Question

given: e is the midpoint of \\( \overline { a c } \\), \\( d e = e c \\) prove: \\( \overline { d e } \cong \overline { a e } \\) identify the reason that justifies each statement. 25. \if then, \\( \angle 1 \cong \angle 2 \\)\ \if \\( c d = e f \\), then \\( \overline { c d } \cong \overline { e f } \\)\ a. definition of congruence a. definition of congruence b. angle addition postulate b. reflexive property c. linear pair theorem c. symmetric property d. vertical angles theorem d. definition of midpoint \if \\( \angle a \\) and \\( \angle b \\) are complementary, then \\( m \angle a + m \angle b = 90 ^ { \circ } \\)\ given: \if \\( l \perp m \\), then \\( \angle 1 \\) is a right angle\ a. complement theorem a. definition of complementary angles b. definition of a right angle b. definition of a right angle c. congruent complements theorem c. definition of perpendicular d. definition of complementary angles d. complement theorem

Explanation:

Step1: Fill in the table

Statements

1. \(E\) is the midpoint of \(\overline{AC}\)
2. \(AE = EC\) (by the definition of midpoint)
3. \(DE=EC\) (given)
4. \(AE = DE\) (transitive property of equality: if \(a = b\) and \(b = c\), then \(a = c\))
5. \(\overline{DE}\cong\overline{AE}\) (by the definition of congruence)

Step2: Analyze multiple - choice questions

First multiple - choice question

• For \(\angle1\cong\angle2\) (vertical angles), the reason is the Vertical Angles Theorem.
• For \(CD = EF\) then \(\overline{CD}\cong\overline{EF}\), the reason is the Definition of Congruence.

Second multiple - choice question

• For \(\angle A\) and \(\angle B\) are complementary, then \(m\angle A+m\angle B = 90^{\circ}\), the reason is the Definition of Complementary Angles.
• For \(l\perp m\), then \(\angle1\) is a right angle, the reason is the Definition of Perpendicular.

Answer:

Table

Statements	Reasons
2. \(AE = EC\)	2. Definition of Midpoint
3. \(DE = EC\)	3. Given
4. \(AE = DE\)	4. Transitive Property of Equality
5. \(\overline{DE}\cong\overline{AE}\)	5. Definition of Congruence

Multiple - choice questions

• First question: For \(\angle1\cong\angle2\), the answer is D. Vertical Angles Theorem; for \(CD = EF\) then \(\overline{CD}\cong\overline{EF}\), the answer is A. Definition of Congruence.
• Second question: For \(\angle A\) and \(\angle B\) are complementary, then \(m\angle A+m\angle B = 90^{\circ}\), the answer is D. Definition of Complementary Angles; for \(l\perp m\), then \(\angle1\) is a right angle, the answer is C. Definition of Perpendicular.