Parallelogram

Total Page：16

File Type：pdf, Size：1020Kb

Prove: Parallelogram 1. If both pairs of opposite sides of a quadrilateral are parallel, then it’s a ||-ogram + ⇒ ||-ogram 2. If both pairs of opposite sides of a quadrilateral are congruent, then it’s a ||-ogram + ⇒ ||-ogram 3. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then it’s a ||-ogram. + ⇒ ||-ogram (Note: + ⇒ ||-ogram) 4. If the diagonals of a quadrilateral bisect each other, then it’s a ||-ogram. + ⇒ ||-ogram 5. If both pairs of opposite angles of a quadrilateral are congruent, then it’s a ||-ogram + ⇒ ||-ogram Prove: Rectangle Prove: Rhombus 1. 1. Rectangle + Rhombus ⇒ ⇒ If all four ∠‘s of a quadrilateral are right ∠‘s, then it is a rectanlge If the diaqonals of a quad are ⊥ bisectors of each other, then it is a rhombus 2. 2 . ||-gram Rhombus parallelogram + ⇒ Rectangle + ⇒ If a parallelogram has at least one right ∠, then it is a rectanlge If a parallelogram has a pair of consecutive sides congruent, then it is a rhombus ||-gram + Rhombus 3. parallelogram + ⇒ Rectangle 3. ⇒ If a parallelogram has a diagonal that bisects two of its angles, then it is a rhombus If a parallelogram has congruent diagonals, then it is a rectanlge 4. Rhombus ⇒ Prove: Square If a quadrilateral has 4 congruent sides, then it is a rhombus 5. Rectangle + Rhombus Square ||-gram + ⇒ Rhombus ⇒ If a quadrilateral is both a rectangle and a rhombus, then it is a square If a parallelogram has perpendicular diagonals,, then it is a rhombus Prove: Kite Prove: Isosceles Trapezoid 1. Kite 1 . + Trapezoid + Isosceles ⇒ Trapezoid ⇒ If a trapezoid has congruent non-parallel sides, then it is an isosceles trapezoid If a quadrilateral has two disjoint pairs of consecutive sides congruent, then it is a kite 2 . Isosceles Trapezoid or + Trapezoid 2 . Kite ⇒ + If a trapezoid has congruent upper or lower base angles, then it is an isosceles trapezoid ⇒ 3 . Isosceles Trapezoid + If a quadrilateral has one diagonal that is the ⊥ bisector of the other, then it is a kite ⇒ Trapezoid If a trapezoid has congruent diagonals, then it is an isosceles trapezoid.

Copy Link

Recommended publications

Quadrilateral Theorems
Quadrilateral Theorems Properties of Quadrilaterals: If a quadrilateral is a TRAPEZOID then, 1. at least one pair of opposite sides are parallel(bases) If a quadrilateral is an ISOSCELES TRAPEZOID then, 1. At least one pair of opposite sides are parallel (bases) 2. the non-parallel sides are congruent 3. both pairs of base angles are congruent 4. diagonals are congruent If a quadrilateral is a PARALLELOGRAM then, 1. opposite sides are congruent 2. opposite sides are parallel 3. opposite angles are congruent 4. consecutive angles are supplementary 5. the diagonals bisect each other If a quadrilateral is a RECTANGLE then, 1. All properties of Parallelogram PLUS 2. All the angles are right angles 3. The diagonals are congruent If a quadrilateral is a RHOMBUS then, 1. All properties of Parallelogram PLUS 2. the diagonals bisect the vertices 3. the diagonals are perpendicular to each other 4. all four sides are congruent If a quadrilateral is a SQUARE then, 1. All properties of Parallelogram PLUS 2. All properties of Rhombus PLUS 3. All properties of Rectangle Proving a Trapezoid: If a QUADRILATERAL has at least one pair of parallel sides, then it is a trapezoid. Proving an Isosceles Trapezoid: 1st prove it’s a TRAPEZOID If a TRAPEZOID has ____(insert choice from below) ______then it is an isosceles trapezoid. 1. congruent non-parallel sides 2. congruent diagonals 3. congruent base angles Proving a Parallelogram: If a quadrilateral has ____(insert choice from below) ______then it is a parallelogram. 1. both pairs of opposite sides parallel 2. both pairs of opposite sides ≅ 3.
[Show full text]
Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022)
9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep FlexiPrep Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem (For CBSE, ICSE, IAS, NET, NRA 2022) Get unlimited access to the best preparation resource for competitive exams : get questions, notes, tests, video lectures and more- for all subjects of your exam. A quadrilateral is a 4-sided polygon bounded by 4 finite line segments. The word ‘quadrilateral’ is composed of two Latin words, Quadric meaning ‘four’ and latus meaning ‘side’ . It is a two-dimensional figure having four sides (or edges) and four vertices. A circle is the locus of all points in a plane which are equidistant from a fixed point. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form vertices of a cyclic quadrilateral. It can be visualized as a quadrilateral which is inscribed in a circle, i.e.. all four vertices of the quadrilateral lie on the circumference of the circle. What is a Cyclic Quadrilateral? In the figure given below, the quadrilateral ABCD is cyclic. ©FlexiPrep. Report ©violations ＠https://tips.fbi.gov/ 1 of 5 9/22/2021 Cyclic Quadrilateral: Cyclic Quadrilateral Theorem and Properties of Cyclic Quadrilateral Theorem- FlexiPrep Let us do an activity. Take a circle and choose any 4 points on the circumference of the circle. Join these points to form a quadrilateral. Now measure the angles formed at the vertices of the cyclic quadrilateral.
[Show full text]
Properties of Equidiagonal Quadrilaterals (2014)
Forum Geometricorum Volume 14 (2014) 129–144. FORUM GEOM ISSN 1534-1178 Properties of Equidiagonal Quadrilaterals Martin Josefsson Abstract. We prove eight necessary and sufficient conditions for a convex quadri- lateral to have congruent diagonals, and one dual connection between equidiag- onal and orthodiagonal quadrilaterals. Quadrilaterals with both congruent and perpendicular diagonals are also discussed, including a proposal for what they may be called and how to calculate their area in several ways. Finally we derive a cubic equation for calculating the lengths of the congruent diagonals. 1. Introduction One class of quadrilaterals that have received little interest in the geometrical literature are the equidiagonal quadrilaterals. They are defined to be quadrilat- erals with congruent diagonals. Three well known special cases of them are the isosceles trapezoid, the rectangle and the square, but there are other as well. Fur- thermore, there exists many equidiagonal quadrilaterals that besides congruent di- agonals have no special properties. Take any convex quadrilateral ABCD and move the vertex D along the line BD into a position D such that AC = BD. Then ABCD is an equidiagonal quadrilateral (see Figure 1). C D D A B Figure 1. An equidiagonal quadrilateral ABCD Before we begin to study equidiagonal quadrilaterals, let us define our notations. In a convex quadrilateral ABCD, the sides are labeled a = AB, b = BC, c = CD and d = DA, and the diagonals are p = AC and q = BD. We use θ for the angle between the diagonals. The line segments connecting the midpoints of opposite sides of a quadrilateral are called the bimedians and are denoted m and n, where m connects the midpoints of the sides a and c.
[Show full text]
Midsegment of a Trapezoid Parallelogram
Vocabulary Flash Cards base angles of a trapezoid bases of a trapezoid Chapter 7 (p. 402) Chapter 7 (p. 402) diagonal equiangular polygon Chapter 7 (p. 364) Chapter 7 (p. 365) equilateral polygon isosceles trapezoid Chapter 7 (p. 365) Chapter 7 (p. 402) kite legs of a trapezoid Chapter 7 (p. 405) Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards The parallel sides of a trapezoid Either pair of consecutive angles whose common side is a base of a trapezoid A polygon in which all angles are congruent A segment that joins two nonconsecutive vertices of a polygon A trapezoid with congruent legs A polygon in which all sides are congruent The nonparallel sides of a trapezoid A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards midsegment of a trapezoid parallelogram Chapter 7 (p. 404) Chapter 7 (p. 372) rectangle regular polygon Chapter 7 (p. 392) Chapter 7 (p. 365) rhombus square Chapter 7 (p. 392) Chapter 7 (p. 392) trapezoid Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards A quadrilateral with both pairs of opposite sides The segment that connects the midpoints of the parallel legs of a trapezoid PQRS A convex polygon that is both equilateral and A parallelogram with four right angles equiangular A parallelogram with four congruent sides and four A parallelogram with four congruent sides right angles A quadrilateral with exactly one pair of parallel sides Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved.
[Show full text]
Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related.
[Show full text]
Cyclic Quadrilaterals
GI_PAGES19-42 3/13/03 7:02 PM Page 1 Cyclic Quadrilaterals Definition: Cyclic quadrilateral—a quadrilateral inscribed in a circle (Figure 1). Construct and Investigate: 1. Construct a circle on the Voyage™ 200 with Cabri screen, and label its center O. Using the Polygon tool, construct quadrilateral ABCD where A, B, C, and D are on circle O. By the definition given Figure 1 above, ABCD is a cyclic quadrilateral (Figure 1). Cyclic quadrilaterals have many interesting and surprising properties. Use the Voyage 200 with Cabri tools to investigate the properties of cyclic quadrilateral ABCD. See whether you can discover several relationships that appear to be true regardless of the size of the circle or the location of A, B, C, and D on the circle. 2. Measure the lengths of the sides and diagonals of quadrilateral ABCD. See whether you can discover a relationship that is always true of these six measurements for all cyclic quadrilaterals. This relationship has been known for 1800 years and is called Ptolemy’s Theorem after Alexandrian mathematician Claudius Ptolemaeus (A.D. 85 to 165). 3. Determine which quadrilaterals from the quadrilateral hierarchy can be cyclic quadrilaterals (Figure 2). 4. Over 1300 years ago, the Hindu mathematician Brahmagupta discovered that the area of a cyclic Figure 2 quadrilateral can be determined by the formula: A = (s – a)(s – b)(s – c)(s – d) where a, b, c, and d are the lengths of the sides of the a + b + c + d quadrilateral and s is the semiperimeter given by s = 2 . Using cyclic quadrilaterals, verify these relationships.
[Show full text]
2 Dimensional Figures
Study Island Copyright © 2021 Edmentum - All rights reserved. Generation Date: 07/26/2021 Generated By: Jennifer Fleming 1. An equilateral triangle is always an example of a/an: I. isosceles triangle II. scalene triangle III. right triangle A. I only B. II and III C. I and II D. II only 2. Which of the following is true about a parallelogram? Parallelograms always have four congruent sides. A. The diagonals of a parallelogram always bisect each other. B. Only two sides of a parallelogram are parallel. C. Opposite angles of a parallelogram are not congruent. D. 3. What is the difference between a rectangle and a rhombus? A rectangle has opposite sides parallel and a rhombus has no sides parallel. A. A rectangle has four right angles and a rhombus has opposite angles of equal measure. B. A rhombus has four right angles and a rectangle has opposite angles of equal measure. C. A rectangle has opposite sides of equal measure and a rhombus has no sides of equal measure. D. 4. What is a property of all rectangles? The four sides of a rectangle have equal length. A. The opposite sides of a rectangle are parallel. B. The diagonals of a rectangle do not have equal length. C. A rectangle only has two right angles. D. 5. An obtuse triangle is sometimes an example of a/an: I. scalene triangle II. isosceles triangle III. equilateral triangle IV. right triangle A. I or II B. I, II, or III C. III or IV D. II or III 6. What is the main difference between squares and rhombuses? Squares always have four interior angles which each measure 90°; rhombuses do not.
[Show full text]
Cyclic and Bicentric Quadrilaterals G
Cyclic and Bicentric Quadrilaterals G. T. Springer Email: [email protected] Hewlett-Packard Calculators and Educational Software Abstract. In this hands-on workshop, participants will use the HP Prime graphing calculator and its dynamic geometry app to explore some of the many properties of cyclic and bicentric quadrilaterals. The workshop will start with a brief introduction to the HP Prime and an overview of its features to get novice participants oriented. Participants will then use ready-to-hand constructions of cyclic and bicentric quadrilaterals to explore. Part 1: Cyclic Quadrilaterals The instructor will send you an HP Prime app called CyclicQuad for this part of the activity. A cyclic quadrilateral is a convex quadrilateral that has a circumscribed circle. 1. Press ! to open the App Library and select the CyclicQuad app. The construction consists DEGH, a cyclic quadrilateral circumscribed by circle A. 2. Tap and drag any of the points D, E, G, or H to change the quadrilateral. Which of the following can DEGH never be? • Square • Rhombus (non-square) • Rectangle (non-square) • Parallelogram (non-rhombus) • Isosceles trapezoid • Kite Just move the points of the quadrilateral around enough to convince yourself for each one. Notice HDE and HE are both inscribed angles that subtend the entirety of the circle; ≮ ≮ likewise with DHG and DEG. This leads us to a defining characteristic of cyclic ≮ ≮ quadrilaterals. Make a conjecture. A quadrilateral is cyclic if and only if… 3. Make DEGH into a kite, similar to that shown to the right. Tap segment HE and press E to select it. Now use U and D to move the diagonal vertically.
[Show full text]
Quadrilateral Geometry
Quadrilateral Geometry MA 341 – Topics in Geometry Lecture 19 Varignon’s Theorem I The quadrilateral formed by joining the midpoints of consecutive sides of any quadrilateral is a parallelogram. PQRS is a parallelogram. 12-Oct-2011 MA 341 2 Proof Q B PQ || BD RS || BD A PQ || RS. PR D S C 12-Oct-2011 MA 341 3 Proof Q B QR || AC PS || AC A QR || PS. PR D S C 12-Oct-2011 MA 341 4 Proof Q B PQRS is a parallelogram. A PR D S C 12-Oct-2011 MA 341 5 Starting with any quadrilateral gives us a parallelogram What type of quadrilateral will give us a square? a rhombus? a rectangle? 12-Oct-2011 MA 341 6 Varignon’s Corollary: Rectangle The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are perpendicular is a rectangle. PQRS is a parallelogram Each side is parallel to one of the diagonals Diagonals perpendicular sides of paralle logram are perpendicular prlllparallelog rmram is a rrctectan gl e. 12-Oct-2011 MA 341 7 Varignon’s Corollary: Rhombus The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are congruent is a rhombus. PQRS is a parallelogram Each side is half of one of the diagonals Diagonals congruent sides of parallelogram are congruent 12-Oct-2011 MA 341 8 Varignon’s Corollary: Square The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are congruent and perpendicular is a square. 12-Oct-2011 MA 341 9 Quadrilateral Centers Each quadrilateral gives rise to 4 triangles using the diagonals.
[Show full text]
Rhombus Rectangle Square Trapezoid Kite NOTES
Geometry Notes G.9 Rhombus, Rectangle, Square, Trapezoid, Kite Mrs. Grieser Name: _________________________________________ Date: _________________ Block: _______ Rhombuses, Rectangles, Squares The Venn diagram below describes the relationship between different kinds of parallelograms: A rhombus is a parallelogram with four congruent sides A rectangle is a parallelogram with four right angles A square is a parallelogram with four congruent sides and four right angles Corollaries: A quadrilateral is a rhombus IFF it has four congruent sides A quadrilateral is a rectangles IFF it has four right angles A quadrilateral is a square IFF it is a rhombus and a rectangle. Since rhombuses, squares, and rectangles are parallelograms, they have all the properties of parallelograms (opposite sides parallel, opposite angles congruent, diagonals bisect each other, etc.) In addition… Rhombus Rectangle Square 4 congruent sides 4 right angles 4 congruent sides diagonals bisect angles diagonals congruent diagonals bisect each other diagonals perpendicular diagonals perpendicular 4 right angles diagonals congruent Theorems: A parallelogram is a rhombus IFF its diagonals are perpendicular. A parallelogram is a rhombus IFF each diagonal bisects a pair of opposite angles. A parallelogram is a rectangle IFF its diagonals are congruent. Examples: 1) Given rhombus DEFG, are the statements 2) Classify the parallelogram and find missing sometimes, always, or never true: values: a) b) a) D F b) D E c) DG GF 3) Given rhombus WXYZ 4) Given rectangle PQRS and and mXZY 34, find: mRPS 62 and QS=18, a) mWZV b) WY c) XY find: a) mQPR b) mPTQ c) ST Geometry Notes G.9 Rhombus, Rectangle, Square, Trapezoid, Kite Mrs.
[Show full text]
Two-Dimensional Figures a Plane Is a Flat Surface That Extends Infinitely in All Directions
NAME CLASS DATE Two-Dimensional Figures A plane is a flat surface that extends infinitely in all directions. A parallelogram like the one below is often used to model a plane, but remember that a plane—unlike a parallelogram—has no boundaries or sides. A plane figure or two-dimensional figure is a figure that lies completely in one plane. When you draw, either by hand or with a computer program, you draw two-dimensional figures. Blueprints are two-dimensional models of real-life objects. Polygons are closed, two-dimensional figures formed by three or more line segments that intersect only at their endpoints. These figures are polygons. These figures are not polygons. This is not a polygon A heart is not a polygon A circle is not a polygon because it is an open because it is has curves. because it is made of figure. a curve. Polygons are named by the number of sides and angles they have. A polygon always has the same number of sides as angles. Listed on the next page are the most common polygons. Each of the polygons shown is a regular polygon. All the angles of a regular polygon have the same measure and all the sides are the same length. SpringBoard® Course 1 Math Skills Workshop 89 Unit 5 • Getting Ready Practice MSW_C1_SE.indb 89 20/07/19 1:05 PM Two-Dimensional Figures (continued) Triangle Quadrilateral Pentagon Hexagon 3 sides; 3 angles 4 sides; 4 angles 5 sides; 5 angles 6 sides; 6 angles Heptagon Octagon Nonagon Decagon 7 sides; 7 angles 8 sides; 8 angles 9 sides; 9 angles 10 sides; 10 angles EXAMPLE A Classify the polygon.
[Show full text]
Conditions of Special Parallelograms 399 Activity Assess
Activity Assess 8-6 MODEL & DISCUSS Conditions The sides of the lantern are identical quadrilaterals. of Special Parallelograms A. Construct Arguments How could you check to see whether a side is a parallelogram? Justify your PearsonRealize.com answer. B C B. Does the side appear to be I CAN… identify rhombuses, rectangular? How could rectangles, and squares by you check? the characteristics of their E diagonals. C. Do you think that diagonals of a quadrilateral can be used to determine whether the quadrilateral is a A D rectangle? Explain. Which properties of the diagonals of a parallelogram help you to classify ESSENTIAL QUESTION a parallelogram? CONCEPTUAL UNDERSTANDING EXAMPLE 1 Use Diagonals to Identify Rhombuses Information about diagonals can help to classify A B a parallelogram. In parallelogram ABCD, AC‾ is perpendicular to BD‾ . What else can you conclude about the parallelogram? E D C STUDY TIP A B Parallelograms have several The diagonals of Any angle at E either properties, and some properties a parallelogram forms a linear pair or bisect each other, may not help you solve a _ E is a vertical angle with particular problem. Here, the fact so AE‾ _≅ CE and ∠AEB, so all four angles DE‾ BE . that diagonals bisect each other ≅ D C are right angles. allows the use of SAS. The four triangles are congruent by SAS, so AB‾ ≅ CB‾ ≅ CD‾ ≅ AD‾ . Since ABCD is a parallelogram with four congruent sides, ABCD is a rhombus. Try It! 1. If ∠JHK and ∠JGK are complementary, H what else can you conclude about L GHJK? Explain.
[Show full text]