Question

written exercises
in exercises 1 - 6 you are given a diagram that is marked with given information.
give the reason for each key step of the proof.
a 1. prove: \\( \overline { n e } \cong \overline { o s } \\)
key steps of proof:
a. \\( \triangle r n x \cong \triangle l o y \\)
b. \\( \angle x \cong \angle y \\)
c. \\( \triangle n e x \cong \triangle o s y \\)
d. \\( \overline { n e } \cong \overline { o s } \\)

1. prove: \\( \overline { b e } \cong \overline { d f } \\)

key steps of proof:
a. \\( \triangle a b c \cong \triangle c d a \\)
b. \\( \angle 1 \cong \angle 2 \\)
c. \\( \triangle a b e \cong \triangle c d f \\)
d. \\( \overline { b e } \cong \overline { d f } \\)

Explanation:

1. Prove: \(\overline{NE}\cong\overline{OS}\)

# Explanation:

Step1: Congruent Triangles (a)

Since \(\triangle RNX\cong\triangle LOY\) (given or by SSS, SAS, ASA, AAS, HL criteria, assume from diagram markings).

Step2: Corresponding Parts (b)

If \(\triangle RNX\cong\triangle LOY\), then \(\angle X\cong\angle Y\) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Step3: Congruent Triangles (c)

For \(\triangle NEX\) and \(\triangle OSY\), assume from diagram markings (e.g., if \(NX = OY\) from \(\triangle RNX\cong\triangle LOY\), and other sides or angles marked equal) \(\triangle NEX\cong\triangle OSY\) (by SSS, SAS, ASA, AAS).

Step4: Corresponding Parts (d)

Since \(\triangle NEX\cong\triangle OSY\), then \(\overline{NE}\cong\overline{OS}\) (CPCTC).

2. Prove: \(\overline{BE}\cong\overline{DF}\)

# Explanation:

Step1: Congruent Triangles (a)

\(\triangle ABC\cong\triangle CDA\) (given or by SSS, SAS, ASA, AAS, HL criteria, assume from diagram markings).

Step2: Corresponding Parts (b)

If \(\triangle ABC\cong\triangle CDA\), then \(\angle1\cong\angle2\) (CPCTC).

Step3: Congruent Triangles (c)

For \(\triangle ABE\) and \(\triangle CDF\), assume from diagram markings (e.g., \(AB = CD\) from \(\triangle ABC\cong\triangle CDA\), and other sides or angles marked equal) \(\triangle ABE\cong\triangle CDF\) (by SSS, SAS, ASA, AAS).

Step4: Corresponding Parts (d)

Since \(\triangle ABE\cong\triangle CDF\), then \(\overline{BE}\cong\overline{DF}\) (CPCTC).

Answer:

1.

• a. Assume SSS/SAS/ASA/AAS/HL (from diagram markings)
• b. CPCTC
• c. Assume SSS/SAS/ASA/AAS (from diagram markings)
• d. CPCTC

2.

• a. Assume SSS/SAS/ASA/AAS/HL (from diagram markings)
• b. CPCTC
• c. Assume SSS/SAS/ASA/AAS (from diagram markings)
• d. CPCTC