Trapezoid and Kite Theorems - Lesson Notes PDF

Lesson 2: TRAPEZOID AND KITES Trapezoids and kites are quadrilaterals that are not parallelograms.  01 TRAPEZOID Learning Objectives At the end of the lesson, the students are expected to: a. apply the Midline Theorem to solve problems involving triangles b. prove the theorems of the tr...

Lesson 2: TRAPEZOID AND KITES Trapezoids and kites are quadrilaterals that are not parallelograms.  01 TRAPEZOID Learning Objectives At the end of the lesson, the students are expected to: a. apply the Midline Theorem to solve problems involving triangles b. prove the theorems of the trapezoids and kites c. solve problems involving parallelograms, trapezoids and kites Review A. Let us have a recall on quadrilaterals. Given the following statements, tell whether it is true or false. 1. All quadrilaterals have four sides. 2. Squares are a specific type of rectangle. 3. Parallelograms have two sets of parallel sides. 4. Rectangles have four right angles. 5. Squares have four equal sides. B. Write YES if the statement is true. Otherwise, write NO. 1. The sides that are opposite of a parallelogram are equal in length. 2. Angles that are consecutive in a parallelogram are equal in measure. 3. Opposite angles within a parallelogram has a sum of 180 degrees. 4. The diagonals of a parallelogram intersect at their midpoints. 5. A diagonal divides a parallelogram into two identical triangles. TRAPEZOID is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases, while its nonparallel sides are called legs. trapezoi CARE d bases legs base 1 angles base 2 angles ISOSCELES TRAPEZOIDS is a special type of trapezoid with legs of equal lengths. Trapezoid LOVE is an isosceles trapezoid. Isosceles LOVE trapezoid bases legs diagonals This section will further expose you to the concepts of the Midline Theorem to solve problems involving triangles, master the theorems of trapezoids and kites and apply this knowledge to solve geometric problems involving these shapes and parallelograms. Definition: Connecting the midpoints of two sides of a triangle creates its midsegment (also known as the midline). Hence K and M are its midpoints, respectively. Definition: In a trapezoid, the midsegment is the line segment that joins the midpoints of the nonparallel sides. Definition: Connecting the midpoints of a trapezoid non-parallel sides creates the midsegment. Triangle Midsegment Theorem: A triangle's midsegment is half the length of the third side and is parallel to the third side. Apply the previous theorem to demonstrate that “The three midsegments of a triangle divide it into four congruent triangles.” The same argument can be used to prove the congruence of the other two triangles UEV and TUE. You may want to continue the proof Example 2. The Midline Theorem: The segment joining the midpoints of two sides of a triangle (or the midline of a triangle) is parallel to its third side and is half as long. PROOF: Example 3 Find the length of the midline and base of △BYE. Using the Midline Theorem, Check: Theorem: A trapezoid midsegment is parallel to its bases and is equal to half of the bases' total length. EF = ½ ( BC + AD )  Example on Triangle Midsegment Theorem; In ∆ 𝑀𝐶𝐺, A and I are the midpoints of MG and GC, respectively. Consider each given information and answer the questions that follow. Example on Trapezoid Midsegment Theorem In trapezoid CUTE, the midpoints of line segments CU and ET are M and D, respectively. Take into account all of the information provided, then respond to the following questions Example 2 The length of the median of a trapezoid is 18 cm. Find the length of the bases if the lower base measures 16 cm more than thrice the length of the upper base. Let x be the length of the upper base and 3x + 16 be the length of the lower base. Since the length of the median is one-half the sum of the lengths of the bases, Thus, the upper base measures 5 cm while the lower base measures 3x + 16 = 3( 5 ) + 16 = 31 cm. Example 3 A flower garden is in the shape of a trapezoid with bases 4 m and 5 m. A fence will be constructed from the middle points of the two sides to divide the garden into two. What should the length of this fence be? The line joining the middle points of its two sides is the median of the trapezoid; hence, its length is one-half the sum of the lengths of the bases of the trapezoid. Thus, The length of the fence to be constructed is 4.5 m.  Word Problem Examples 1. The horizontal line at the top of the roof measures 5m and the line at its base measures 15 m. The two lines are 8 m away from each other. What is the area of the trapezoidal roof? Solutions: Given : Area trapezoid = ½h(b1+b2) = ½ (8 m)(5m+15 m) base 1(b1) = 5 m = ½ (8 m)(20 m) base (b2) = 15 m = ½ (160 m)2 height (h) = 8 m = 80 m2 2. Given: Quadrilateral LOVE is a special kind of trapezoid called an isosceles trapezoid, where sides LE and OV are parallel, and sides LO and EV are equal. UR is the median of this trapezoid. a. If LE = 3x – 7cm, OV = 2x + 1cm and UR = 12cm, what are the lengths of the two bases? B. If measure of angle O is 2x – 9 and m ∠L = 2x + 5, what is the measure of angle E ? The length of one base of the trapezoid is double the length of the other base, and the median (UR) measures 6 centimeters, find the length of each leg. Theorems On Kite THEOREM: A kite area is calculated as one-half the product of its diagonal lengths. THEOREM: The intersection of the diagonals of a kite forms a 90-degree angle, and one diagonal bisects the other. Fly’s a Kite! Quadrilateral RUBY is a kite. Refer to the figure provided and answer the questions related to its geometric features 1. Given that the length of UY is double the length of UR, and the perimeter of the figure is 42 centimeters, determine the length of UY. 2. Given: RY=15cm; UB = 8cm. What is the area of kite RUBY? (RY)(UB) 2 15(8) 2 Kite is a quadrilateral with distinct pairs of congruent and adjacent sides. THEOREM. Kite Angles Theorem: The non vertex angles of a kite are congruent. Quadrilateral KITE is a kite PROOF:  THEOREM 1. Kite Diagonals Theorem: The diagonals of a kite are perpendicular. THEOREM 2. Kite Diagonal Bisector Theorem: The diagonal connecting the vertices of the vertex angles of a kite is the perpendicular bisector of the other diagonal. THEOREM 3. Kite Angle Bisector Theorem: The vertex angles of a kite are bisected by its diagonal.  Seatwork A. Evaluate the truth value of each statement below. Indicate "TRUE" for correct statements and "FALSE" for incorrect statements. 1. Isosceles trapezoids have congruent diagonals. 2. The parallel sides of an isosceles trapezoid are the same length. 3. Opposite angles in a kite can add up to 180 degrees. 4. The line that connects the midpoints of the non-parallel sides of a trapezoid is called the mean. 5. The area of a kite is calculated by taking twice the product of its diagonal lengths.  SAYING it is ONE THING, but PROVING it is ANOTHER.  Do you have any questions? Thank you for [email protected] | +91 620 421 838 yourcompany.com listening! CREDITS: This presentation template was created by Slidesgo, including icons by Flaticon, infographics & images by Freepik Please keep this slide for the attribution