Lesson 8-4 Rectangles

5-Minute Check on Lesson 8-3 Transparency 8-4 Click the mouse button or press the Space Bar to display the answers. Determine whether each quadrilateral is a parallelogram. Justify your answer

5-Minute Check on Lesson 8-3 Transparency 8-4 Click the mouse button or press the Space Bar to display the answers. Determine whether each quadrilateral is a parallelogram. Justify your answer Yes, diagonal bisect each other Yes, opposite angles congruent

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

Vocabulary Rectangle – quadrilateral with four right angles.

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

Example 4-1c Answer: 5 Quadrilateral EFGH is a rectangle. If FH = 5x + 4 and GE = 7x – 6, find x. EXAMPLE 2

Example 4-2a Quadrilateral LMNP is a rectangle. Find x. Angle Addition Theorem Answer: 10 Substitution Simplify. Subtract 10 from each side. Divide each side by 8.  MLP is a right angle, so m  MLP = 90° EXAMPLE 3

Quadrilateral LMNP is a rectangle. Find y. EXAMPLE 3 (CONT)

Example 4-2d Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent. Alternate Interior Angles Theorem Divide each side by 6. Substitution Subtract 2 from each side. Simplify. Answer: 5 EXAMPLE 3 (CONT)

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

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 (10-24)