Explain how the SAS criterion can be used to prove similarity in two triangles.(1 point)

Question

Responses

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 at least two angles of the triangles are given and are the same, then the triangles are similar.
If at least two angles of the triangles are given and are the same, then the triangles are similar.

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 all three side lengths are given and proportional to one another, resulting in the same scale factor, then the triangles are similar.

Answer 1

The SAS (Side-Angle-Side) criterion for triangle similarity states that if two triangles have one angle that is included between two pairs of proportional sides, then the triangles are similar.

In other words, if you have two triangles \(ABC\) and \(DEF\), and you can show that:

1. The lengths of two sides of triangle \(ABC\) are proportional to the lengths of the corresponding sides of triangle \(DEF\) (i.e., \(\frac{AB}{DE} = \frac{AC}{DF}\)), and
2. The angle between those two sides in triangle \(ABC\) is equal to the angle between the corresponding sides in triangle \(DEF\) (i.e., \(\angle A = \angle D\)),

then it can be concluded that triangle \(ABC\) is similar to triangle \(DEF\) (denoted as \(ABC \sim DEF\)). This means that their corresponding angles are equal, and their corresponding side lengths are in proportion.

So, in summary, if the two sides of each triangle are proportional and the included angle is the same, then the triangles themselves are similar according to the SAS criterion.

Answer 2

answer?

Answer 3

The SAS criterion states that if two corresponding side lengths of each triangle are proportional and the included angle is equal, then the triangles are similar.

So, the answer to your question is:

If two corresponding side lengths of each triangle are given and are proportional, with a proven included angle measure, the triangles are similar.

Answer 4

A coordinate plane's axes range from negative 10 to 10 by 1-unit increments. Triangles upper A upper B upper C and upper A prime upper B prime upper C prime are drawn in the system.
How can a similarity transformation be used to determine that the AA criterion proves the dilated triangle A′B′C′ is similar to triangle ABC ?

(1 point)
Responses

Reflect the triangle across the x-axis and check for symmetry among the triangles with respect to the x-axis.
Reflect the triangle across the x -axis and check for symmetry among the triangles with respect to the x -axis.

Find the angle measures of the original triangle and the dilated triangle, then compare proportionality of the ratio to the scale factor.
Find the angle measures of the original triangle and the dilated triangle, then compare proportionality of the ratio to the scale factor.

Center the original triangle inside the dilated triangle and see if the side lengths are congruent by using the scale factor.
Center the original triangle inside the dilated triangle and see if the side lengths are congruent by using the scale factor.

Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles.
Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles.

Answer 5

To prove that triangle \(A'B'C'\) is similar to triangle \(ABC\) using the AA (Angle-Angle) criterion through a similarity transformation, we can ensure that the angles of the triangles are equal without needing to analyze side lengths or congruence.

The most appropriate response for your question is:

Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles.

This method allows you to verify that the corresponding angles remain equal, which is the basis for concluding that the triangles are similar according to the AA criterion.

Answer 6

Two triangles are sketched on a coordinate plane using a dilation of 2, sharing the same center ordered pair at the origin. One triangle has angle measures of 34 degrees and 80 degrees. Using the AA criterion, what would be the two corresponding angles of the dilated triangle?(1 point)
Responses

17 degrees and 40 degrees
17 degrees and 40 degrees

68 degrees and 160 degrees
68 degrees and 160 degrees

34 degrees and 80 degrees
34 degrees and 80 degrees

44 degrees and 70 degrees

Answer 7

When a triangle undergoes a dilation, the angles of the triangle do not change; they remain the same as those of the original triangle. Therefore, if one triangle has angle measures of 34 degrees and 80 degrees, the corresponding angles in the dilated triangle will also be 34 degrees and 80 degrees.

Thus, the correct response is:

34 degrees and 80 degrees.

Answer 8

How does a dilation transformation compare to translations, rotations, and reflections?(1 point)
Responses

Dilation transformations are the only ones that do not maintain the same side lengths.
Dilation transformations are the only ones that do not maintain the same side lengths.

All transformations change the size and shape of figures.
All transformations change the size and shape of figures.

All transformations move figures across the coordinate plane.
All transformations move figures across the coordinate plane.

Dilation transformations make figures larger; translations shrink figures.

Answer 9

The correct response is:

Dilation transformations are the only ones that do not maintain the same side lengths.

This is because dilation specifically alters the size of the figure while maintaining its shape (the angles remain the same), whereas translations, rotations, and reflections preserve both the size and shape of the original figure.

Answer 10

When a figure undergoes a reflection, it changes(1 point)
Responses

size only.
size only.

both position and size.
both position and size.

orientation only.
orientation only.

both position and orientation.

Answer 11

The correct response is:

both position and orientation.

When a figure undergoes a reflection, its position changes as it is flipped over a line (the line of reflection), and its orientation changes because the figure is mirrored. However, the size of the figure remains the same.