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5.4 – Rectangles

Key Terms

  • Rectangle – A quadrilateral with four right angles.
    • Rectangles are parallelograms: opposite sides are parallel and congruent.
  • Right Angle – An angle that measures 90°.
    • Right angles are often marked with a small square symbol.
      • Perpendicular lines form right angles.

 

Review

  • Quadrilaterals are four-sided polygons

 

  • Triangles
    • The sum of interior angles of a triangle add up to 180°

 

  • Isosceles Triangles
    • They have at least two congruent sides.
    • The angles opposite those sides are also congruent.

 

  • Properties of Parallelograms
    • Opposite sides are parallel.
    • Opposite sides are congruent.
    • Opposite angles are congruent.
    • Consecutive angles are supplementary.
    • Diagonals bisect each other.

 


Notes

  • ALL rectangles are parallelograms.
  • ALL parallelograms are quadrilaterals.
  • ALL quadrilaterals are polygons.
  • SOME parallelograms are rectangles.
  • SOME quadrilaterals are rectangles.
  • SOME polygons are rectangles.

GeoB 5.4 Rectangle Diagram

 

  • Properties of Rectangles

GeoB 5.4 Rectangle Properties

 

  • Quadrilaterals vs Parallelograms
    • A quadrilateral with four right angles is a rectangle.
    • A quadrilateral with four right angles is a parallelogram (since ALL rectangles are parallelograms).
      • A quadrilateral with one right angle may or may not be a rectangle.  We don’t know.  There is not enough information.  It could be a trapezoid.
    • A parallelogram with at least ONE right angle is a rectangle.
      • Because of the properties of parallelograms, if a parallelogram has one right angle, then the other three angles will also be right angles
        • Parallelograms have supplementary consecutive angles (they add up to 180°)
        • Parallelograms have congruent opposite angles

GeoB 5.4 Consecutie Right Angles

 

  • Isosceles Triangles
    • Rectangles have two sets of congruent isosceles triangles.
    • The two congruent isosceles triangles are vertical (opposite) from one another.

 

  • Diagonals
    • Diagonals of a rectangle are congruent.
    • If the diagonals of a quadrilateral are congruent, then the quadrilateral is a rectangle.

 


  • Proofs
    • To prove a parallelogram Is a rectangle:
      • Show that one angle is a right angle.
      • Show that the diagonals are congruent.
    • To prove that the diagonals of a rectangle are congruent:
      • Show that opposite sides of the quadrilateral are congruent.
      • Show that all right angles are congruent.
      • Show CPCTC: Corresponding parts of congruent triangles are congruent.

 

  • Proof with Diagonals
    • Fill in the template (below the video)

 

Geometric Proof Template

 


Examples

  • Ex 1. A construction worker needs to put a rectangular window in the side of a building.
    • He knows from measuring that the top and bottom of the window have a width of 8 feet and the sides have a length of 15 feet.
    • He also measured one diagonal to be 17 feet. What is the length of the other diagonal?
      • Rule: Diagonals of a rectangle are congruent; so, they would be the same.
      • Answer: 17 feet

 

  • Ex 2. If LMNO is a rectangle, and MO = 36, what is the length of \overline{NP}?
    • Remember: diagonals of a rectangle are congruent
    • MO is a diagonal
    • NP is half of a diagonal, so NP is half of 36

GeoB 5.4 Rectangle DiagonalsAnswer:  18

 

  • Ex 3. If LMNO is a rectangle, and m\angle MON=30, what is the value of x?
    • Remember: triangles add up to 180°
      • Rectangles have 2 sets of congruent isosceles triangles
      • Since triangles have a sum of interior angles of 180°: subtract the base angles from the triangle to find the missing angle (non-base angle).
      • If one triangle has a base angle that measures 30°, then the other base angle measures 30°
      • Since the sum of interior angles of a triangle add up to 180°:
        • 30 + 30 = 60
        • 180 – 60 = 120
      • Notice that x is vertical from the missing angle (120°) in the isosceles triangle.
        • Vertical angles are congruent, so x = 120°

GeoB 5.4 Rectangle Isosceles AnglesAnswer:  x = 120°

 

  • Ex 4. If STUV is a rectangle and m\angle VSU=52, what is the value of x?
    • Remember: rectangles have 2 sets of congruent isosceles triangles.
      • If one triangle has a 52° base angle, then it has a second 52° base angle.
    • Remember: remote exterior angles equal the sum of the two interior remote angles.
      • The two interior remote angles in this example are the base angles. Each is 52°.
      • x is the remote exterior angle, so:
      • 52 + 52 = 104

GeoB 5.4 Rectangle Remote Ext

Answer:  x = 104°

 

  • Ex 5. In quadrilateral OPQR, ∠P is a right angle. Is OPQR a rectangle?
    • Because you need to know all 4 angles of a quadrilateral are right angles to prove that it’s a rectangle, the answer is, “Cannot Be Determined.”
    • If OPQR was a parallelogram (not just a quadrilateral), all you would have needed was just one right angle to prove it was a rectangle.

 

  • Ex 6. In quadrilateral EFGH, \overline{EG}\cong \overline{FH}. Is EFGH a rectangle?
    • We don’t have enough information here. This COULD be a trapezoid.  We don’t know!
    • It would need to be a parallelogram for us to prove it was a rectangle.
    • Since we don’t know what type of quadrilateral it is, the answer is, “Cannot be determined.”

 


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