Question: Which two triangles are congruent by $ASA$? [asy] unitsize(1cm); pair T, U, V; T = (0,0); U =…

Which two triangles are congruent by $ASA$?

[asy] unitsize(1cm); pair T, U, V; T = (0,0); U = (2,0); V = (1,2); draw(T—U—V—cycle); label(“$T$”, T, SW); label(“$U$”, U, SE); label(“$V$”, V, N); [/asy]

[asy] unitsize(1cm); pair H, F, G; H = (0,0); F = (2,0); G = (1,2); draw(H—F—G—cycle); label(“$H$”, H, SW); label(“$F$”, F, SE); label(“$G$”, G, N); [/asy]

[asy] unitsize(1cm); pair A, B, C; A = (0,0); B = (2,0); C = (1,2); draw(A—B—C—cycle); label(“$A$”, A, SW); label(“$B$”, B, SE); label(“$C$”, C, N); [/asy]

Solution

To determine which two triangles are congruent by ASA (Angle-Side-Angle), we need to check for two pairs of corresponding angles and the included side being equal. 1. Examine Triangle \( \triangle TUV \): - Angle \( \angle T \) and Angle \( \angle U \) are marked. - Side \( TU \) is marked between these angles. 2. Examine Triangle \( \triangle FHG \): - Angle \( \angle F \) and Angle \( \angle G \) are marked. - Side \( FG \) is marked between these angles. 3. Examine Triangle \( \triangle ABC \): - Angle \( \angle A \) and Angle \( \angle B \) are marked. - Side \( AB \) is marked between these angles. By comparing the marked angles and sides: - \( \triangle TUV \) has \( \angle T \), \( \angle U \), and side \( TU \). - \( \triangle ABC \) has \( \angle A \), \( \angle B \), and side \( AB \). Triangles \( \triangle TUV \) and \( \triangle ABC \) have two pairs of corresponding angles and the included side equal. Thus, the triangles \(\triangle TUV\) and \(\triangle ABC\) are congruent by ASA.