Questions: 1. If PQ=PQ, then PQ ≅ PQ 2. If K is between J and L, then JK+KL=JL 3. EF ≅ EF 4. If RS=TU, then RS+XY=TU+XY 5. If AB=DE, then DE=AB 6. If Y is the midpoint of XZ, then XY=YZ 7. If FG ≅ HI and HI ≅ JK, then FG ≅ JK 8. If AB+CD=EF+CD, then AB=EF 9. If PQ+RS=TV and RS=WX, then PQ+WX=TV 10. If LP=PN, and LrP, and N are collinear, then P is the midpoint of LN 11. If UV ≅ UV, then UV=UV 12. If CD+DE=CE, then CD=CE-DE

1. If PQ=PQ, then PQ ≅ PQ
2. If K is between J and L, then JK+KL=JL
3. EF ≅ EF
4. If RS=TU, then RS+XY=TU+XY
5. If AB=DE, then DE=AB
6. If Y is the midpoint of XZ, then XY=YZ
7. If FG ≅ HI and HI ≅ JK, then FG ≅ JK
8. If AB+CD=EF+CD, then AB=EF
9. If PQ+RS=TV and RS=WX, then PQ+WX=TV
10. If LP=PN, and LrP, and N are collinear, then P is the midpoint of LN
11. If UV ≅ UV, then UV=UV
12. If CD+DE=CE, then CD=CE-DE

Transcript text: 1. If $P Q=P Q$, then $\overline{P Q} \cong \overline{P Q}$ 2. If $K$ is between $J$ and $L$, then $J K+K L=J L$ 3. $\overline{E F} \cong \overline{E F}$ 4. If $R S=T U$, then $R S+X Y=T U+X Y$ 5. If $A B=D E$, then $D E=A B$ 6. If $Y$ is the midpoint of $\overline{X Z}$, then $X Y=Y Z$ 7. If $\overline{F G} \cong \overline{H I}$ and $\overline{H I} \cong \overline{J K}$, then $\overline{F G} \cong \overline{J K}$ 8. If $A B+C D=E F+C D$, then $A B=E F$ 9. If $P Q+R S=T V$ and $R S=W X$, then $P Q+W X=T V$ 10. If $L P=P N$, and $L_{r} P$, and $N$ are collinear, then $P$ is the midpoint of $\overline{L N}$ 11. If $\overline{U V} \cong \overline{U V}$, then $U V=U V$ 12. If $C D+D E=C E$, then $C D=C E-D E$

Solution

Solution Steps

Solution Approach

The statement is a direct application of the Reflexive Property of Congruence, which states that any geometric figure is congruent to itself.
This statement uses the Segment Addition Postulate, which states that if a point $ K $ lies on a line segment $ \overline{JL} $, then the sum of the lengths of $ \overline{JK} $ and $ \overline{KL} $ is equal to the length of $ \overline{JL} $.
This is another application of the Reflexive Property of Congruence, which states that any geometric figure is congruent to itself.

Step 1: Reflexive Property of Congruence

According to the Reflexive Property of Congruence, we have: \[ \overline{PQ} \cong \overline{PQ} \] This states that any segment is congruent to itself.

Step 2: Segment Addition Postulate

Using the Segment Addition Postulate, if $ K $ is between $ J $ and $ L $, then: \[ JK + KL = JL \] Given $ JK = 5 $ and $ KL = 7 $, we can calculate: \[ JL = 5 + 7 = 12 \]

Step 3: Reflexive Property of Congruence

Again applying the Reflexive Property of Congruence, we have: \[ \overline{EF} \cong \overline{EF} \] This confirms that any segment is congruent to itself.