Given : ABCE is a rectangle. D is the midpoint of CE.
Prove : AD is congruent to BD .
Statements || Reasons
1. ABCE is a rectangle. D is the midpoint of CE . || 1. Given
2. <AED is congruent to BCD || 2. Definition of rectangle
3. AE is congruent to BC || 3. Definition of rectangle
I have to fill in a 4, 5, and 5 statement and reason .
I can't post the link of the shape here, but if you look up the question, it will be there on OpenStudy. The shape of the rectangle with the upside down triangle in it. I actually just copied the question, I'm really stuck and confused. I don't know what to do next. I'm not good at proofs at all! I really need help please!
I know this question is old but I will answer it anyways :D.
4 statement.) Line ED and line CD are congruent
4 reason.) definition of Midpoint
5 statement.) Triangle AED is congruent to triangle BCD
5 reason.) SAS
6 statement.) Line AD is congruent to line BD
6 reason.) CPCTC
I hope this helps you. I'm not sure if I'm right because i was just taking the little quiz thing that this question was on and my teacher hasn't graded it but I'm feeling very confident with it and ive always been good with math so i hope you trust my answer :)
Helped me so much, thank you loads :)
4. AD is congruent to BD || 4. Definition of midpoint (D is the midpoint of CE)
5. Triangle AED is congruent to triangle BCD || 5. Side-Angle-Side (SAS) congruency theorem
6. AD is congruent to BD || 6. Corresponding parts of congruent triangles are congruent
Reasoning:
- Statement 1 is given.
- Statement 2 is based on the definition of a rectangle, which states that all angles in a rectangle are right angles.
- Statement 3 is based on the definition of a rectangle, which states that opposite sides of a rectangle are congruent.
- Statement 4 follows from the given information that D is the midpoint of CE, so AD and BD are congruent.
- Statement 5 uses the SAS congruency theorem, which states that if two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent.
- Statement 6 follows from statement 5, as corresponding parts of congruent triangles are congruent. Therefore, AD is congruent to BD.
4. AD is congruent to BD || 4. Definition of midpoint (D is the midpoint of CE)
5. Triangle AED is congruent to triangle BCD || 5. SSS (side-side-side) congruence postulate
Explanation:
To prove that AD is congruent to BD, we can use the fact that D is the midpoint of CE. According to the definition of a midpoint, it means that D divides CE into two equal parts. Therefore, we can conclude that AD is congruent to BD.
To further prove the congruence of the triangles AED and BCD, we can use the fact that ABCE is a rectangle. Since ABCE is a rectangle, we know that opposite sides are parallel and congruent. So, AE is congruent to BC, as mentioned in statement 3.
Now, by using the definition of midpoint in statement 1 and the fact that AE is congruent to BC, we can conclude that AD is congruent to BD (statement 4).
Finally, to establish the congruence of triangles AED and BCD, we can apply the SSS (side-side-side) congruence postulate. By saying that AD is congruent to BD (statement 4), and AE is congruent to BC (statement 3), we have the necessary side lengths to prove that triangle AED is congruent to triangle BCD (statement 5).
To prove that AD is congruent to BD, we need to show that the lengths of AD and BD are equal.
Here's how we can complete the proof:
4. Triangle AED is congruent to triangle BCD. || 4. ASA (Angle-Side-Angle) congruence postulate - Since AE is congruent to BC (given) and <AED is congruent to <BCD (opposite angles of a rectangle are congruent), we can conclude that triangle AED is congruent to triangle BCD.
5. AD is congruent to BD. || 5. Corresponding parts of congruent triangles are congruent. Since triangle AED is congruent to triangle BCD (proved in statement 4), the corresponding sides AD and BD must also be congruent.
Thus, by providing the reasons and steps outlined above, we have proven that AD is congruent to BD based on the given information.