Let's analyze each question to determine whether the triangles are similar, and if they are, identify the similarity criterion (AA~, SSS~, SAS~), or conclude that the triangles are not similar. I'll go through each part systematically:

1.

Triangles $\triangle TSR$ and $\triangle NPQ$.

• Given:

  • $\triangle TSR$: $\angle T = 55^\circ, \angle S = 37^\circ, \angle R = 88^\circ$ (angle sum).
  • $\triangle NPQ$: $\angle N = 17^\circ, \angle P = 20^\circ, \angle Q = 143^\circ$.

• Observation: Angles in $\triangle NPQ$ do not match the angles in $\triangle TSR$. No pair of angles are equal, so the triangles are not similar.

4.

Triangles $\triangle NFH$ and $\triangle ESH$.

• Given:

  • $\triangle NFH$: $\angle N = 29^\circ, \angle F = 106^\circ, \angle H = 45^\circ$.
  • $\triangle ESH$: $\angle E = 45^\circ, \angle S = 29^\circ, \angle H = 106^\circ$.

• Observation: All corresponding angles are equal: $\angle N = \angle S, \angle F = \angle H, \angle H = \angle E$. By the AA similarity criterion, $\triangle NFH \sim \triangle ESH$.

• Similarity Statement: $\triangle NFH \sim \triangle ESH$ by AA~.

5.

Triangles $\triangle QNR$ and $\triangle MLN$.

• Given:

  • Sides of $\triangle QNR$: $QR = 56, NR = 48, QN = unknown$.
  • Sides of $\triangle MLN$: $ML = 70, LN = 60, MN = unknown$.

• Observation: Ratios of corresponding sides ($QR/LN, NR/ML$) do not match. Since one of the sides is also unknown, further data is required to confirm similarity. Based on available information, the triangles are not similar.

8.

Triangles $\triangle JKL$ and $\triangle KLM$.

• Given:

  • $JK = 6, KL = 21, LM = 10$, with a shared side $KL = 21$.

• Observation: The ratios of corresponding sides in $\triangle JKL$ and $\triangle KLM$ are not proportional ($JK/KL \neq KL/LM$). The triangles are not similar.

If you want to verify any specific part in detail or have more questions about similarity criteria, let me know!

5 Questions to Explore Further:

1. How is the AA similarity criterion used to confirm triangle similarity?
2. How do you calculate missing angles in a triangle when two angles are known?
3. What is the significance of side ratios in SSS and SAS similarity?
4. Why is it important for all corresponding angles or sides to match in triangle similarity?
5. How do you handle situations where one side length is unknown?

Tip:

Always check if at least two corresponding angles are equal (AA~ criterion) before proceeding to analyze sides for triangle similarity.