name the property that justifies each statement.
5. ∠z ≅ ∠z
6. 2(3x + 5) = 6x + 10
7. if 12x = 84, then x = 7.
8. if $\overline{st} \cong \overline{qr}$, then $\overline{qr} \cong \overline{st}$.
9. if $m\angle a = 15$, then $3m\angle a = 45$.
10. $xy = xy$
11. if $3x + 14 = 80$, then $3x = 66$.
12. if $kl = mn$, then $mn = kl$.
13. if $2x + y = 5$ and $x = y$, then $2x + x = 5$.
14. if $ab - bc = 12$, then $ab = 12 + bc$.
15. if $\angle 1 \cong \angle 2$ and $\angle 2 \cong \angle 3$, then $\angle 1 \cong \angle 3$.

Question

Accepted Answer

Let's solve each problem by identifying the property that justifies the statement:

### Problem 5: $\angle Z \cong \angle Z$
# Explanation:
## Step1: Recall properties of congruence.
The Reflexive Property of Congruence states that any geometric figure (angle, segment, etc.) is congruent to itself. So $\angle Z \cong \angle Z$ is justified by the Reflexive Property of Congruence.
# Answer:
Reflexive Property of Congruence

### Problem 6: $2(3x + 5) = 6x + 10$
# Explanation:
## Step1: Recall algebraic properties.
The Distributive Property states that $a(b + c)=ab + ac$. Here, $a = 2$, $b=3x$ and $c = 5$. So $2(3x+5)=2	imes3x+2	imes5 = 6x + 10$, which is the Distributive Property.
# Answer:
Distributive Property

### Problem 7: If $12x=84$, then $x = 7$
# Explanation:
## Step1: Recall properties of equality.
The Division Property of Equality states that if $a=b$, then $\frac{a}{c}=\frac{b}{c}$ (where $c
eq0$). Here, we divide both sides of $12x = 84$ by $12$: $\frac{12x}{12}=\frac{84}{12}$, so $x = 7$. So it is the Division Property of Equality.
# Answer:
Division Property of Equality

### Problem 8: If $\overline{ST}\cong\overline{QR}$, then $\overline{QR}\cong\overline{ST}$
# Explanation:
## Step1: Recall properties of congruence.
The Symmetric Property of Congruence states that if $\overline{AB}\cong\overline{CD}$, then $\overline{CD}\cong\overline{AB}$. So if $\overline{ST}\cong\overline{QR}$, then $\overline{QR}\cong\overline{ST}$ is justified by the Symmetric Property of Congruence.
# Answer:
Symmetric Property of Congruence

### Problem 9: If $m\angle A=15$, then $3m\angle A = 45$
# Explanation:
## Step1: Recall properties of equality.
The Multiplication Property of Equality states that if $a = b$, then $ac=bc$. Here, we multiply both sides of $m\angle A=15$ by $3$: $3	imes m\angle A=3	imes15$, so $3m\angle A = 45$. So it is the Multiplication Property of Equality.
# Answer:
Multiplication Property of Equality

### Problem 10: $XY = XY$
# Explanation:
## Step1: Recall properties of equality.
The Reflexive Property of Equality states that any quantity is equal to itself. So $XY=XY$ is justified by the Reflexive Property of Equality.
# Answer:
Reflexive Property of Equality

### Problem 11: If $3x + 14=80$, then $3x=66$
# Explanation:
## Step1: Recall properties of equality.
The Subtraction Property of Equality states that if $a + b=c$, then $a=c - b$. Here, we subtract $14$ from both sides of $3x + 14=80$: $3x+14 - 14=80 - 14$, so $3x=66$. So it is the Subtraction Property of Equality.
# Answer:
Subtraction Property of Equality

### Problem 12: If $KL = MN$, then $MN = KL$
# Explanation:
## Step1: Recall properties of equality.
The Symmetric Property of Equality states that if $a = b$, then $b=a$. So if $KL = MN$, then $MN = KL$ is justified by the Symmetric Property of Equality.
# Answer:
Symmetric Property of Equality

### Problem 13: If $2x + y=5$ and $x = y$, then $2x + x=5$
# Explanation:
## Step1: Recall properties of equality.
The Substitution Property of Equality states that if $a = b$, then $a$ can be substituted for $b$ (or vice - versa) in any equation. Here, since $x = y$, we substitute $x$ for $y$ in the equation $2x + y=5$, getting $2x + x=5$. So it is the Substitution Property of Equality.
# Answer:
Substitution Property of Equality

### Problem 14: If $AB - BC=12$, then $AB=12 + BC$
# Explanation:
## Step1: Recall properties of equality.
The Addition Property of Equality states that if $a - b=c$, then $a=c + b$. Here, we add $BC$ to both sides of $AB - BC=12$: $AB - BC+BC=12 + BC$, so $AB=12 + BC$. So it is the Addition Property of Equality.
# Answer:
Addition Property of Equality

### Problem 15: If $\angle1\cong\angle2$ and $\angle2\cong\angle3$, then $\angle1\cong\angle3$
# Explanation:
## Step1: Recall properties of congruence.
The Transitive Property of Congruence states that if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$. So if $\angle1\cong\angle2$ and $\angle2\cong\angle3$, then $\angle1\cong\angle3$ is justified by the Transitive Property of Congruence.
# Answer:
Transitive Property of Congruence

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QUESTION IMAGE

Question

Explanation:

Problem 5: $\angle Z \cong \angle Z$

Step1: Recall properties of congruence.

Step1: Recall algebraic properties.

Step1: Recall properties of equality.

Answer:

Problem 6: $2(3x + 5) = 6x + 10$