Question

en δpqr, m es el punto medio del segmento lateral pq y n es el punto medio del segmento pr la tabla a continuación es una demostración incompleta que demuestra que mn || qr 27 ¿qué afirmaciones faltan en la prueba? a 4. δ pmetronorte ~ δ prq 5. ∠ pnortemetro ≅ ∠ pqr b 4. δ pmetronorte ~ δ prq 5. ∠ pnortemetro ≅ ∠ prq c 4. δ pmetronorte ~ δ pqr 5. ∠ pnortemetro ≅ ∠ prq d 4. δ pmetronorte ~ δ pqr 5. ∠ pnortemetro ≅ ∠ pqr

Explanation:

To solve this problem, we analyze the triangle and the midline theorem (or similar triangles).

Step 1: Identify the triangles

Since \( M \) is the midpoint of \( PQ \) and \( N \) is the midpoint of \( PR \), by the midline theorem (or the AA similarity criterion), \( \triangle PMN \sim \triangle PQR \) (because \( \angle P \) is common and \( \frac{PM}{PQ} = \frac{PN}{PR} = \frac{1}{2} \), so SAS similarity or AA similarity applies).

Step 2: Corresponding angles

If \( \triangle PMN \sim \triangle PQR \), then the corresponding angles are equal. So \( \angle PMN \cong \angle PQR \) (corresponding angles of similar triangles).

Now, let's check the options:

• Option A: Triangles are not \( \triangle PMN \) and \( \triangle PRQ \) (wrong correspondence).
• Option B: Triangles are not \( \triangle PMN \) and \( \triangle PRQ \) (wrong correspondence).
• Option C: Angle correspondence is wrong (\( \angle PMN \) should correspond to \( \angle PQR \), not \( \angle PRQ \)).
• Option D: \( \triangle PMN \sim \triangle PQR \) (correct similarity) and \( \angle PMN \cong \angle PQR \) (correct corresponding angles).

Answer:

D. 4. \( \triangle PMN \sim \triangle PQR \)

1. \( \angle PMN \cong \angle PQR \)