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Understanding Rectangles: Properties, Justifications, and Parallelograms

This lesson explores rectangles and parallelograms by examining their properties and characteristics. Students will learn to determine whether a quadrilateral is a parallelogram and how to apply the distance and slope formulas using given vertices. The lesson emphasizes the importance of congruent diagonals in parallelograms, justifying that if the diagonals are congruent, the shape is a rectangle. It also includes practice problems and real-world applications to reinforce these concepts, preparing students for standardized test questions.

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Understanding Rectangles: Properties, Justifications, and Parallelograms

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  1. Lesson 8-4 Rectangles

  2. Transparency 8-4 5-Minute Check on Lesson 8-3 • Determine whether each quadrilateral is a parallelogram. Justify your answer. • 2. • Determine whether the quadrilateral with the given vertices is a parallelogram using the method indicated. • 3. A(,), B(,), C(,), D(,) Distance formula • 4. R(,), S(,), T(,), U(,) Slope formula • 5. Which set of statements will prove LMNO a parallelogram? L M Standardized Test Practice: N O LM // NO and LO  MN LO // MN and LO  MN A B LM  LO and ON  MN LO  MN and LO  ON C D Click the mouse button or press the Space Bar to display the answers.

  3. Transparency 8-4 5-Minute Check on Lesson 8-3 • Determine whether each quadrilateral is a parallelogram. Justify your answer. • 2. • Determine whether the quadrilateral with the given vertices is a parallelogram using the method indicated. • 3. A(,), B(,), C(,), D(,) Distance formula • 4. R(,), S(,), T(,), U(,) Slope formula • 5. Which set of statements will prove LMNO a parallelogram? Yes, diagonal bisect each other Yes, opposite angles congruent Yes, opposite sides equal No, RS not // UT L M Standardized Test Practice: N O LM // NO and LO  MN LO // MN and LO  MN A B LM  LO and ON  MN LO  MN and LO  ON C D Click the mouse button or press the Space Bar to display the answers.

  4. Objectives • Recognize and apply properties of rectangles • A rectangle is a quadrilateral with four right angles and congruent diagonals • Determine whether parallelograms are rectangles • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle

  5. Vocabulary • Rectangle – quadrilateral with four right angles.

  6. Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites Trapezoids IsoscelesTrapezoids Rectangles Rhombi Squares

  7. The diagonals of a rectangle are congruent, so Example 4-1a Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x - 4 find x. Definition of congruent segments Substitution Subtract 6x from each side. Add 4 to each side. Answer: 8

  8. Example 4-1c Quadrilateral EFGH is a rectangle. If FH = 5x + 4 and GE = 7x – 6, find x. Answer: 5

  9. Solve for x and y in the following rectangles A B Hint: Special Right Triangles x 60° 8 30° D C y A B x 2y + 8 4y -12 3x - 8 D C Hint: p is perimeter 2x A B P = 36 feet x x D 2x C x Hint: 2 Equations, 2 Variables  Substitution A B 3y 3x -9 2y D C

  10. Example 4-2a Quadrilateral LMNP is a rectangle. Findx. MLP is a right angle, so mMLP = 90° Angle Addition Theorem Substitution Simplify. Subtract 10 from each side. Divide each side by 8. Answer: 10

  11. Quadrilateral LMNP is a rectangle. Findy.

  12. Example 4-2d Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent. Alternate Interior Angles Theorem Substitution Simplify. Subtract 2 from each side. Divide each side by 6. Answer: 5

  13. Example 4-2e Quadrilateral EFGH is a rectangle. a. Findx. b. Find y. Answer: 11 Answer: 7

  14. Kyle is building a barn for his horse. He measures the diagonals of the door opening to make sure that they bisect each other and they are congruent. How does he know that the corners are angles? We know that A parallelogram with congruent diagonals is a rectangle. Therefore, the corners are angles. Answer: Example 4-3a

  15. Quadrilateral Characteristics Summary Convex Quadrilaterals 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Parallelograms Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to basesMedian = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Rhombi IsoscelesTrapezoids All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Angles all 90° Diagonals congruent Legs are congruent Base angle pairs congruent Diagonals are congruent Squares Diagonals divide into 4 congruent triangles

  16. Summary & Homework • Summary: • A rectangle is a quadrilateral with four right angles and congruent diagonals • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle • Homework: • pg 428-429; 10-13, 16-20, 42

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