15.) Segment \( D E \) is a midsegment of \( \triangle A B C \). Refer to the diagram to determine the coordinates of \( D \) and \( E \). 13. In \( \triangle Q R S, m \angle Q=55^{\circ} \), and \( m \angle R=86^{\circ} \). In \( \triangle T U V, m \angle T=55^{\circ} \), and \( m \angle V=39^{\circ} \). Is \( \triangle Q R S \sim \triangle T U V \) ? Explain. 14. The ratio of the angle measura.

Sure, let's address each of your questions in order. Please note that for some questions, additional information or a diagram is necessary to provide a complete answer. --- **13. Similarity of Triangles \( \triangle QRS \) and \( \triangle TUV \)** To determine whether \( \triangle QRS \) is similar to \( \triangle TUV \), we'll analyze their corresponding angles. **Given:** - In \( \triangle QRS \): - \( m \angle Q = 55^\circ \) - \( m \angle R = 86^\circ \) - In \( \triangle TUV \): - \( m \angle T = 55^\circ \) - \( m \angle V = 39^\circ \) **Steps:** 1. **Find the Third Angle in Each Triangle:** - For \( \triangle QRS \): \[ m \angle S = 180^\circ - (m \angle Q + m \angle R) = 180^\circ - (55^\circ + 86^\circ) = 39^\circ \] - For \( \triangle TUV \): \[ m \angle U = 180^\circ - (m \angle T + m \angle V) = 180^\circ - (55^\circ + 39^\circ) = 86^\circ \] 2. **Compare Corresponding Angles:** - \( \angle Q \) corresponds to \( \angle T \): Both are \( 55^\circ \). - \( \angle R \) corresponds to \( \angle U \): Both are \( 86^\circ \). - \( \angle S \) corresponds to \( \angle V \): Both are \( 39^\circ \). 3. **Conclusion:** Since all corresponding angles of \( \triangle QRS \) and \( \triangle TUV \) are congruent, the triangles are **similar** by the **Angle-Angle-Angle (AAA) Similarity Postulate**. --- **14. Ratio of the Angle Measures** *It appears that the question for item 14 is incomplete. Could you please provide the full question or additional details so I can assist you effectively?* --- **15. Coordinates of Points \( D \) and \( E \) on Midsegment \( DE \) of \( \triangle ABC \)** *This question refers to a diagram that depicts \( \triangle ABC \) with segment \( DE \) as its midsegment. To determine the coordinates of points \( D \) and \( E \), I would need the coordinates of the vertices \( A \), \( B \), and \( C \) of \( \triangle ABC \). Additionally, knowing which sides \( D \) and \( E \) are the midpoints of would be essential.* **General Approach to Find Midsegment Coordinates:** 1. **Identify the Midpoints:** - A midsegment connects the midpoints of two sides of a triangle. 2. **Calculate the Midpoints:** - If \( D \) is the midpoint of side \( AB \) with coordinates \( A(x_1, y_1) \) and \( B(x_2, y_2) \): \[ D\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] - If \( E \) is the midpoint of side \( AC \) with coordinates \( A(x_1, y_1) \) and \( C(x_3, y_3) \): \[ E\left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \] 3. **Determine the Midsegment Coordinates:** - Once you have the coordinates of \( D \) and \( E \), those are the endpoints of the midsegment \( DE \). **Example:** Suppose: - \( A(2, 3) \) - \( B(4, 7) \) - \( C(6, 3) \) Then: - Midpoint \( D \) of \( AB \): \[ D\left( \frac{2 + 4}{2}, \frac{3 + 7}{2} \right) = D(3, 5) \] - Midpoint \( E \) of \( AC \): \[ E\left( \frac{2 + 6}{2}, \frac{3 + 3}{2} \right) = E(4, 3) \] - Therefore, \( DE \) has coordinates \( D(3, 5) \) and \( E(4, 3) \). *Please provide the specific coordinates or the diagram related to \( \triangle ABC \) so I can give a precise answer.* --- Feel free to provide the additional information needed for questions 14 and 15, and I'll be happy to assist further!