name
problem set 1: properties of real numbers
identify the property that justifies each statement
1. ( mangle abc = mangle xyz )
( mangle abc - mangle rst = mangle xyz - mangle rst )
subtraction property of equality
2. ( mqt = mfu )
( mqt + mwx = mfu + mwx )
3. ( angle jkl cong angle jkl )
4. ( gh = mn ) and ( mn = op )
so ( gh = op )
5. ( mxy = 4 , ext{cm} ) and ( mbc = 4 , ext{cm} )
so ( mxy = mbc )
6. ( pr cong pr )
7. ( gh = jk )
( gh - rs = jk - rs )
8. ( mangle 1 = 134^circ ) and ( mangle 2 = 134^circ )
so ( mangle 1 = mangle 2 )
9. ( mangle abc = mangle def )
( mangle abc + mangle qrs = mangle def + mangle qrs )
10. ( gh = gh )
11. ( ed = 3 , ext{in} ) and ( pq = 3 , ext{in} )
so ( ed = pq )
12. ( angle efg cong angle lmn ) and ( angle lmn cong angle spt ),
so ( angle efg cong angle spt )

Question

name
problem set 1: properties of real numbers
> identify the property that justifies each statement
1. ( mangle abc = mangle xyz )
( mangle abc - mangle rst = mangle xyz - mangle rst )
subtraction property of equality
2. ( mqt = mfu )
( mqt + mwx = mfu + mwx )
3. ( angle jkl cong angle jkl )
4. ( gh = mn ) and ( mn = op )
so ( gh = op )
5. ( mxy = 4 , 	ext{cm} ) and ( mbc = 4 , 	ext{cm} )
so ( mxy = mbc )
6. ( pr cong pr )
7. ( gh = jk )
( gh - rs = jk - rs )
8. ( mangle 1 = 134^circ ) and ( mangle 2 = 134^circ )
so ( mangle 1 = mangle 2 )
9. ( mangle abc = mangle def )
( mangle abc + mangle qrs = mangle def + mangle qrs )
10. ( gh = gh )
11. ( ed = 3 , 	ext{in} ) and ( pq = 3 , 	ext{in} )
so ( ed = pq )
12. ( angle efg cong angle lmn ) and ( angle lmn cong angle spt ),
so ( angle efg cong angle spt )

Accepted Answer

Let's solve each problem by identifying the property of real numbers (or equality/congruence) that justifies each statement:

### Problem 3: $\angle JKL \cong \angle JKL$
# Explanation:
## Step 1: Recall the property.
A geometric figure is congruent to itself. This is the **Reflexive Property of Congruence** (for angles, in this case). The Reflexive Property states that any quantity (angle, segment, etc.) is congruent (or equal) to itself.
# Answer: Reflexive Property of Congruence

### Problem 4: $GH \cong MN$ and $MN \cong OP$, so $GH \cong OP$
# Explanation:
## Step 1: Recall the property.
If one quantity is congruent to a second, and the second is congruent to a third, then the first is congruent to the third. This is the **Transitive Property of Congruence** (for segments, in this case). The Transitive Property for congruence (or equality) states that if $ a \cong b $ and $ b \cong c $, then $ a \cong c $.
# Answer: Transitive Property of Congruence

### Problem 5: $m\overline{XY} = 4 \, \text{cm}$ and $m\overline{BC} = 4 \, \text{cm}$, so $m\overline{XY} = m\overline{BC}$
# Explanation:
## Step 1: Recall the property.
If two quantities are equal to the same quantity, then they are equal to each other. This is the **Transitive Property of Equality** (or we can also think of it as the Substitution Property, but more accurately, since both equal 4 cm, they equal each other). The Transitive Property of Equality states that if $ a = b $ and $ b = c $, then $ a = c $. Here, $ a = m\overline{XY} $, $ b = 4 \, \text{cm} $, $ c = m\overline{BC} $.
# Answer: Transitive Property of Equality (or Substitution Property, but Transitive is more precise here)

### Problem 6: $\overline{PR} \cong \overline{PR}$
# Explanation:
## Step 1: Recall the property.
A segment is congruent to itself. This is the **Reflexive Property of Congruence** (for segments, in this case). The Reflexive Property states that any geometric figure (segment, angle, etc.) is congruent to itself.
# Answer: Reflexive Property of Congruence

### Problem 7: $GH \cong JK$, so $GH - RS = JK - RS$
# Explanation:
## Step 1: Recall the property.
If two segments are congruent (equal in length), then subtracting the same segment (length) from both will keep them congruent (equal). This is the **Subtraction Property of Equality** (applied to congruent segments, since congruent segments have equal lengths). The Subtraction Property of Equality states that if $ a = b $, then $ a - c = b - c $. Here, $ a = GH $, $ b = JK $, $ c = RS $ (in terms of length, since $ GH \cong JK $ implies $ GH = JK $ in length).
# Answer: Subtraction Property of Equality (for Congruent Segments, using their length equality)

### Problem 8: $m\angle 1 = 134^\circ$ and $m\angle 2 = 134^\circ$, so $m\angle 1 = m\angle 2$
# Explanation:
## Step 1: Recall the property.
If two angles have the same measure, then they are equal. This is the **Transitive Property of Equality** (or more simply, since both equal $ 134^\circ $, they equal each other). The Transitive Property of Equality states that if $ a = b $ and $ b = c $, then $ a = c $. Here, $ a = m\angle 1 $, $ b = 134^\circ $, $ c = m\angle 2 $.
# Answer: Transitive Property of Equality (or Substitution Property, but Transitive is more precise here)

### Problem 9: $m\angle ABC = m\angle DEF$, so $m\angle ABC + m\angle QRS = m\angle DEF + m\angle QRS$
# Explanation:
## Step 1: Recall the property.
If two angles are equal (in measure), then adding the same angle (measure) to both will keep them equal. This is the **Addition Property of Equality**. The Addition Property of Equality states that if $ a = b $, then $ a + c = b + c $. Here, $ a = m\angle ABC $, $ b = m\angle DEF $, $ c = m\angle QRS $.
# Answer: Addition Property of Equality

### Problem 10: $GH = GH$
# Explanation:
## Step 1: Recall the property.
A segment has the same length as itself. This is the **Reflexive Property of Equality** (for segments, in terms of length). The Reflexive Property of Equality states that any quantity is equal to itself.
# Answer: Reflexive Property of Equality

### Problem 11: $ED = 3 \, \text{in}$ and $PQ = 3 \, \text{in}$, so $ED = PQ$
# Explanation:
## Step 1: Recall the property.
If two segments have the same length (both 3 in), then they are equal in length. This is the **Transitive Property of Equality** (or more simply, since both equal 3 in, they equal each other). The Transitive Property of Equality states that if $ a = b $ and $ b = c $, then $ a = c $. Here, $ a = ED $, $ b = 3 \, \text{in} $, $ c = PQ $.
# Answer: Transitive Property of Equality (or Substitution Property, but Transitive is more precise here)

### Problem 12: $\angle EFG \cong \angle LMN$ and $\angle LMN \cong \angle SPT$, so $\angle EFG \cong \angle SPT$
# Explanation:
## Step 1: Recall the property.
If one angle is congruent to a second, and the second is congruent to a third, then the first is congruent to the third. This is the **Transitive Property of Congruence** (for angles, in this case). The Transitive Property for congruence states that if $ a \cong b $ and $ b \cong c $, then $ a \cong c $.
# Answer: Transitive Property of Congruence

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QUESTION IMAGE

Question

Explanation:

Problem 3: $\angle JKL \cong \angle JKL$

Step 1: Recall the property.

Step 1: Recall the property.

Step 1: Recall the property.

Answer:

Problem 4: $GH \cong MN$ and $MN \cong OP$, so $GH \cong OP$