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The problem in the image provides two options. Let's solve both:

### Problem 1: 
**AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that AD bisects ∠A.**

#### Solution:

1. Since \( AB = AC \), triangle ABC is isosceles, meaning that the angles opposite these sides, i.e., \( \angle B = \angle C \).

2. The altitude \( AD \) is drawn from vertex \( A \) to the base \( BC \), making \( AD \) perpendicular to \( BC \). Therefore, \( \angle ADB = \angle ADC = 90^\circ \).

3. In triangle \( ABD \) and triangle \( ACD \):
   - \( AB = AC \) (given).
   - \( AD = AD \) (common side).
   - \( \angle ADB = \angle ADC = 90^\circ \) (altitude property).

   Therefore, by the **RHS congruence** criterion, \( \triangle ABD \cong \triangle ACD \).

4. Since the two triangles are congruent, \( \angle BAD = \angle CAD \). This shows that \( AD \) bisects \( \angle A \).

### Problem 2:
**In the figure below, the diagonal PR of quadrilateral PQRS bisects \( \angle QPS \) and \( \angle QRS \). Prove that QR = SR.**

#### Solution:

1. The diagonals \( PR \) bisect \( \angle QPS \) and \( \angle QRS \), so:
   - \( \angle QPR = \angle SPR \) (since \( PR \) bisects \( \angle QPS \)).
   - \( \angle QRP = \angle SRP \) (since \( PR \) bisects \( \angle QRS \)).

2. In triangles \( QPR \) and \( SPR \):
   - \( \angle QPR = \angle SPR \) (given).
   - \( \angle QRP = \angle SRP \) (given).
   - \( PR = PR \) (common side).

   Therefore, by the **ASA congruence** criterion, \( \triangle QPR \cong \triangle SPR \).

3. Since the triangles are congruent, we conclude that \( QR = SR \).

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. What are the properties of an isosceles triangle?
2. How does congruence work in triangle proofs?
3. What is the significance of the altitude in an isosceles triangle?
4. Can a diagonal in a quadrilateral always bisect opposite angles?
5. What are the different criteria for triangle congruence (SSS, SAS, ASA, RHS)?

**Tip:** In geometry, always look for congruent triangles to prove equal lengths or bisected angles!

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that AD bisects ∠A.
OR
In the figure below, the diagonal PR of quadrilateral PQRS bisects ∠QPS and ∠QRS. Prove that QR = SR.

Solve geometry problems involving isosceles triangles and congruent triangles. Learn how to prove that AD bisects angle A in a triangle and that QR equals SR in a quadrilateral. This solution covers key concepts such as congruence, altitude properties, and angle bisectors, suitable for students in Grades 9-10.

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