Question

1. given: k is the mid - point of $overline{jl}$, m is the mid - point of $overline{ln}$, $jk = mn$. prove: $overline{kl}congoverline{lm}$. statements reasons 1. k is the mid - point of $overline{jl}$, m is the mid - point of $overline{ln}$ 1. given 2. $jk = kl$, $lm = mn$ 2. 3. $jk = mn$ 3. given 4. $mn = kl$, $lm = mn$ 4. 5. $lm = kl$ 5. 6. $kl = lm$ 6. 7. $overline{kl}congoverline{lm}$ 7. 5. given: $overline{xy}congoverline{uv}$, $overline{yz}congoverline{tu}$. prove: $overline{xz}congoverline{tv}$. statements reasons 1. $overline{xy}congoverline{uv}$, $overline{yz}congoverline{tu}$ 1. 2. $xy = uv$, $yz = tu$ 2. 3. $xy + yz = xz$, $tu+uv = tv$ 3. 4. $uv + yz = xz$, $yz + uv = tv$ 4. 5. $xz = tv$ 5. 6. $overline{xz}congoverline{tv}$ 6. 6. given: $overline{yw}congoverline{yz}$, $overline{xy}congoverline{vy}$. prove: $overline{xz}congoverline{vw}$. statements reasons 1. $overline{wy}congoverline{yz}$, $overline{xy}congoverline{vy}$ 1. 2. $wy = yz$, $xy = vy$ 2. 3. $xy + yz = xz$ 3. 4. $vy + yw = xz$ 4. 5. $vy + yw = vw$ 5. 6. $xz = vw$ 6. 7. $overline{xz}congoverline{vw}$ 7. © gina wilson (all things algebra), 2014

Explanation:

1.

Step1: Definition of mid - point

If \(K\) is the mid - point of \(\overline{JL}\), then \(JK = KL\) by the definition of a mid - point. If \(M\) is the mid - point of \(\overline{LN}\), then \(LM=MN\) by the definition of a mid - point.

Step2: Substitution

Since \(JK = MN\) (given) and \(JK = KL\), \(LM = MN\), we can substitute to get \(MN=KL\) and \(LM = MN\).

Step3: Transitive property

If \(MN = KL\) and \(LM = MN\), then by the transitive property of equality, \(LM = KL\).

Step4: Symmetric property

If \(LM = KL\), then by the symmetric property of equality, \(KL = LM\).

Step5: Definition of congruent segments

If \(KL = LM\), then \(\overline{KL}\cong\overline{LM}\) by the definition of congruent segments.

Step1: Given

The given information \(\overline{XY}\cong\overline{UV}\) and \(\overline{YZ}\cong\overline{TU}\) implies \(XY = UV\) and \(YZ = TU\) by the definition of congruent segments.

Step2: Segment addition postulate

By the segment addition postulate, \(XY + YZ=XZ\) and \(TU + UV = TV\).

Step3: Substitution

Since \(XY = UV\) and \(YZ = TU\), we can substitute to get \(UV+YZ = XZ\) and \(YZ + UV=TV\).

Step4: Transitive property

By the transitive property of equality, if \(UV + YZ=XZ\) and \(YZ + UV = TV\), then \(XZ = TV\).

Step5: Definition of congruent segments

If \(XZ = TV\), then \(\overline{XZ}\cong\overline{TV}\) by the definition of congruent segments.

Step1: Given

The given \(\overline{WY}\cong\overline{YZ}\) and \(\overline{XY}\cong\overline{VY}\) implies \(WY = YZ\) and \(XY = VY\) by the definition of congruent segments.

Step2: Segment addition postulate

By the segment addition postulate, \(XY + YZ=XZ\).

Step3: Substitution

Since \(WY = YZ\) and \(XY = VY\), we substitute to get \(VY+YW = XZ\).

Step4: Segment addition postulate

By the segment addition postulate, \(VY + YW=VW\).

Step5: Transitive property

By the transitive property of equality, if \(VY + YW=XZ\) and \(VY + YW = VW\), then \(XZ = VW\).

Step6: Definition of congruent segments

If \(XZ = VW\), then \(\overline{XZ}\cong\overline{VW}\) by the definition of congruent segments.

Answer:

1. Definition of mid - point; 4. Substitution; 5. Transitive property of equality; 6. Symmetric property of equality; 7. Definition of congruent segments

5.