Can someone please check this?
given: line AE and line BD bisect each other.
prove: triangle ACB is congruent to triangle ECD
Statements
1. Line AE and line BD bisect each other
2. Line AC is congruent to line EC, line DC is congruent to line BC
3. Angle ACB is congruent to angle ECD ???
4. Triangle ACB is congruent to triangle ECD
Reasons
1. Given
2. Definition of segment bisector
3. Vertical angles are congruent
4. SSS???
am I right?
No, you are not right. The third statement should be "Angle ACB is congruent to angle DEC" and the fourth reason should be "Angles-Side-Angle (ASA) Congruence Postulate".
Your reasoning for statements 1 and 2 is correct. However, your reasoning for statement 3 is not accurate.
The correct reason for statement 3 would be the Angle Bisector Theorem. Since line AE and line BD bisect each other, it means that angle ACB is bisected by line AE and angle ECD is bisected by line BD. According to the Angle Bisector Theorem, when a line bisects an angle, it divides the opposite side into segments that are proportional to the other two sides of the triangle.
Therefore, the correct reason for statement 3 is the Angle Bisector Theorem.
Your reasoning for statement 4, SSS (Side-Side-Side) is also incorrect. The congruence of triangle ACB and triangle ECD is not proven by SSS, but rather by ASA (Angle-Side-Angle).
The correct reason for statement 4 is ASA, because we know that angle ACB is congruent to angle ECD (from statement 3), and line AC is congruent to line EC and line DC is congruent to line BC (from statement 2).
So, here is the revised proof:
Statements
1. Line AE and line BD bisect each other
2. Line AC is congruent to line EC, line DC is congruent to line BC
3. Angle ACB is congruent to angle ECD
4. Triangle ACB is congruent to triangle ECD
Reasons
1. Given
2. Definition of segment bisector
3. The Angle Bisector Theorem
4. ASA (Angle-Side-Angle)
Your reasoning is mostly correct, but there is one small mistake in your Reasons section. Let me explain how to correctly prove the congruence of triangle ACB and triangle ECD.
To prove that triangle ACB is congruent to triangle ECD based on the given information and statements, you can use the following steps:
1. Given: Line AE and line BD bisect each other.
2. Reason: Given.
3. By definition, when a line bisects another line, it divides it into two equal parts. So, line AC is congruent to line EC, and line DC is congruent to line BC.
4. Reason: Definition of segment bisector.
5. When two lines intersect, the vertically opposite angles are congruent. Therefore, angle ACB is congruent to angle ECD.
6. Reason: Vertical angles are congruent.
7. Finally, by using the SSS (side-side-side) congruence postulate, we can conclude that triangle ACB is congruent to triangle ECD because their corresponding sides and angles are congruent.
8. Reason: SSS congruence postulate.
Therefore, your proof would look like this:
Proof:
1. Given: Line AE and line BD bisect each other.
2. Line AC is congruent to line EC, line DC is congruent to line BC (By definition of segment bisector).
3. Angle ACB is congruent to angle ECD (Vertical angles are congruent).
4. Triangle ACB is congruent to triangle ECD (By SSS congruence postulate).
Hope this helps!