**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.

- Right angles are often marked with a small square symbol.

**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.

- Properties of Rectangles

- 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

- Because of the properties of parallelograms, if a parallelogram has one right angle, then the other three angles will also be 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.

- To prove a parallelogram Is a rectangle:

- 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.
- Answer: 17 feet

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

Answer: 18

- Ex 3. If LMNO is a rectangle, and , 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°

- Remember: triangles add up to 180°

Answer: x = 120°

- Ex 4. If STUV is a rectangle and , 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

- Remember: rectangles have 2 sets of congruent isosceles triangles.

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, . 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.”