Question

proof complete the proof.
given: b is the midpoint of $overline{ae}$, b is the midpoint of $overline{cd}$.
prove: $\triangle abdcong\triangle ebc$
statements reasons

1. b is the midpoint of $overline{ae}$. 1.
2. $angle b$ 2. definition of midpoint
3. b is the midpoint of $overline{cd}$. 3.
4. 4. definition of midpoint
5. $angle abdcongangle ebc$ 5.
6. $\triangle abdcong\triangle ebc$ 6.

proof complete the proof.
given: $overline{om}perpoverline{ln}$, $overline{ol}congoverline{on}$
prove: $\triangle omlcong\triangle omn$
statements reasons

1. $overline{om}perpoverline{ln}$ 1. given
2. 2. if 2 angles are $perp$, then they form 4 right $\triangle$
3. 3. right angle congruence theorem
4. $overline{ml}congoverline{mn}$ 4.
5. $overline{om}congoverline{om}$ 5.
6. $\triangle omlcong\triangle omn$ 6.

Explanation:

Step1: Mid - point property for first proof

Since \(B\) is the mid - point of \(\overline{AE}\), by the definition of mid - point, \(AB = EB\).

1. \(B\) is the mid - point of \(\overline{AE}\) (Given)
2. \(AB=EB\) (Definition of mid - point)
3. \(B\) is the mid - point of \(\overline{CD}\) (Given)
4. \(BD = BC\) (Definition of mid - point)
5. \(\angle ABD\cong\angle EBC\) (Vertical angles are congruent)
6. \(\triangle ABD\cong\triangle EBC\) (Side - Angle - Side (SAS) congruence criterion)

Step2: Right - triangle congruence for second proof

1. \(\overline{OM}\perp\overline{LN}\) (Given)
2. \(\angle OML=\angle OMN = 90^{\circ}\) (If two lines are perpendicular, they form four right angles)
3. \(\overline{OL}\cong\overline{ON}\) (Given)
4. \(\overline{ML}\cong\overline{MN}\) (The perpendicular from the vertex of an isosceles triangle to the base bisects the base)
5. \(\overline{OM}\cong\overline{OM}\) (Reflexive property of congruence)
6. \(\triangle OML\cong\triangle OMN\) (Hypotenuse - Leg (HL) congruence criterion for right - triangles)

Answer:

For the first proof:

1. Given
2. \(AB = EB\)
3. Given
4. \(BD = BC\)
5. Vertical angles are congruent
6. SAS

For the second proof:

1. Given
2. \(\angle OML=\angle OMN = 90^{\circ}\)
3. Given
4. The perpendicular from the vertex of an isosceles triangle to the base bisects the base
5. Reflexive property of congruence
6. HL