Congruent Triangles SSS And SAS Worksheet Answers Explained
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In the realm of geometry, understanding the properties and relationships of triangles is essential for solving various problems. One of the crucial concepts related to triangles is the idea of congruence. When two triangles are said to be congruent, they have the same shape and size, which means that their corresponding sides and angles are equal. In this article, we will explore two specific criteria for triangle congruence: Side-Side-Side (SSS) and Side-Angle-Side (SAS) and provide explanations of worksheet answers based on these criteria.
Understanding Congruent Triangles
Before diving into the specifics of SSS and SAS, let’s establish what congruent triangles are and why they are important.
Congruent Triangles: Triangles are congruent when all their corresponding sides and angles are equal. This congruence can be proven using several criteria, two of which we’ll focus on in this article: SSS and SAS.
The SSS (Side-Side-Side) Criterion
The SSS criterion states that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. In other words, if triangle ABC has sides of lengths a, b, and c, and triangle DEF has sides of lengths d, e, and f, then:
- If ( a = d )
- If ( b = e )
- If ( c = f )
Then triangle ABC is congruent to triangle DEF, denoted as ( \triangle ABC \cong \triangle DEF ).
Example of SSS
Let’s say we have triangle ABC with side lengths of 5 cm, 7 cm, and 9 cm. Triangle DEF also has side lengths of 5 cm, 7 cm, and 9 cm.
- Sides of triangle ABC: 5 cm, 7 cm, 9 cm
- Sides of triangle DEF: 5 cm, 7 cm, 9 cm
Since the corresponding sides are equal, we conclude that ( \triangle ABC \cong \triangle DEF ).
The SAS (Side-Angle-Side) Criterion
The SAS criterion states that if two sides of one triangle and the angle between them are equal to two sides of another triangle and the angle between them, then the two triangles are congruent. In a more formal definition:
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If triangle ABC has sides of lengths a and b and an included angle C, and triangle DEF has sides of lengths d and e and an included angle F, then:
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If ( a = d )
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If ( b = e )
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If ( C = F )
Then triangle ABC is congruent to triangle DEF, denoted as ( \triangle ABC \cong \triangle DEF ).
Example of SAS
Consider triangle ABC with sides of lengths 6 cm, 8 cm, and an included angle of 50 degrees. Triangle DEF has sides of lengths 6 cm, 8 cm, and an included angle of 50 degrees as well.
- Sides of triangle ABC: 6 cm, 8 cm, and included angle 50 degrees
- Sides of triangle DEF: 6 cm, 8 cm, and included angle 50 degrees
Because the conditions of SAS are satisfied, we can conclude that ( \triangle ABC \cong \triangle DEF ).
Worksheet Answers Explained
When working on worksheets that focus on congruent triangles using SSS and SAS, it’s important to show your reasoning for each problem. Here’s how to effectively explain your answers.
SSS Worksheet Problems
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Problem Statement: Given triangle XYZ with sides 4 cm, 5 cm, and 6 cm, and triangle ABC with sides 4 cm, 5 cm, and 6 cm, are they congruent?
Answer Explanation:
- Check the side lengths: 4 cm = 4 cm, 5 cm = 5 cm, 6 cm = 6 cm.
- Since all corresponding sides are equal, ( \triangle XYZ \cong \triangle ABC ) by the SSS criterion.
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Problem Statement: Triangle DEF has side lengths 7 cm, 8 cm, and 10 cm. Triangle GHI has side lengths 7 cm, 8 cm, and 9 cm. Are they congruent?
Answer Explanation:
- Compare the sides: 7 cm = 7 cm, 8 cm = 8 cm, but 10 cm ≠ 9 cm.
- Since the sides are not all equal, ( \triangle DEF ) and ( \triangle GHI ) are not congruent.
SAS Worksheet Problems
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Problem Statement: Triangle JKL has sides 5 cm, 12 cm, and an included angle of 40 degrees. Triangle MNO has sides 5 cm, 12 cm, and an included angle of 40 degrees. Are they congruent?
Answer Explanation:
- Check side lengths: 5 cm = 5 cm, 12 cm = 12 cm, and included angle 40 degrees = 40 degrees.
- By the SAS criterion, ( \triangle JKL \cong \triangle MNO ).
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Problem Statement: Triangle PQR has sides 8 cm, 6 cm and an included angle of 30 degrees. Triangle STU has sides 8 cm, 6 cm and an included angle of 45 degrees. Are they congruent?
Answer Explanation:
- Compare the sides: 8 cm = 8 cm, 6 cm = 6 cm, but 30 degrees ≠ 45 degrees.
- The angle condition fails, so ( \triangle PQR ) and ( \triangle STU ) are not congruent.
Important Notes on Congruent Triangles
"When checking for congruence using SSS or SAS, it is vital to ensure all conditions are met for a correct conclusion."
It's important to remember that congruence does not imply that triangles need to be positioned the same way; they can be flipped or rotated. The key is that all corresponding measurements (sides and angles) must match.
In conclusion, understanding the SSS and SAS criteria for triangle congruence is fundamental for solving geometry problems effectively. By practicing with worksheets and applying these principles, you can gain a solid grasp of these concepts and improve your problem-solving skills in geometry.