What are the five ways to confirm that triangles are congruent?
Understand the Problem
The question is asking about methods for proving that two triangles are congruent, specifically focusing on using the Side-Side-Side (SSS) theorem along with other criteria for triangle congruence.
Answer
$\triangle GIH \cong \triangle JLK$ by SSS.
Answer for screen readers
The triangles $\triangle GIH$ and $\triangle JLK$ are congruent by the SSS criterion.
Steps to Solve
- Identify Corresponding Sides and Angles
Examine the two triangles, $\triangle GIH$ and $\triangle JLK$. Identify the lengths of their corresponding sides and angles. For the SSS congruence criterion, three pairs of sides must be equal.
- List the Measures of the Sides
Assume the measures of the sides as follows:
- $GI = JL$
- $IH = LK$
- $GH = JK$
Confirm the values of these sides to ensure they are equal.
- State the SSS Congruence Criterion
According to the SSS criterion, if all three corresponding sides of two triangles are congruent, then the triangles are congruent. Therefore, if: $$ GI = JL, IH = LK, GH = JK $$ then it can be concluded that: $$ \triangle GIH \cong \triangle JLK $$
- Conclude the Congruence of the Triangles
Based on the measures of sides found in previous steps, write the final conclusion that $\triangle GIH$ is congruent to $\triangle JLK$ by SSS.
The triangles $\triangle GIH$ and $\triangle JLK$ are congruent by the SSS criterion.
More Information
The Side-Side-Side (SSS) theorem is a basic congruence criterion in geometry. It states that if three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent. This property is fundamental in proving geometric figures' congruence and is widely used in various mathematical applications.
Tips
- Assuming Congruence Without Verification: Students sometimes claim triangles are congruent without checking all three pairs of corresponding sides. Always verify the measures before concluding.
- Mismatch of Corresponding Parts: It's crucial to match the respective sides and angles of the triangles correctly; otherwise, the conclusion of congruence might be incorrect.
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