Angle-Side."
- The SAS Congruence Theorem states that if two triangles have two pairs of corresponding sides that are congruent (equal in length), and the angle between those pairs of sides is also congruent, then the triangles are congruent. This means the triangles are identical in shape and size, though they may be oriented differently.
2. Exploring the given options:
- Option 1: If two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent.
- Option 2: If a triangle has an angle between two congruent sides, then it is isosceles.
- Option 3: If two triangles have one pair of congruent angles and two pairs of congruent sides, then the triangles are congruent.
- Option 4: If two triangles have one pair of congruent angles and at least one pair of congruent sides, then the triangles are congruent.
3. Analyzing each option:
- Option 1 correctly summarizes the SAS Congruence Theorem. It specifies that the congruent angle must be between the two pairs of congruent sides, which is essential for the theorem to apply.
- Option 2 is not related to triangle congruence. It describes a property of isosceles triangles, which is different from the SAS condition.
- Option 3 seems similar, but it is not specific about the position of the congruent angle. The angle must be between the two pairs of congruent sides, which this option does not specifically mention.
- Option 4 is incorrect for SAS as it requires the angle to be specifically between the two pairs of congruent sides rather than being just any angle.
4. Conclusion:
- Clearly, the correct statement of the SAS Congruence Theorem is encapsulated in Option 1. Hence, according to the theorem, if two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent.
Therefore, the correct answer is Option 1: "If two triangles have one pair of congruent angles between two pairs of congruent sides, then the triangles are congruent."
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