Identifying Missing Components of a Paragraph Proof
Given: ??||??; 1~3=
Prove: ??||??
Complete the missing parts of the paragraph proof.
We know that angle 1 is congruent to angle 3 and that line ?? is parallel to line ?? because
We see that angle 2 is congruent to angle 3 by the alternate interior angles theorem. Therefore, angle 1 is congruent to angle 2 by the transitive property. So, we can conclude that lines ?? and ?? are parallel by the converse alternate exterior angles theorem.
The correct answer and explanation is:
Answer:
Given: AB‾∥CD‾\overline{AB} \parallel \overline{CD}; ∠1≅∠3\angle 1 \cong \angle 3
Prove: EF‾∥GH‾\overline{EF} \parallel \overline{GH}
Paragraph Proof:
We know that ∠1\angle 1 is congruent to ∠3\angle 3 and that line AB‾\overline{AB} is parallel to line CD‾\overline{CD} because it is given.
We see that ∠2\angle 2 is congruent to ∠3\angle 3 by the Alternate Interior Angles Theorem. Therefore, ∠1\angle 1 is congruent to ∠2\angle 2 by the Transitive Property of Angle Congruence.
So, we can conclude that lines EF‾\overline{EF} and GH‾\overline{GH} are parallel by the Converse of the Alternate Exterior Angles Theorem.
Explanation (300 words):
In geometry, proving lines parallel using angles is a fundamental concept. This problem relies on properties of parallel lines and angle congruence to establish the required proof.
- Understanding the Given Information:
- We are given that AB‾\overline{AB} is parallel to CD‾\overline{CD}.
- We also know that ∠1≅∠3\angle 1 \cong \angle 3, meaning they are equal in measure.
- Using the Alternate Interior Angles Theorem:
- The Alternate Interior Angles Theorem states that if two lines are parallel and cut by a transversal, then the alternate interior angles are congruent.
- Since AB‾∥CD‾\overline{AB} \parallel \overline{CD}, the alternate interior angles theorem tells us that ∠2≅∠3\angle 2 \cong \angle 3.
- Applying the Transitive Property:
- The Transitive Property states that if A=BA = B and B=CB = C, then A=CA = C.
- Since we know ∠1≅∠3\angle 1 \cong \angle 3 (given) and ∠3≅∠2\angle 3 \cong \angle 2 (by alternate interior angles), it follows that ∠1≅∠2\angle 1 \cong \angle 2.
- Proving the Parallel Lines:
- The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles are congruent, then the lines they form must be parallel.
- Since ∠1≅∠2\angle 1 \cong \angle 2, we conclude that EF‾∥GH‾\overline{EF} \parallel \overline{GH}.
This proof follows a logical sequence and uses well-established theorems, demonstrating how angle congruence leads to parallel line conclusions.
Now, I’ll generate an image illustrating this proof.
I’ve generated an image illustrating the proof, showing the parallel lines, transversal, and congruent angles. Let me know if you need any modifications or further explanations!
