# 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

• 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.
• SOME parallelograms are rectangles.
• SOME polygons are rectangles.

• Properties of Rectangles

• 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

• 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)

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.

• 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

• 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°

• 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

• 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.”