In the diagram, a∥b
and e∥f
, forming a parallelogram. Write a proof that proves that opposite sides of the parallelogram are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements.
Two-Column Proof:
1. Given: a∥b and e∥f form a parallelogram
2. Angle a = angle f (alternate interior angles)
3. Angle b = angle e (alternate interior angles)
4. Line a is congruent to line e (corresponding parts of congruent triangles are congruent)
5. Line b is congruent to line f (corresponding parts of congruent triangles are congruent)
6. Opposite sides of the parallelogram are congruent (definition of a parallelogram)
Paragraph Proof:
Given that a∥b and e∥f form a parallelogram, we can prove that opposite sides of the parallelogram are congruent. By the alternate interior angles theorem, we know that angle a is congruent to angle f and that angle b is congruent to angle e. Using these congruent angles, we can create two congruent triangles by drawing a segment connecting points on opposite sides. By the corresponding parts of congruent triangles are congruent theorem, we can conclude that line a is congruent to line e and line b is congruent to line f. Therefore, opposite sides of the parallelogram are congruent, which is the definition of a parallelogram.
Two-Column Proof:
1. Given: LMNO is a rectangle
2. LM is congruent to ON (opposite sides of a rectangle are congruent)
3. NO is congruent to LO (opposite sides of a rectangle are congruent)
4. Triangle LMO is congruent to triangle NOL (SAS congruence)
5. OM is congruent to NL (corresponding parts of congruent triangles are congruent)
6. Diagonals LM and NO are congruent (definition of congruent segments)
Paragraph Proof:
Given that LMNO is a rectangle, we can prove that the diagonals of the rectangle are congruent. Opposite sides of a rectangle are congruent, so LM is congruent to ON and NO is congruent to LO. By using the SAS congruence (Side-Angle-Side) theorem, we can conclude that triangle LMO is congruent to triangle NOL. By the corresponding parts of congruent triangles are congruent theorem, OM is congruent to NL. Therefore, the diagonals LM and NO are congruent by the definition of congruent segments.
Prove that the diagonals of rectangle LMNO
are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements
To prove that opposite sides of the parallelogram are congruent, we can use the properties of parallel lines and the definition of a parallelogram.
Let's use a two-column proof to demonstrate this:
Statement | Reason
-----------------------------------------------------------
1. a∥b | Given
2. e∥f | Given
3. ∠a = ∠b | Corresponding angles are congruent when a∥b
4. ∠e = ∠f | Corresponding angles are congruent when e∥f
5. ∠a + ∠e = 180° | Angles in a straight line form a straight angle (180°)
6. ∠b + ∠f = 180° | Angles in a straight line form a straight angle (180°)
7. ∠a + ∠e = ∠b + ∠f | Transitive property of equality (from statements 5 and 6)
8. ∠a + ∠e = ∠a + ∠e | Reflexive property of equality
9. ∠b + ∠f = ∠a + ∠e | Substitution property (from statements 7 and 8)
10. ∠a = ∠b = ∠e = ∠f | Subtracting ∠e from both sides (from statement 9)
11. AB = CD | Opposite sides of the parallelogram are intercepted by congruent angles (from statements 3 and 10)
12. BC = AD | Opposite sides of the parallelogram are intercepted by congruent angles (from statements 4 and 10)
Therefore, we have proven that opposite sides of a parallelogram are congruent using the given information and the properties of parallel lines and angles.