Question

question 1 of 10

which descriptions from the list below accurately describe the relationship between \\(\triangle qrs\\) and \\(\triangle tuv\\)? check all that apply.

a. same shape
b. similar
c. same size
d. congruent

Explanation:

Compare corresponding angles

We check the interior angles of \(\triangle QRS\) and \(\triangle TUV\).

• For \(\triangle QRS\): \(m\angle Q = 56^\circ\), \(m\angle R = 83^\circ\), \(m\angle S = 41^\circ\).
• For \(\triangle TUV\): \(m\angle T = 56^\circ\), \(m\angle U = 83^\circ\), \(m\angle V = 41^\circ\).

Since all three corresponding angles are equal, the triangles have the same shape.

Check for similarity

By the Angle-Angle (AA) similarity criterion, two triangles with equal corresponding angles are similar.
Thus, \(\triangle QRS \sim \triangle TUV\).
This confirms they are similar and have the same shape.

Compare corresponding side lengths

We check the ratio of corresponding sides to see if they are congruent:

• \(\frac{TU}{QR} = \frac{72}{24} = 3\)
• \(\frac{UV}{RS} = \frac{90}{30} = 3\)
• \(\frac{TV}{QS} = \frac{108}{36} = 3\)

Since the scale factor is \(3
eq 1\), the triangles do not have the same size and are not congruent.

Answer:

• A. Same shape (Correct answer)
• B. Similar (Correct answer)
• C. Same size
• D. Congruent