https://file.aiquickdraw.com/mathbot/file/17286640009534g56hbd5.jpg

Let's analyze the image and solve the problem step by step. The task is to identify each property shown and classify them either as a property of congruence or equality.

### Problem Breakdown:
1. **(a)** \( x(y + z) = xy + xz \): This represents the **Distributive Property** (of multiplication over addition).
2. **(b)** If \( a = b \), then \( a + c = b + c \): This is the **Addition Property of Equality**.
3. **(c)** If \( x = 9 \), then \( 4x - 1 = 4(9) - 1 \): This represents **Substitution Property of Equality**, where you replace \( x \) with 9 in the equation.
4. **(d)** If \( \angle 1 \cong \angle 3 \) and \( \angle 3 \cong \angle 5 \), then \( \angle 1 \cong \angle 5 \): This is the **Transitive Property of Congruence**.
5. **(e)** If \( D = F \), then \( DE = FE \): This is the **Substitution Property of Equality**, replacing \( D \) with \( F \) to show equality in distances.
6. **(f)** \( ST \cong ST \): This is the **Reflexive Property of Congruence**, as something is always congruent to itself.

### Summary of Properties:
- **9.a**: Distributive Property
- **9.b**: Addition Property of Equality
- **9.c**: Substitution Property of Equality
- **9.d**: Transitive Property of Congruence
- **9.e**: Substitution Property of Equality
- **9.f**: Reflexive Property of Congruence

Would you like further details on any of these properties, or explanations on how they apply? 

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Here are 5 additional related questions to consider:
1. What is the difference between the transitive property of congruence and the transitive property of equality?
2. Can the distributive property be applied to subtraction as well? If so, how?
3. How does the reflexive property work in the context of congruence and equality?
4. When would the substitution property of equality be necessary in solving an algebraic equation?
5. Can the addition property of equality be used in geometric proofs?

### Tip:
When solving algebraic equations or proving congruence, it helps to recognize properties like transitivity or reflexivity to simplify your work.

Name the property for each shown below. Make sure to identify each as a property of congruence or equality.

This problem focuses on identifying different mathematical properties such as the distributive property, substitution, and transitive properties. It involves both congruence and equality, with examples to help understand each concept. Suitable for students in Grades 8-10.

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