4-5 NAME $$ \qquad $$ DATE $$ \qquad $$ Practice Student Edition Pages 214-221 More Congruent Triangles Draw and label triangles MNO and XYZ. Indicate the additional pairs of corresponding parts that would have to be proved congruent in order to use the given postulate or theorem to prove the triangles congruent. 1. $$ \angle N \cong \angle Y $$ and $$ \overline{N O} \cong \overline{Y Z} $$ by ASA 2. $$ \angle O \cong \angle Z $$ and $$ \angle M \cong \angle X $$ by AAS 3. $$ \angle O \cong \angle Z $$ and $$ \overline{M O} \cong \overline{X Z} $$ by AAS 4. $$ \angle N \cong \angle Y $$ and $$ \angle M \cong \angle X $$ by ASA

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1. For triangles $$MNO$$ and $$XYZ$$ with the correspondence $$M\to X$$, $$N\to Y$$, and $$O\to Z$$:

- In item 1 the given congruent parts are  
  $$\angle N\cong\angle Y$$ and $$\overline{NO}\cong\overline{YZ}$$.  
  Using the ASA postulate, the congruent side must be between the two congruent angles.  
  Since $$\overline{NO}$$ is between vertices $$N$$ and $$O$$, the missing pair is the angle at the other endpoint of the side:  
  $$
  \angle O\cong\angle Z.
  $$

2. In item 2 the given parts are  
   $$\angle O\cong\angle Z$$ and $$\angle M\cong\angle X$$.  
   For an AAS proof, a non‐included side corresponding to the two given angles must also be congruent.  
   In triangle $$MNO$$, the side opposite the third angle (at $$N$$) is $$\overline{MO}$$.  
   Its corresponding side in triangle $$XYZ$$ is $$\overline{XZ}$$. Thus, the additional required congruence is:
   $$
   \overline{MO}\cong\overline{XZ}.
   $$

3. In item 3 the given parts are  
   $$\angle O\cong\angle Z$$ and $$\overline{MO}\cong\overline{XZ}$$.  
   For AAS the missing part is one of the angles not adjacent to the given side.  
   The other given vertex in triangle $$MNO$$ is $$M$$ so that its corresponding angle is $$\angle M$$ in triangle $$XYZ$$.  
   Therefore, the additional pair is:
   $$
   \angle M\cong\angle X.
   $$

4. In item 4 the given parts are  
   $$\angle N\cong\angle Y$$ and $$\angle M\cong\angle X$$.  
   For an ASA proof the side included between the two given angles must be congruent.  
   In triangle $$MNO$$, the side between $$\angle M$$ and $$\angle N$$ is $$\overline{MN}$$,  
   and its corresponding side in triangle $$XYZ$$ is $$\overline{XY}$$.  
   Hence, the additional congruent pair is:
   $$
   \overline{MN}\cong\overline{XY}.
   $$
1. For triangles $$MNO$$ and $$XYZ$$: - Given: $$\angle N \cong \angle Y$$ and $$\overline{NO} \cong \overline{YZ}$$ - Additional congruent pair needed: $$\angle O \cong \angle Z$$ 2. For triangles $$MNO$$ and $$XYZ$$: - Given: $$\angle O \cong \angle Z$$ and $$\angle M \cong \angle X$$ - Additional congruent pair needed: $$\overline{MO} \cong \overline{XZ}$$ 3. For triangles $$MNO$$ and $$XYZ$$: - Given: $$\angle O \cong \angle Z$$ and $$\overline{MO} \cong \overline{XZ}$$ - Additional congruent pair needed: $$\angle M \cong \angle X$$ 4. For triangles $$MNO$$ and $$XYZ$$: - Given: $$\angle N \cong \angle Y$$ and $$\angle M \cong \angle X$$ - Additional congruent pair needed: $$\overline{MN} \cong \overline{XY}$$

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