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Rectangle Definition: A rectangle is a parallelogram that has a right angle. Corollary 4.3.1: All angles of a rectangle are right angles. Theorem 4.3.2: The diagonals of a rectangle are congruent. A rectangle is a parallelogram:
- Opposite sides are congruent (they equal each other).
- Opposite angles are congruent (they equal each other).
- Consecutive angles are supplementary (they add up to 180)
- Diagonals bisect each other (they cut each other in half)
- Diagonals are congruent (they equal each other)
- All four angles are 90. Example 1: Given: Rectangle AGPZ a. GT = 6. Find AZ. b. mAGP = 34. Find mGZA. c. AT = 3x + 13 and TG = 5x - 21. Find GP. A T Z G P
Example 2: Given the rectangle M N Q P a. If QP = 9 and NP = 6, find NQ and MP. b. If MQ = x , MP = 51 and QP = 2 x, find x and the length of QP. Example 3: Given : Rectangle WXYZ with diagonals WY and XZ. Prove: m 1 m 2 W X V 1 2 Z Y Statements Reasons
- Rectangle WXYZ with diagonals WY and XZ 1. Given
2. 2 The diagonals of a rectangle are
3. WZ^ XY 3. Opposite sides of a rectangle are
4. ZY ZY 4.
5. XZY WYZ 5.
Example 6 : AGZP is a square with GT= 12. Find AZ. Rhombus Definition: A rhombus is a parallelogram with two congruent adjacent sides. Corollary 4.3.4: All sides of a rhombus are congruent. Theorem 4.3.5: The diagonals of a rhombus are perpendicular. Squares and Rhombi A square is a quadrilateral with 4 right angles and 4 congruent sides. A rhombus is also a quadrilateral, but its characterized by 4 congruent sides; a rhombus does NOT have four congruent angles. The properties of a parallelogram apply to both squares and rhombi. A rhombus however has two special properties:
- The diagonals of a rhombus are perpendicular (they form right angles)
- Each diagonal of a rhombus bisects a pair of opposite angles (the angles are cut in half). A G T P Z
Example 7 : Given a rhombus ABCD A D B a. If DC = 6.3 , find the perimeter of ABCD. C b. If DB = 8 and AC = 6, find DC. Example 8 : ABCD is a rhombus. mADB = 27. Find the mADC. Example 9: FISH is a rhombus with FI= 6x + 2 and SI = 8x - 4. Find FH. D A B C F S I H
