Question

Please help. i’ve been struggling with proofs and no ones been able to help me

Answer 1

See explanation.

Explanation:

Statement Reasons

1)
`\overline{DC} \cong \overline{BC}`1)Given

2)
`\angle CED \cong \angleD` 2)Given

3)
`\overline{CE} \cong \overline{CD}`

and
`\triangle CED` is isosceles 3)Converse of the Base Angle Thm.

4)
`\overline{CE} \cong \overline{CE}`4) Reflexive Property

5)
`\overline{CE} \cong \overline{BC}` 5) Transitive Property

6)
`\angle CBE \cong \angle CEB`

and
`\triangle CBE` is isosceles 6) Base Angle Theorem

I'm going to write my statements and reasons in order below just in case it isn't readable on your computer from above.

Statements:

1)
`\overline{DC} \cong \overline{BC}`

2)
`\angle CED \cong \angleD`

3)
`\overline{CE} \cong \overline{CD}`

and
`\triangle CED` is isosceles

4)
`\overline{CE} \cong \overline{CE}`

5)
`\overline{CE} \cong \overline{BC}`

6)
`\angle CBE \cong \angle CEB`

and
`\triangle CBE` is isosceles

Reasons:

1) Given

2) Given

3) Converse of the Base Angle Theorem

4) Reflexive property

5) Transitive property

6) Base Angle Theorem

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Whenever I start a two-column proof, I state me givens.

I applied the converse of the base angle theorem because we were given two angles in the triangle CED that were congruent. By this theorem, if two angles are congruent, then their opposite legs are congruent. So also by this theorem; if you can apply it, then you have an isosceles triangle.

Both triangles shared the side CE is why I used the reflexive property.

I knew CE and CD were congruent by the theorem I stated in reasons: 3 (converse of the base angle theorem).

I also had that in my given that DC=BC. So by transitive property, I could conclude that CE=BC.

This is when I finally used the base angle theorem to conclude that the opposite angles of those congruent legs in triangle CBE were congruent.

The base angle theorem says if two legs in a triangle are congruent, then the opposite angles of those legs are congruent. So you can conclude from this theorem; if is applicable which it was here, that the triangle is an isosceles.