HIGH SCHOOL Geometry Soowook Lee
Chapter 4 Quadrilaterals This chapter will cover basic quadrilaterals, including parallelograms, trapezoids, rhombi, rectangles, squares, kites, and cyclic quadrilaterals.
Section 1 Parallelograms (properties) February 11, 2012 Parallelogram WE WILL DISCUSS... A quadrilateral with both pairs of opposite sides parallel. 1. Definition 2. Discovery of Properties Figure 1. Various Parallelograms A parallelogram is a quadrilateral with 3. Summary A B 4. Related Problems opposite sides parallel. It is the "parent" of some other quadrilaterals, which are obtained by adding restrictions of various D C kinds Parallelogram ABCD Exercise 1. Let A(-2, 3), B(5, 3), C(0, -1), and D(a, -1) be vertices of a parallelogram on a cartesian plane. Find the value of a. Video solution to Ex. 1 2
Exercise 2. In parallelogram ABCD, diagonal AC is drawn. Prove that Triangle ABC is congruent to Triangle ADC. Use the definition of a parallelogram. Knowing the triangles are congruent form Ex. 2, what can you observe about corresponding parts of the triangles from the parallelogram? List all you observe. Figure 2. Parallelogram ABCD with diagonal AC Figure 3. Properties of a Parallelogram (part 1) A B A B D C D C Swipe to see the next step A diagonal forms two congruent triangles in a parallelogram. Video solution to Ex. 2 So far, we have... 1. Opposite sides are parallel. (Definition) 2. Opposite sides are congruent. 3. Opposite angles are congruent. 4. A diagonal divides the parallelogram into two congruent triangles. 3
Exercise 3. In parallelogram ABCD, diagonal AC and BD are drawn, intersecting at E. Using the previously discussed properties of a parallelogram, show that Triangle ABE is congruent to Triangle CDE. Reflecting on the congruent triangles from Ex. 3, what can you conclude about diagonals in parallelogram? List a couple of things you observe. Figure 4. Parallelogram ABCD with diagonals AC and BD Figure 5. Properties of Parallelogram (part 2) A B A B E E D C D C Swipe to see the next steps By congruent corresponding parts in congruent triangles, we see that diagonals are bisected. Video solution to Ex. 3 More properties, 5. Diagonals bisect each other. 6. Diagonals divides a parallelogram into four triangles with equal area. *(See Prob. 2) 7. Consecutive interior angles are supplementary. *(See Prob. 1) 4
Problems 1. Prove that consecutive interior angles in a parallelogram are supplementary. (hint: Definition of a parallelogram) 4. In the diagram below of parallelogram MATH,,, and. A T What is the perimeter of the parallelogram? 2. In parallelogram ABCD, diagonals AC and BD are drawn, intersecting at E. Show that the diagonals create four triangles with equal areas. M a. 1 b. 3 c. 8 d. 16 H 3. In the diagram below of parallelogram ABCD with diagonals AC and BD,, and. What is the measure of A B? E 1 2 D C 5. Which statement is not true about every parallelogram? a. Two pairs of opposite sides are congruent. b. Two pairs of opposite angles are congruent. c. Consecutive interior angles are congruent. d. The diagonals bisect each other. a. 15 o b. 20 o c. 25 o d. 70 o 5
6. In parallelogram JETS, JE =10 and ET = 8. What is a possible length of diagonal JT? a. 2 b. 12 c. 18 d. 20 8. Parallelogram QWER has coordinates Q (2, 2), W (-1, 6), E (1, 9), and R (4, 5). Prove that diagonals of QWER bisect each other. 7. In parallelogram GEOM with G (2, 0), E (8, 0), and O (6, 4), A, B, C, and D are midpoints of sides GE, EO, OM, and MG, respectively. Show that ABCD is also a parallelogram. Then, find the ratio of areas between parallelograms GEOM and ABCD. 6
Section 2 Parallelogram (proofs) WE WILL DISCUSS... 1. How we can show that a quadrilateral is a parallelogram, using the definition and properties. 2. Methods of proving a quadrilateral is a parallelogram. 3. Related Problems How do we prove that a quadrilateral is a parallelogram? We can show this by satisfying the definition of a parallelogram. As we have discussed in the previous section, a parallelogram is a quadrilateral when two pairs of opposite sides are parallel. Exercise 1. Quadrilateral ABCD has two pairs of congruent opposite sides, that is AB = CD and AD = BC. Show that ABCD is a parallelogram. Figure 1. Quadrilateral ABCD A D B C Swipe for a hint 7
As we can see from the previous exercise, a property of a parallelogram can be used to prove that a quadrilateral is a parallelogram. Let us examine other properties to see if they are enough to show that a quadrilateral is a parallelogram. Exercise 2. Quadrilateral EFGH has two pairs of congruent opposite interior angles. Show that EFGH is a parallelogram. So far, we have seen that properties of a parallelogram are useful tool to show that a quadrilateral is a parallelogram. The next exercise will use another property of a parallelogram. Exercise 3. Quadrilateral IJKL has diagonals bisecting each other. Show that IJKL is a parallelogram. Figure 3. Quadrilateral with bisecting diagonals Figure 2. Quadrilateral with congruent opposite angles E F I J H G L K swipe for a hint swipe for a hint 8
So far, we know that a quadrilateral is a parallelogram if... 1. two pairs of opposite sides are parallel. 2. two pairs of opposite sides are congruent. 3. two pairs of opposite angles are congruent. 4. two diagonals bisect each other. 5. one pair of opposite sides are congruent and parallel. (See prob. 1) 9
Problems 1. Show that when a quadrilateral has a pair of congruent and parallel sides, it is a parallelogram. 3. Show that why a pair of congruent opposite sides and another pair of parallel opposite sides in a quadrilateral are not sufficient enough to prove that it is a parallelogram. 2. Quadrilateral QWER has coordinates Q (2, 2), W (-1, 6), E (4, 5), and R (1, 9). Show that it is a parallelogram. 4. A parallelogram has coordinates (1, 3), (-2, 6), and (0, -2). Find all possible coordinates for the last vertex. 10
5. Quadrilateral FOUR has coordinates F (1, 6), O (7, 4), U (-3, 0), and R (-7, 2). Show that midpoints of the quadrilateral form a parallelogram. 6. Given: Quadrilateral ABCD, diagonal AFEC,,,, Prove: ABCD is a parallelogram. B 1 E C A F 2 D 11
Section 3 Rhombi (properties) WE WILL DISCUSS... 1. Definition 2. Discovery of Properties 3. Summary 4. Related Problems Rhombus A quadrilateral with all four sides equal in length; an equilateral quadrilateral A rhombus is known as a special parallelogram. Can you show why a quadrilateral with four congruent sides is a parallelogram? Exercise 1. Prove that a quadrilateral with four sides with equal length is a parallelogram. Figure 1. an equilateral quadrilateral B A C D 12
Since a rhombus is a parallelogram, all properties of a parallelogram are valid in a rhombus. Let s continue to discover more properties of a rhombus. Exercise 2. Rhombus ABCD has a diagonal AC. Using the definition of a rhombus, show that triangle ABC and ADC are congruent. What can you observe about the diagonals from having both triangles congruent to each other? See the diagrams below. Figure 3. Property of a rhombus B Figure 2. Rhombus with a diagonal A C B D A C Due to congruent triangles, corresponding angles are congruent. D swipe for a hint So far, we know that a rhombus... 1. has all congruent sides. (Definition) 2. is a parallelogram. Therefore, it has all the properties of a parallelogram. 3. Diagonals bisect vertex angles. 13
Exercise 3. In rhombus JUNE, diagonals JN and UE intersect M. Show that triangle JMU is congruent to triangle NMU. Figure 4. A rhombus with two diagonals U Since the consecutive triangles are congruent, what can you conclude about angle JMU and angle NMU? They are congruent and a linear pair. This implies that the given angles are right angles. Slides 1. Properties of a rhombus J M N E swipe for a hint 14
Problems 1. lengths of diagonals of a rhombus are 10 and 24. What is the perimeter of the rhombus? 3. Two congruent equilateral triangles with side of length of 4 units are overlapped as shown below, creating a rhombus in the middle. Vertex of each triangle is tangent to the midpoint of a side of the other triangle. What is the perimeter of the rhombus? 2. In rhombus ABCD, EC = 6 and measure of angle BCE is 60 degrees. Find the perimeter and the area of the rhombus. A D E B C 4. In the diagram below of rhombus ABCD,. What is? 1) 40 2) 45 3) 50 4) 80 15
5. In rhombus ABCD, the measures in inches of AB is 3x + 2 and BC is x + 12. Find the number of inches in the length of DC. 7. Rhombus FEGH has coordinates of F(3, 2), E(0, 6), and G(-3, 2). Find the coordinates of H. 6. An equilateral triangle and a rhombus with an interior angle of 60 o have the perimeters of 12 inches. What is the ratio of areas between the triangle and the rhombus? 16
Section 4 Rhombi (proofs) WE WILL DISCUSS... 1. How we can show that a quadrilateral is a rhombus, using the definition and properties. 2. Methods of proving a quadrilateral is a rhombus. 3. Related Problems How do we know if a quadrilateral is a rhombus? Once again, we can justify it by definition of a rhombus; a quadrilateral with four congruent sides. We will discuss a few cases, where properties of a rhombus can be used to prove a quadrilateral is a rhombus. Exercise 1. In parallelogram ABCD, AB is congruent to BC. Show that ABCD is a rhombus. Figure 1. Parallelogram ABCD A parallelogram with congruent consecutive sides 17
As we can see from Ex. 1, when a parallelogram has congruent consecutive sides, it is a rhombus. Exercise 2. In quadrilateral JUNE, diagonals JN and UE bisect each other and are perpendicular to each other. Show that JUNE is a rhombus. Exercise 3. In parallelogram JULY, diagonals JL and UY are bisectors of vertex angles. Show that JULY is a rhombus. Figure 3. A parallelogram with angle bisecting diagonals Figure 2. A quadrilateral with perpendicular bisecting diagonals From Ex. 2, we see that when diagonals are perpendicular bisectors of each other, the quadrilateral is a rhombus. Let us look at another property of a rhombus. From Ex. 3, we see that a parallelogram with angle bisecting diagonals is a rhombus. So, in general, we can prove that a quadrilateral is a rhombus with a property of a rhombus in conjunction with a property of a parallelogram. 18
In summary, Methods of proving a quadrilateral is a rhombus Problems 1. In parallelogram PEOR, PE and OE are extended to V and S respectively. If VO = SP and VE is perpendicular to VO and ES to SP. Prove that PEOR is a rhombus. 2. 19
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Section 5 Rectangles WE WILL DISCUSS... 1. Definition 2. Discovery of Properties 3. Summary 4. Related Problems What is a rectangle? It a a quadrilateral with four right angles. Since all angles are equal in measures, it can be also called an equiangular quadrilateral. Exercise 1. Show that a rectangle is a parallelogram. Figure 1. A rectangle is a parallelogram This shows that properties of a 21
parallelogram is valid in a rectangle. What property can you observe from diagonals? Figure 3. Properties of a Rectangle Let us look at the following Exercise. Figure 2. Rectangle with diagonals Exercise 2. Rectangle ABCD has diagonals AC and BD, intersecting at M. Show that triangle ABC and BAD are congruent. 22
Properties of a rectangle 1. All properties from a parallelogram 2. All interior angles are congruent. 3. Diagonals are congruent. Since the triangles are congruent, we can see that diagonals are congruent. This is a property of a rectangle. In summary, 23
Section 6 Rectangles (proofs) WE WILL DISCUSS... 1. How we can show that a quadrilateral is a rectangle, using the definition and properties. 2. Methods of proving a quadrilateral is a rectangle. 3. Related Problems How can we prove that a quadrilateral is a rectangle? We can show that it is a rectangle by satisfying the definition, a quadrilateral with four congruent interior angles. Exercise 1. Parallelogram RITE has a right angle at vertex R. Show that it is a rectangle. Figure 1. A rectangle is a parallelogram. As we can see from this exercise, a parallelogram with a right angle is a rectangle. 24
Exercise 2. A quadrilateral has two congruent diagonals bisecting each other. Show that the quadrilateral is a rectangle. Figure 2. Quadrilateral with congruent bisecting diagonals From Ex. 2, we see that bisecting diagonals imply that the quadrilateral is a parallelogram. And congruent diagonals imply that the parallelogram is a rectangle. 25
Parallel a. (of straight lines) lying in the same plane but never meeting no matter how far extended. b. (of planes) having common perpendiculars. c. (of a single line, plane, etc.) equidistant from another or others (usually followed by to or with ). Related Glossary Terms Drag related terms here Index Find Term
Parallelogram A quadrilateral with both pairs of opposite sides parallel Related Glossary Terms Quadrilateral Index Find Term
Quadrilateral a polygon with four sides Related Glossary Terms Parallelogram Index Find Term