Question
Explain how the SAS criterion can be used to prove similarity in two triangles.(1 point)
Responses
If given a scale factor, you can determine if triangles are similar.
If given a scale factor, you can determine if triangles are similar.
If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
If at least two angles of the triangles are given and are the same, then the triangles are similar.
Responses
If given a scale factor, you can determine if triangles are similar.
If given a scale factor, you can determine if triangles are similar.
If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.
If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
If all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.
If at least two angles of the triangles are given and are the same, then the triangles are similar.
Answers
GPT-4o mini
The SAS (Side-Angle-Side) criterion for triangle similarity states that if two triangles have two corresponding sides that are in proportion and the included angle between those sides is congruent, then the two triangles are similar.
Here's a breakdown of this criterion:
1. **Proportional Sides**: You compare the lengths of two sides from each triangle. For triangles \( ABC \) and \( DEF \), if:
\[
\frac{AB}{DE} = \frac{AC}{DF}
\]
then the sides are proportional.
2. **Included Angle**: You must also establish that the angle formed between those two sides is the same in both triangles, meaning:
\[
\angle A \cong \angle D
\]
3. **Conclusion**: If both conditions are satisfied, you can conclude that the triangles \( ABC \) and \( DEF \) are similar (denoted as \( \triangle ABC \sim \triangle DEF \)).
Therefore, if two corresponding side lengths of each triangle are proportional and the included angle measure is proven to be equal, then the triangles are indeed similar.
Here's a breakdown of this criterion:
1. **Proportional Sides**: You compare the lengths of two sides from each triangle. For triangles \( ABC \) and \( DEF \), if:
\[
\frac{AB}{DE} = \frac{AC}{DF}
\]
then the sides are proportional.
2. **Included Angle**: You must also establish that the angle formed between those two sides is the same in both triangles, meaning:
\[
\angle A \cong \angle D
\]
3. **Conclusion**: If both conditions are satisfied, you can conclude that the triangles \( ABC \) and \( DEF \) are similar (denoted as \( \triangle ABC \sim \triangle DEF \)).
Therefore, if two corresponding side lengths of each triangle are proportional and the included angle measure is proven to be equal, then the triangles are indeed similar.