Question
The four-sided geometric figure pictured is called a parallelogram. One feature of parallelograms is that opposite sides have equal lengths.
A parallelogram is shown. It is folded from one point to another to form 2 triangles.
The dotted line splits the parallelogram into two triangles. What is true about the congruency of the two triangles?
More information is needed.
The triangles can be proven congruent using SSS.
The triangles can be proven congruent using HL.
The triangles are not congruent.
A parallelogram is shown. It is folded from one point to another to form 2 triangles.
The dotted line splits the parallelogram into two triangles. What is true about the congruency of the two triangles?
More information is needed.
The triangles can be proven congruent using SSS.
The triangles can be proven congruent using HL.
The triangles are not congruent.
Answers
GPT-4o mini
To determine the congruency of the two triangles formed by folding the parallelogram along the dotted line, we can analyze the sides and angles of the triangles.
1. **Opposite Sides of a Parallelogram**: Since the figure is a parallelogram, we know that opposite sides are equal in length. Let's denote the vertices of the parallelogram as \( A, B, C, D \) such that \( AB \parallel CD \) and \( AD \parallel BC \). Therefore, we have:
- \( AB = CD \)
- \( AD = BC \)
2. **Folding the Parallelogram**: When we fold the parallelogram along the dotted line (let's say from point \( A \) to point \( C \)), we split it into two triangles: \( \triangle ABC \) and \( \triangle ACD \).
3. **Side Lengths**: The sides of these triangles have the following relationships:
- \( AB = CD \) (by the properties of the parallelogram)
- \( AC \) is common to both triangles.
- \( AD = BC \) (also by the properties of the parallelogram)
Thus, we can apply the Side-Side-Side (SSS) congruence criterion, which states that if three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent.
In conclusion, the two triangles can be proven congruent using the SSS (Side-Side-Side) congruence criterion.
The correct answer is: **The triangles can be proven congruent using SSS.**
1. **Opposite Sides of a Parallelogram**: Since the figure is a parallelogram, we know that opposite sides are equal in length. Let's denote the vertices of the parallelogram as \( A, B, C, D \) such that \( AB \parallel CD \) and \( AD \parallel BC \). Therefore, we have:
- \( AB = CD \)
- \( AD = BC \)
2. **Folding the Parallelogram**: When we fold the parallelogram along the dotted line (let's say from point \( A \) to point \( C \)), we split it into two triangles: \( \triangle ABC \) and \( \triangle ACD \).
3. **Side Lengths**: The sides of these triangles have the following relationships:
- \( AB = CD \) (by the properties of the parallelogram)
- \( AC \) is common to both triangles.
- \( AD = BC \) (also by the properties of the parallelogram)
Thus, we can apply the Side-Side-Side (SSS) congruence criterion, which states that if three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent.
In conclusion, the two triangles can be proven congruent using the SSS (Side-Side-Side) congruence criterion.
The correct answer is: **The triangles can be proven congruent using SSS.**