Question

Write and solve your own proof.

Given: Angle B is congruent to Angle D, Angle DAE is congruent to Angle BEA.
Prove: Triangle ABC is congruent to Triangle EDC.
Thank you!

Answer 1

By using the SAS congruence postulate we can prove that △ABC≅△EDC.

We are given that ∠B≅∠D and ∠DAE≅∠BEA.

Since vertical angles are always congruent, we have ∠DAB=∠CED.

Therefore, △ABD≅△ECD by the AAS congruence postulate.

By CPCTC, we know that AB=DE and AD=CE.

∠DAE≅∠BEA | Given

∠DAB=∠CED | Vertical angles

△ABD≅△ECD | AAS congruence

AB=DE | CPCTC

AD=CE | CPCTC

△ABC≅△EDC | SAS congruence

Therefore, △ABC≅△EDC.

Finally, we can use the SAS congruence postulate to conclude that △ABC≅△EDC.

The SAS (Side-Angle-Side) Congruence Postulate states that:

• If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
• This postulate is fundamental in geometry for proving the congruence of triangles. It allows us to establish that two triangles are identical in size and shape based on specific matching sides and angles.

Here are some additional points regarding the SAS Congruence Postulate:

• The postulate is reversible. If two triangles are congruent, then their corresponding sides and included angles are also congruent.
• The postulate only applies to triangles. It cannot be used to prove the congruence of other polygons.
• The order of the sides and the included angle does not matter. We can write the postulate as ASA, AAS, or SAS, and the meaning remains the same.

Answer 2

1.
`\angle B \cong \angle D` 1. Given.

2.
`\angle DAE \cong \angle BEA` 2. Given.

3.
`\triangle AEC` is isosceles 3. by statement 2 and definition of triangle isosceles.

4.
`AC \cong EC` 4. by definition of triangle isosceles.

5.
`\angle ACB \cong \angle ECD` 5. by vertical angles.

6.

`\angle B + \angle ACB + \angle BAC = 180\°\\\angle D + \angle ECD + \angle DEC = 180\°` 6. by internal angles of a triangle.

7.
`\angle B + \angle ACB + \angle BAC =\angle D + \angle ECD + \angle DEC \\\angle BAC = \angle DEC` 7. by statement 1, 5 and 6.

8.
`\triangle ABC \cong EDC`. 8. by AAA postulate.

Remember, the AAA postulate is about the congruence between thre pairs of corresponding angles.