Select ALL the correct answers.
Which situations describe similar but not congruent triangles?
1. Triangle TUV is rotated 180° clockwise about the origin and translated 3 units up to create ΔT'U'V'.
2. The three angles of ΔGHI have the same measures as the three angles of ΔQRS, and the side lengths of ΔGHI are twice the corresponding side lengths of ΔQRS.
3.Two of the angles of ΔABC have the same measures as two of the angles of ΔDEF, and all of the corresponding side lengths of the two triangles are different.
4. Triangle PQR is dilated about the origin by a scale factor of and reflected across the y-axis to create ΔP'Q'R'.
5. The three side lengths of ΔJKL are the same as the three corresponding side lengths of ΔXYZ, and one of the angles of ΔJKL has the same measure as one of the angles of ΔXYZ.
2. The three angles of ΔGHI have the same measures as the three angles of ΔQRS, and the side lengths of ΔGHI are twice the corresponding side lengths of ΔQRS.
4. Triangle PQR is dilated about the origin by a scale factor of and reflected across the y-axis to create ΔP'Q'R'.
The correct answers are:
2. The three angles of ΔGHI have the same measures as the three angles of ΔQRS, and the side lengths of ΔGHI are twice the corresponding side lengths of ΔQRS.
3. Two of the angles of ΔABC have the same measures as two of the angles of ΔDEF, and all of the corresponding side lengths of the two triangles are different.
5. The three side lengths of ΔJKL are the same as the three corresponding side lengths of ΔXYZ, and one of the angles of ΔJKL has the same measure as one of the angles of ΔXYZ.
To determine which situations describe similar but not congruent triangles, we need to understand the definitions of similar and congruent triangles.
Similar triangles have the same shape but may differ in size. This means that their corresponding angles are congruent, but their corresponding side lengths are proportional.
Congruent triangles have the exact same shape and size. This means that their corresponding angles are congruent, and their corresponding side lengths are equal.
Let's go through each situation one by one:
1. Triangle TUV is rotated 180° clockwise about the origin and translated 3 units up to create ΔT'U'V'.
This situation does not describe similar triangles because rotating a triangle does not change its shape, and translating does not affect the angles or side lengths of the triangle.
2. The three angles of ΔGHI have the same measures as the three angles of ΔQRS, and the side lengths of ΔGHI are twice the corresponding side lengths of ΔQRS.
This situation describes similar triangles. When the angles have the same measures, it implies that the corresponding angles are congruent. Additionally, when the side lengths are in a constant ratio (in this case, twice the length), it implies similarity.
3. Two of the angles of ΔABC have the same measures as two of the angles of ΔDEF, and all of the corresponding side lengths of the two triangles are different.
This situation does not describe similar triangles because for triangles to be similar, all corresponding angles must have the same measure, not just two. Additionally, if all corresponding side lengths are different, it means they are not proportional, which further indicates that the triangles are not similar.
4. Triangle PQR is dilated about the origin by a scale factor of k and reflected across the y-axis to create ΔP'Q'R'.
This situation describes similar triangles. Dilating a figure (in this case, by a scale factor of k) does not change the shape, and reflecting across an axis also preserves the shape. Therefore, the corresponding angles remain congruent, and the corresponding side lengths are proportional.
5. The three side lengths of ΔJKL are the same as the three corresponding side lengths of ΔXYZ, and one of the angles of ΔJKL has the same measure as one of the angles of ΔXYZ.
This situation describes similar triangles. When the side lengths are equal and one angle has the same measure, it indicates similarity. However, it is important to note that having one angle with the same measure does not guarantee similarity. It is the combination of equal side lengths and congruent angles that determine similarity.
So, the correct answers are: 2, 4, and 5.