prove that a cyclic quadrilateral is a rectangle?
Answers
Answer:
This is your answer for the above question
Answer:
There are three ways to prove that a quadrilateral is a rectangle. Note that the second and third methods require that you first show (or be given) that the quadrilateral in question is a parallelogram:
If all angles in a quadrilateral are right angles, then it’s a rectangle (reverse of the rectangle definition). (Actually, you only need to show that three angles are right angles — if they are, the fourth one is automatically a right angle as well.)
If the diagonals of a parallelogram are congruent, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).
If a parallelogram contains a right angle, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).
Step-by-step explanation:
Congruent supplementary angles are right angles: If two angles are both supplementary and congruent, then they’re right angles. This idea makes sense because 90° + 90° = 180°.
Okay, so here’s the proof:
Statement 1:
Reason for statement 1: Given.
Statement 2:
Reason for statement 2: If same-side exterior angles are supplementary, then lines are parallel.
Statement 3:
Reason for statement 3: If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.
Statement 4:
Reason for statement 4: If two angles are supplementary to the same angle, then they’re congruent.
Statement 5:
Reason for statement 5: Given.
Statement 6:
Reason for statement 6: If two angles are both supplementary and congruent, then they’re right angles.
Statement 7:
Reason for statement 7: If lines form a right angle, then they’re perpendicular.
Statement 8:
Reason for statement 8: If lines are perpendicular, then they form right angles.
Statement 9:
Reason for statement 9: If a parallelogram contains a right angle, then it’s a rectangle.
Statement 10:
Reason for statement 10: The diagonals of a rectangle are congruent.