It appears you have uploaded an image containing some geometry problems. I'll analyze the content of the image and solve the relevant questions.

Analysis of the Questions:

Question 3:

Two triangles (∆HAT and ∆PMT) share intersecting lines, and pairs of corresponding angles are marked congruent.

• The triangles can be proven congruent by ASA (Angle-Side-Angle) since:
  • Two angles (∠HAT ≅ ∠PMT, and ∠HTA ≅ ∠PTM) are marked congruent.
  • The side HT ≅ PT is shared between the two triangles.

Question 5:

Two triangles (∆WAT and ∆RET) are connected via a shared side (segment TE), and congruent angles and sides are marked.

• The triangles can be proven congruent by ASA:
  • ∠AWT ≅ ∠ERT (marked).
  • ∠WTA ≅ ∠RET (marked).
  • Side TE is shared (common side).

Question 7:

Two triangles (∆ABD and ∆CBE) intersect at point B, and pairs of sides and angles are marked congruent.

• The triangles can be proven congruent by SAS (Side-Angle-Side):
  • Side AB ≅ CB and side BD ≅ BE.
  • The included angle ∠ABD ≅ ∠CBE is marked.

Summary of Proofs Used:

• Question 3: ASA (Angle-Side-Angle).
• Question 5: ASA (Angle-Side-Angle).
• Question 7: SAS (Side-Angle-Side).

Would you like further details about the reasoning or help with anything specific?

Relative Questions for Further Understanding:

1. What are the conditions for congruence in triangles besides ASA and SAS?
2. Can the triangles in Question 3 be proven congruent using another theorem (e.g., AAS or HL)?
3. What is the difference between SAS and SSA in proving triangle congruence?
4. Why is the order of sides and angles crucial when naming congruence theorems?
5. How can coordinate geometry be applied to prove the congruence of triangles?

Tip:

Always check the congruence of corresponding parts (angles and sides) when proving triangles congruent, ensuring you align the vertices correctly.