Math 9: Quadrilateral PDF

MATH 9: QUADRILATERAL Objectives: determines the conditions that make a quadrilateral a parallelogram uses properties to find measures of angles, sides and other quantities involving parallelograms proves theorems on the different kinds of parallelogram (rectangle, rhombus, square). Vo...

MATH 9: QUADRILATERAL Objectives: determines the conditions that make a quadrilateral a parallelogram uses properties to find measures of angles, sides and other quantities involving parallelograms proves theorems on the different kinds of parallelogram (rectangle, rhombus, square). Vocabulary Bank A quadrilateral is a closed plane figure consisting of four line segments or sides and four angles. A quadrilateral is said to be convex if the diagonals intersect. If the diagonals do not intersect, it is considered a non-convex quadrilateral. Recall: Determine if the following lines are parallel or not. Figures a c, and e are examples of parallel lines. Figures b and d are called intersecting lines. NAME MY PARTS! Directions: Write the appropriate term found in the box below for each of the following. Use quadrilateral LOVE. Sides Vertices Angles Opposite Angles Consecutive Sides Diagonals Consecutive Angles Opposite Sides ____________1. ∠L and ∠E ____________2. VO and EV ____________3. L and V ____________4. EL and VO ____________5. ∠L and ∠O ____________6. point L, point O, point V and point E ____________7. EO ____________8. ∠O and ∠E ____________9. VL ____________10. LO and EV Kinds of Quadrilaterals The figure is a family tree of quadrilaterals. LESSON 1: PARALLELOGRAM  What is a parallelogram? What are the properties of a parallelogram?  Parallelogram is a quadrilateral with both pairs of opposite sides are parallel. Properties of a Parallelogram Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.  A quadrilateral is a parallelogram if and only if it satisfies the following conditions: a. The pairs of opposite sides in a parallelogram are congruent, which means: RU ≅ BY and RB ≅ UY A quadrilateral is called a parallelogram if it contains two sides that are opposite of equal length, which must also be parallel. If RU ≅ BY then RU || BY 𝑜𝑟 If RB ≅ UY then RB || UY  A quadrilateral is a parallelogram if and only if it satisfies the following conditions: b. The two sets of opposing angles are congruent, thus: ∠R ≅ ∠ Y and ∠B ≅ ∠U A quadrilateral must be a parallelogram if it contains two pairs of opposite angles of the same measure.  A quadrilateral is a parallelogram if and only if it satisfies the following conditions: c. Any two angles that share the same side in a parallelogram are supplementary, thus: 𝑚∠R + 𝑚∠U = 180 𝑚∠Y + 𝑚∠U = 180 𝑚∠Y + 𝑚∠B = 180 𝑚∠B + 𝑚∠R = 180 If a quadrilateral has two adjacent angles that are supplementary (add up to 180 degrees), then it must be a parallelogram.  A quadrilateral is a parallelogram if and only if it satisfies the following conditions: d. The two diagonals bisect each other, thus: RY 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 BU A quadrilateral must be a parallelogram if its diagonals cross at their midpoints.  A quadrilateral is a parallelogram if and only if it satisfies the following conditions: e. Another characteristic is that a diagonal splits a parallelogram into two matching triangles. ∆𝑅YU ≅ ∆YRB  Special Parallelograms Classification Figure Characteristic Rhombus A parallelogram with four congruent sides A parallelogram with four right Rectangle angles A parallelogram with four congruent Square sides and four congruent angles Let's list the conditions that ensure a quadrilateral is a parallelogram. THEOREMS: Equal opposite sides are a sufficient for a quadrilateral to be a parallelogram A single pair of congruent and parallel sides is sufficient to classify a quadrilateral as a parallelogram. Diagonals that bisect each other are sufficient to identify a quadrilateral as a parallelogram. A quadrilateral with equal opposite angles is always a parallelogram. A quadrilateral with any two consecutive angles adding up to 180 degrees is a parallelogram. Example 1: Given RUBY is a parallelogram, find a. 𝑚∠U b. 𝑚∠B c. 𝑚∠Y Solution: a. Consecutive angles are supplementary. 𝑚∠R + 𝑚∠U = 180°, 115° + 𝑚∠U = 180°, 𝑚∠U = 180° − 115°, 𝑚∠U = 65° b. Opposite angles are congruent. 𝑚∠R = 𝑚∠B, 115° = 𝑚∠B 𝑚∠B = 115° c. Consecutive angles are supplementary. 𝑚∠R + 𝑚∠Y = 180°, 115° + 𝑚∠Y = 180° 𝑚∠Y = 180° − 115° 𝑚∠Y = 65 Example 2: ROAD is a parallelogram, RM =2y-1, MA = y+2, MO = 3x-4 and DM = 2x+1. Evaluate the expression x and y. Solution: Diagonals of the parallelogram bisect each other. RM = MA MO = DM 2y -1 = y + 2 3x – 4 = 2x +1 2y – y – 1 = 2 3x -2x -4 = 1 y=2+1 x=1+4 y=3 x=5 In this section, you will prove the theorems on Square, Rectangle, and Rhombus. THEOREM 1. If a parallelogram contains a right angle, then it is a rectangle with four right angles. Given: ARCK is a rectangle Prove: CA ≅ KR After confirming that a rectangle's diagonals are congruent, let's attempt to use this theorem in the following instances: Example: Given rectangle RUBY, with |RB| = 2x + 28 cm and |UY| = 4(5x - 2) cm. a. Evaluate the expression of x. b. What are the lengths of line segments RB and UY? Solution: a.|RB| = |UY| b. |RB| = 2x + 28 c. |UY| = |RB|, 2x +28 = 4(5x-2) |RB| = 2(2) +28 |UY| = 𝟑𝟐 cm 2x +28 = 20x -8 |RB| = 4 +28 28+8 = 20x-2x |RB| = 32 cm 36 = 18x x=2 THEOREM 2. In a rhombus, the diagonals are perpendicular and bisect each other. Given: Rhombus RUBY Prove: RB ⊥ UY  THEOREM 3. Every diagonals in a rhombus bisects opposite angles. Given: Rhombus: VORE Prove: ∠5 ≅ ∠6 ∠7 ≅ ∠8 Example PARK is a rhombus m∠𝐴𝐾𝑅 = 36°, Find the following: a. m∠P𝐾𝐴 b. m∠P𝐴𝑅 c. m∠𝐾P𝐴 Solution: a. m∠P𝐾𝐴 = 36 By theorem 3 each diagonal bisects opposite angles m∠𝐴𝐾𝑅 = 𝑚 ∠P𝐾𝐴. Thus, m∠P𝐾𝐴 = 36°  Example PARK is a rhombus m∠𝐴𝐾𝑅 = 36°, Find the following: a. m∠P𝐾𝐴 b. m∠P𝐴𝑅 c. m∠𝐾P𝐴 b. m ∠P𝐴𝑅 Solution: In a parallelogram, opposite angles are m∠P𝐴𝑅 = 𝑚∠P𝐾𝑅 congruent m ∠P𝐾𝑅 = m∠P𝐾𝐴 + m ∠𝐴𝐾𝑅 Angle Addition postulate m∠P𝐾𝑅 = 360 + 360 Substitution m∠P𝐾𝑅 = 720 Since m∠PKR and m∠P𝐴𝑅 are congruent, therefore m∠P𝐴𝑅 = 72°  Example PARK is a rhombus m∠𝐴𝐾𝑅 = 36°, Find the following: a. m∠P𝐾𝐴 b. m∠P𝐴𝑅 c. m∠𝐾P𝐴 c. m ∠𝐾P𝐴 Parallelogram consecutive angles are Solution: supplementary. 𝑚∠P𝐾𝑅 + 𝑚∠𝐾P𝐴 =180° 72° + m∠𝐾P𝐴 =180° Substitution (-72°) + m ∠𝐾P𝐴 = 180° +( -72°) APE m∠𝐾P𝐴 = 108° Thus, the measure of m∠𝐾P𝐴 = 108° THEOREM 4. The diagonals of a square are of equal length and intersect at right angles. Given: Square RUBY Prove: RY= BU Example: 1.Square ABCD with diagonals AC = y + 8 and DB = 3y -12. Find BD. THEOREM 5. A square’s diagonals are perpendicular.  Seatwork A. Fill in the blanks with the appropriate term/s to make each condition of parallelogram true. 1. A quadrilateral with equal opposite sides is always a ________________. 2. A quadrilateral with __________________ and parallel sides is a parallelogram 3. A quadrilateral with diagonals that ___________ each other is a parallelogram. 4. A quadrilateral with ___________________ opposite angles is always a parallelogram. 5. A quadrilateral with any two consecutive angles adding up to _________ degrees is a parallelogram.