The task in the image is to prove that each pair of triangles in the diagrams is congruent, providing the necessary reasons and writing the vertices in matching order. Below is a breakdown of each part in the provided diagrams:

a)

• Diagram description: A circle with two radii $OA$ and $OB$, and chord $AB$. Two triangles are formed: $\triangle OAB$ and $\triangle OBA$.
• Proof of congruence:
  1. $OA = OB$ (radii of the same circle).
  2. $AB = AB$ (common side).
  3. $\angle OAB = \angle OBA$ (isosceles triangle property).
  • Reason for congruence: $\triangle OAB \cong \triangle OBA$ by SAS (Side-Angle-Side).

b)

• Diagram description: Two triangles are drawn within a quadrilateral, and there are marked equal segments and angles.
• Proof of congruence:
  1. $AB = AB$ (common side).
  2. $\angle A = \angle A$ (common angle).
  3. $AC = AD$ (given equal segments).
  • Reason for congruence: $\triangle ABC \cong \triangle ABD$ by SAS (Side-Angle-Side).

c)

• Diagram description: Two overlapping triangles share vertices and sides marked with equal lengths and arrows indicating opposite orientations.
• Proof of congruence:
  1. $AC = AE$ (given equal segments).
  2. $BC = DE$ (given equal segments).
  3. $\angle CAB = \angle EAD$ (vertically opposite angles are equal).
  • Reason for congruence: $\triangle CAB \cong \triangle EAD$ by SAS (Side-Angle-Side).

d)

• Diagram description: Two triangles with shared sides and marked equal angles.
• Proof of congruence:
  1. $AB = AB$ (common side).
  2. $\angle ABC = \angle ABD$ (angles are marked equal).
  3. $BC = BD$ (given equal segments).
  • Reason for congruence: $\triangle ABC \cong \triangle ABD$ by SAS (Side-Angle-Side).

e)

• Diagram description: A kite-shaped quadrilateral split into two triangles with equal sides marked.
• Proof of congruence:
  1. $AC = AC$ (common side).
  2. $AB = AD$ (given equal segments).
  3. $BC = CD$ (given equal segments).
  • Reason for congruence: $\triangle ABC \cong \triangle ADC$ by SSS (Side-Side-Side).

f)

• Diagram description: Two triangles with equal angles marked and shared segments.
• Proof of congruence:
  1. $AB = AB$ (common side).
  2. $\angle BAC = \angle DAC$ (marked equal).
  3. $\angle ACB = \angle ACD$ (marked equal).
  • Reason for congruence: $\triangle ABC \cong \triangle ADC$ by ASA (Angle-Side-Angle).

Questions for Further Exploration:

1. How does the congruence condition $ASA$ differ from $SAS$?
2. In part e, could the property of diagonals of a kite help reinforce the congruence proof?
3. In part f, if we only know two angles are equal, can we deduce anything about the third angle?
4. Can any pair of congruent triangles imply the corresponding quadrilaterals are also congruent?
5. What is the role of common sides in establishing triangle congruence?

Tip:

When proving triangle congruence, always identify the shared sides or angles first, as these simplify the proof significantly.