De nition: A diagonal - Berkeley City College

1 Definition: Aquadrilateralis a polygon with 4 sides. Adiagonalof a quadrilat-eral is a line segment whose end-points are opposite vertices of the picture below,ABCDis a quadrilateral,AC,BDare the two name a quadrilateral by naming the four vertices in consecutive order. So wecan name the quadrilateral asABCD, or quadrilateralBCDA, : ATrapezoidis a quadrilateral with a pair of parallel pair of parallel sides (AB||DC) are called thebasesof the trapazoid, andthe non-parallel sides (DA,CB) form thelegsof the the two legs of the trapazoid are congruent to each other, then we have anisoceles :The base angles of an isoceles trapazoid are the above isoceles trapazoid, A = BThe converse of this statement is also true: If the base angles of a trapazoid iscongruent, then the trapazoid is.

2 Aparallelogramis a quadrilateral wherebothpairs of oppositesides are parallel. We use the symbolto represent a ,AB||DC,AD|| :Opposite sides of a parallelogram are : GivenABCD, we must prove thatAB =DCandAD =BC. Wecontruct the diagonal ,AC, of the a diagonal toABCD1. ||DC,AD||BC2. Def. of3. DCA =BAC, DAC = BCA3. Alternate Interior =AC4. =4 ACD5. =CD,AD =CB6. CPCTCThe converse of this statement is also true. That is, if both pairs of oppositesides of a quadrilateral are congruent, then the quadrilateral is a :The diagonals of a parallelogram bisect each : GivenABCD, let the diagonalsACandBDintersect atE, we mustprove thatAE =CEandBE = diagonals toABCD1.

3 ||DC,AD||BC2. Def. of3. DCE =BAE, CDE = ABE3. Alternate Interior =AB4. opposite sides ofare = =4 CDE5. =CE,BE =DE6. ,BDbisect each other7. Def. of segment bisectorThe converse is also true: If the diagonals of a quadrilateral bisect each other,then the quadrilateral is a :Opposite angles of a parallelogram are congruent to each other. InABCD, A = C, and B = , if both pairs of opposite angles of a quadrilateral are congruent toeach other, then the quadrilateral is a a parallelogram with all four angles being right angles. In aparallelogram, if one angle is a right angle, then all four angles are right (why?)

4 ADBCABCDis a :The two diagonals of a rectangle are rectangleABCD,AC = a parallelogram with all four sides congruent to each a rhombus, which meansAB =BC =CD =DA. A rhombus has adiamond-like :The diagonals of a rhombus are perpendicular to each : Given rhombusABCD, let the diagonalsAC,BDintersect atE, we mustprove thatAC diagonals to rhombusABCD1. =BC2. Def. of =CE3. Diagonals ofbisect each =BE4. =4 BCE5. SSS6. BEA = BEC6. BD7. Def. of perpendicular linesAsquareis a parallelogram with four congruent sides and four right angles. Inother words, a square is a rectangleanda a square, which means that A, B, C, and Dare all right addition,AB =BC =CD =DAUnderstand that rectangles, rhombus, squares are all parallelograms.

5 Thereforethey all have properties that a parallelogram has. Any theorem that is true abouta parallelogram can be applied to a rectangle, rhombus, or square. These specialparallelograms, of course, have more specific properties that may not be sharedby other parallelograms. We use a table to indicate the properties that are truefor each kind of figure:PropertiesParallelogramRectangleR hombusSquareOpposite sides ParallelyesyesyesyesOpposite sides CongruentyesyesyesyesDiagonals bisect each otheryesyesyesyesOpposite angles are congruentyesyesyesyesDiagonals are congruentnoyesnoyesAll four angles are rightnoyesnoyesDiagonals are perpendicularnonoyesyesAll four sides congruentnonoyesyesTo prove that a parallelogram is a rectangle, we need to prove that one of itsinterior angle is right.

6 We can also try to prove that its diagonals are prove that a parallelogram is a rhombus, we need to prove that its four sidesare congruent. We can also try to prove that its diagonals are prove that a parallelogram is a square, we need to prove that it is a rectangleand a :If three or more parallel lines cut off congruent segments on onetransversal, then they cut off congruent segments on all other picture below,AB||CD||EF. IfHGis a transversal cutoff into equal parts bythe three parallel lines, thenKJwill also be cut-off into equal parts by the threeparallel : In the picture below, given linesAB||CD||EF, andLM =MN, we needto prove thatRQ =P Q.

7 We will do so by introducing a new line, the linethroughQparallel ||CD||EF,LM =NM1. Given2. ConstructV TthroughQparallel toLN2. Parallel Qare parallelograms3. Def. of =QV,LM =T Q4. Opposite sides ofare = Q =T Q5. Substitution6. RV Q = P T Q, V RQ = T P Q6. Alternate Interior Q =4P T Q7. Q =T Q8. CPCTCT heorem:If a line is drawn from the midpoint of one side of a triangle andparallel to a second side, then that line bisects the third picture below,Mis the midpoint ofAB. If we construct a line throughMparallel toAC, then this line will intersectBCatN, whereNis the midpointofBCAMCNBThe converse of this theorem is also true.

8 If a line connects the midpoints of twosides of a triangle, then the line is parallel to the third side. In addition, thelength of this line is half of the length of the third the picture above, ifMis the midpoint ofABandNis the midpoint ofCB,thenMN||AC, andMN=12 ACTheorem:The three medians of a triangle intersect at a point (thecentroidof the triangle). This point is two-thirds of the distance from any vertex to themidpoint of the opposite the above, ifAF,CE, andBDare medians of4 ABC, then they intersect ata single point,M, andCM= 2ME,AM= 2MF,BM= 2MD.