QUADRILATERALS.pptx
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This document defines and describes types of quadrilaterals including parallelograms, trapezoids, kites, rectangles, rhombuses, and squares. It provides properties and theorems for each shape and examples to solve problems involving midlines, midsegments, isosceles trapezoids, and kites. Seatwork and homework problems are also presented.
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1. A parallelogram is a quadrilateral with two properties: opposite sides are parallel and opposite angles are congruent.
2. There are several types of parallelograms including rectangles, squares, and rhombi.
3. Rectangles have four right angles in addition to the properties of parallelograms. Squares have four congruent sides and four right angles. Rhombi have four congruent sides.
Application of the Properties of Parallelogram
Properties of parallelogram applies to rectangles, rhombi and squares.
In a parallelogram,
Opposite sides of a parallelogram are parallel.
A diagonal of a parallelogram divides it into two congruent triangles.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
If one angle of a parallelogram is right, then all the angles are right.
Consecutive angles of a parallelogram are supplementary.
Diagonals of a parallelogram bisect each other.
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- 2. TYPES OF QUADRILATERALS 1. Parallelogram- quadrilateral with 2 pairs of parallel sides 2.Trapezoid- quadrilateral with 1 pair of parallel sides 3. Kite- quadrilateral with no pair of parallel sides
- 3. PARALLELOGRAMS Quadrilateral with 2 pairs of parallel sides Parallel- lines that do not meet Quadrilateral MATH is a parallelogram M A S H T
- 4. PROPERTIES OF PARALLELOGRAMS SIDES 1. Opposite sides are congruent. ANGLES 2. Opposite angles are congruent. 3. Consecutive angles are supplementary. DIAGONALS 4. Diagonals bisect each other. 5. Diagonal forms two congruent triangles.
- 6. THEOREMS ON RECTANGLE 1. All angles are right angles. 2. Diagonals are congruent. Given: QuadrilateralCARE is a rectangle. C A S E R
- 7. THEOREMS ON RHOMBUS Rhombus- parallelogram whose sides are congruent 1. Diagonals are perpendicular 2. Diagonals bisect opposite angles. Given: Quadrilateral HEAR is a rhombus. H E T R A
- 8. SQUARE -Sides are congruent, therefore it is a rhombus. -A type of parallelogram whose diagonals are perpendicular and congruent.
- 9. WHAT MAKES THE SQUARE EXTRA SPECIAL?
- 11. SEATWORK: I. Name all quadrilaterals using the legend below with the given properties: P- parallelogram Re- rectangle Rh- rhombus and S- square 1. Diagonals are congruent. 2. Diagonals bisect each other. 3. Diagonals are perpendicular. 4. One pair of opposite sides are parallel. 5. All sides are congruent.
- 12. SEATWORK: II.Write always true, sometimes true or never true. 1. A square is a rectangle. 2. A rhombus is a square. 3. A parallelogram is a square. 4. A rectangle is a rhombus. 5. A quadrilateral is a parallelogram.
- 13. HOMEWORK: 1 2 3
- 14. MIDLINE A line joining two midpoints of the sides of a triangle. Parallel to the base Half the measure of the base ML= 𝐵 2
- 15. EXAMPLE In ∆MGC, A and I are midpoints of MG and CG, respectively.Answer the following: 1. Given:AI= 10.5, find MC. 2. Given: CG= 32, find GI. 3. Given:AG= 7, CI=8, find MG+CG. 4. Given: AI= 3x-2 and MC= 9x-13, 1. Solve for x 2. Find AI 3. Find MC M C I G A
- 16. SEATWORK In ∆MGC, A and I are midpoints of MG and CG, respectively.Answer the following: Given: MG≅CG, AG= 2y-1, IC= y+5 1. Solve for y 2. Find AG 3. Find CG M C I G A
- 17. MIDSEGMENT A line joining two midpoints of the sides of a trapezoid Also known as median Parallel to the bases Half the measure of the bases MS= 𝑏1+𝑏2 2
- 18. EXAMPLE 1. If IJ= x, HG= 8 and EF= 12, find x. 2.If HG= x, IJ= 16 and EF= 22, find x. H F J G I E
- 19. SEATWORK If HG= 3y-2, EF= 2y+4 and IJ= 8.5, find: a. Value of y b. HG and EF H F J G I E
- 20. ISOSCELESTRAPEZOID 1.Base angles are congruent 2.Opposite angles are supplementary 3.Diagonals are congruent
- 21. EXAMPLE: Consider quadrilateral MATH is a trapezoid, answer the following questions: 1. If ∠𝐻𝑀𝐴= 115°, find ∠𝑇𝐴M. 2. If ∠𝑀𝐻𝑇 = 3𝑥 + 10 𝑎𝑛𝑑 ∠𝑀𝐴𝑇 = 2𝑥 − 5, find the measure of the two angles. M A T H
- 22. SEATWORK: Using the same given as the example, if AH= 4y-3 and MT= 2y+5, find: a.The property used to solve the problem b.Value of y c. Length of AH
- 23. KITE A quadrilateral whose pair of adjacent sides are equal Properties: 1. Diagonals are perpendicular 2. Area of a kite is half the product of its diagonals
- 24. EXAMPLE Quadrilateral PLAY is a kite. P 1. PA= 12, LY= 6. Find the area. L 2. Area= 135, LY= 9. Find PA. S 3. If PL= 6, find PY. Y 4. Find m∠𝑃𝑆𝐿. 5. If m∠LAS = 30° , find m∠𝐴𝐿𝑆. A
- 25. QUIZ: A quadrilateral POST is an isosceles trapezoid with OS ∥ PT. ER is a median. 1-2. Draw the figure and label its parts. 3-5. If OS= 3x-2, PT= 2x+10 and ER= 14, how long is OS and PT? 6-7. If angle P= 2x+5 and angle O= 3x-10, what is angleT? A quadrilateral LIKE is a kite where LI≅IK and LE ≅KE. 8-9. Draw the figure and label its parts. 10-11. If LE is twice LI and the perimeter is 21, find LE. 12-15. If IE= x-1 and LK= x+2, how long is IE and LK if the area is 44?