Let's analyze the proof step by step to fill in the missing statement and reason.

The problem is asking to prove that $\triangle ABC$ is an isosceles triangle, given that $AC \perp BD$ and $\triangle ABD \cong \triangle CBD$.

Given:

• $AC \perp BD$
• $\triangle ABD \cong \triangle CBD$

Proof:

We want to show $\triangle ABC$ is isosceles, which means that two sides of the triangle are congruent.

Let’s review each step:

1. Step 1 (Given):

   • Statement: $AC \perp BD$ and $\triangle ABD \cong \triangle CBD$
   • Reason: Given

2. Step 2:

   • Statement: $\angle ADB$ is a right angle.
   • Reason: Perpendicular lines form right angles.

3. Step 3:

   • Statement: $\angle CDB$ is a right angle.
   • Reason: Perpendicular lines form right angles.

4. Step 4:

   • Statement: $\angle ADB \cong \angle CDB$
   • Reason: All right angles are congruent.

Now, the missing statement and reason at Step 5:

Missing Step 5:

• Statement: $BD \cong BD$
• Reason: Reflexive property of congruence (a segment is congruent to itself).

6. Step 6:

   • Statement: $\triangle ABD \cong \triangle CBD$
   • Reason: ASA (Angle-Side-Angle postulate)

7. Step 7:

   • Statement: $AB \cong BC$
   • Reason: Corresponding parts of congruent triangles are congruent (CPCTC).

8. Step 8:

   • Statement: $\triangle ABC$ is an isosceles triangle.
   • Reason: The triangle has two congruent sides.

This completes the proof. If you have any further questions, let me know!

5 Related Questions:

1. Why does the reflexive property apply to line segment $BD$?
2. How does the ASA postulate work in triangle congruence?
3. What are some other ways to prove that a triangle is isosceles?
4. How can CPCTC be used in other types of geometric proofs?
5. Why do perpendicular lines always form right angles?

Tip:

Always identify common sides or angles in overlapping triangles to apply properties like the reflexive property for congruence.