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 } $
2. 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 } $

Question

Accepted Answer

# Explanation:
## 1. Prove: $\overline{NE}\cong\overline{OS}$
### a. $\triangle RNX\cong\triangle LOY$
By SSS (Side - Side - Side) congruence criterion (since the sides of the triangles are marked as equal in the diagram).
### b. $\angle X\cong\angle Y$
Corresponding parts of congruent triangles are congruent (CPCTC).
### c. $\triangle NEX\cong\triangle OSY$
By ASA (Angle - Side - Angle) congruence criterion (we know $\angle X\cong\angle Y$ from part b, and from the diagram, we can assume some side - angle - side relationships based on the markings).
### d. $\overline{NE}\cong\overline{OS}$
Corresponding parts of congruent triangles are congruent (CPCTC).
## 2. Prove: $\overline{BE}\cong\overline{DF}$
### a. $\triangle ABC\cong\triangle CDA$
By SSS (Side - Side - Side) congruence criterion (as the sides of the triangles are marked as equal in the diagram).
### b. $\angle1\cong\angle2$
Corresponding parts of congruent triangles are congruent (CPCTC).
### c. $\triangle ABE\cong\triangle CDF$
By AAS (Angle - Angle - Side) congruence criterion (we know $\angle1\cong\angle2$ from part b, and from the diagram, we can assume some right - angle and side relationships).
### d. $\overline{BE}\cong\overline{DF}$
Corresponding parts of congruent triangles are congruent (CPCTC).

# Answer:
1.
    - a. SSS
    - b. CPCTC
    - c. ASA
    - d. CPCTC
2.
    - a. SSS
    - b. CPCTC
    - c. AAS
    - d. CPCTC

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Question

Explanation:

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

a. \(\triangle RNX\cong\triangle LOY\)

b. \(\angle X\cong\angle Y\)

c. \(\triangle NEX\cong\triangle OSY\)

d. \(\overline{NE}\cong\overline{OS}\)

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

a. \(\triangle ABC\cong\triangle CDA\)

b. \(\angle1\cong\angle2\)

c. \(\triangle ABE\cong\triangle CDF\)

d. \(\overline{BE}\cong\overline{DF}\)

Answer: