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Questions and Answers
What does the SSS Congruence Theorem state?
What does the SSS Congruence Theorem state?
In the process of proving the SSS theorem, what forms when L and L₁ are connected?
In the process of proving the SSS theorem, what forms when L and L₁ are connected?
Which condition is unnecessary when applying the SSS Congruence Theorem?
Which condition is unnecessary when applying the SSS Congruence Theorem?
How is congruence established when proving that angles in a quadrilateral are equal using the SSS Congruence Theorem?
How is congruence established when proving that angles in a quadrilateral are equal using the SSS Congruence Theorem?
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In the example provided, how is the congruence of triangles established involving the common side TL?
In the example provided, how is the congruence of triangles established involving the common side TL?
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Study Notes
Proving the Third Side-Side-Side (SSS) Congruence Theorem
- The SSS Congruence Theorem states that if three sides of one triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.
- To prove this theorem, two triangles, KLM and K₁L₁M₁, are superimposed with K coinciding with K₁, M coinciding with M₁, and L and L₁ positioned on opposite sides of the line KM.
- Connecting L and L₁ creates two isosceles triangles: LKL₁ and LML₁.
- Since KL = KL₁ and ML = ML₁, these isosceles triangles are identified by definition.
- In an isosceles triangle, the base angles are equal, so ∠1 = ∠2 and ∠3 = ∠4.
- Therefore, ∠1 + ∠3 = ∠2 + ∠4, indicating that ∠ KLM and ∠ K₁L₁M₁ are equal.
- Based on the theorem's statement, LK = L₁K₁ and LM = L₁M₁ and ∠KLM = ∠K₁L₁M₁, proving that ΔKLM ≅ ΔK₁L₁M₁ using the SAS Congruence Theorem.
The Third Side-Side-Side (SSS) Theorem in Practice
- Three different placement scenarios of the intersection point of segment LL₁ with the combined side of the triangle KM K₁M₁ (or its extension) are possible, but all follow the same proving process.
- Identify drawings depicting triangles congruent due to the SSS Congruence Theorem.
- Drawings 3 and 4 are correctly paired.
- The provided example shortens problem conditions; ∠A = ∠D is redundant while AC = DE is necessary for SSS congruence.
- The text shows two triangles with two equal sides and the third displaced by equal segments. Proving congruence relies on the common side TL for both triangles. Using FT + TL = LM + TL results in FL = MT, confirming the SSS Congruence Theorem.
Applying the Third Side-Side-Side (SSS) Congruence Theorem to Prove Angle Congruence
- The SSS Congruence Theorem is used to prove angle congruence within a quadrilateral.
- To prove that ∠E = ∠F in a quadrilateral with equal sides DE = DF and KE = KF, connect points D and K to form two triangles: DEK and DFK.
- The drawn line segment DK is a common side for both triangles.
- Since DE = DF and KE = KF, these triangles are congruent by the SSS Theorem.
- Therefore, ∠E = ∠F.
The Third Side-Side-Side (SSS) Congruence Theorem in Summary
- The SSS Congruence Theorem is a key geometric tool. If all three sides of one triangle are equal to the corresponding sides of a second triangle, the triangles are congruent.
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Description
This quiz focuses on the SSS Congruence Theorem, which demonstrates that if the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent. It includes proofs and practical applications of the theorem to solidify understanding.
