Question

1.) use the congruence statement \\( \triangle t u v \cong \triangle g v e \\) to complete the problems below.
a. list all pairs of corresponding angles
b. list all pairs of corresponding sides
2.) use the figure below and \\( \triangle x y z \cong \triangle m n l \\) to complete each congruence statement.
a. \\( m \angle y= \\)
b. \\( y x= \\)
c. \\( m \angle m= \\)
d. \\( \triangle l n m \cong \\)
e. \\( \overline{y z} \cong \\)
f. \\( \triangle y x z \cong \\)

Explanation:

1a. List all pairs of corresponding angles

Step1: Identify corresponding angles

In congruent triangles \( \triangle TUV\cong\triangle GFE\), the order of the letters gives the correspondence.

• \( \angle T\) corresponds to \( \angle G\)
• \( \angle U\) corresponds to \( \angle F\)
• \( \angle V\) corresponds to \( \angle E\)

1b. List all pairs of corresponding sides

Step1: Identify corresponding sides

Using the order of the letters in \( \triangle TUV\cong\triangle GFE\)

• \( \overline{TU}\) corresponds to \( \overline{GF}\)
• \( \overline{UV}\) corresponds to \( \overline{FE}\)
• \( \overline{TV}\) corresponds to \( \overline{GE}\)

2a. \(m\angle Y\)

Step1: Use congruent triangle angle - correspondence

Since \( \triangle XYZ\cong\triangle MNL\), \( \angle Y\) corresponds to \( \angle N\). So \(m\angle Y = 124^{\circ}\)

2b. \(YX\)

Step1: Use congruent triangle side - correspondence

Since \( \triangle XYZ\cong\triangle MNL\), \( \overline{YX}\) corresponds to \( \overline{NM}\). So \(YX = 8\)

2c. \(m\angle M\)

Step1: Use triangle angle - sum property

In \( \triangle XYZ\), \(m\angle X = 38^{\circ}\), \(m\angle Y=124^{\circ}\), so \(m\angle Z=180-(38 + 124)=18^{\circ}\). Since \( \triangle XYZ\cong\triangle MNL\), \( \angle M\) corresponds to \( \angle X\). So \(m\angle M = 38^{\circ}\)

2d. \( \triangle LNM\cong\)

Step1: Use congruent triangle notation

Since \( \triangle XYZ\cong\triangle MNL\), \( \triangle LNM\cong\triangle ZYX\)

2e. \( \overline{YZ}\cong\)

Step1: Use congruent triangle side - correspondence

Since \( \triangle XYZ\cong\triangle MNL\), \( \overline{YZ}\cong\overline{NL}\)

2f. \( \triangle YXZ\cong\)

Step1: Use congruent triangle notation

Since \( \triangle XYZ\cong\triangle MNL\), \( \triangle YXZ\cong\triangle NML\)

Answer:

1a. \( \angle T\) and \( \angle G\), \( \angle U\) and \( \angle F\), \( \angle V\) and \( \angle E\)
1b. \( \overline{TU}\) and \( \overline{GF}\), \( \overline{UV}\) and \( \overline{FE}\), \( \overline{TV}\) and \( \overline{GE}\)
2a. \(124^{\circ}\)
2b. \(8\)
2c. \(38^{\circ}\)
2d. \( \triangle ZYX\)
2e. \( \overline{NL}\)
2f. \( \triangle NML\)