You are watching: Diagonals of a rhombus bisect each other
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: watch Source, ShowAbout ikleyn: Diagonals the a rhombus bisect its anglesLet me remind you that a rhombus is a parallelogram which has actually all the sides of the very same length.As a parallelogram, the rhombus has all the properties of a parallelogram:- the contrary sides are parallel;- opposing sides room of same length;- the diagonals bisect every other;- the opposite angles space congruent;- the amount of any kind of two consecutive angle is equal to 180�.Rhombis (plural the rhombus) have added properties.Theorem 1In a rhombus, the diagonals are the angle bisectors. ProofLet the square ABCD it is in the rhombus (Figure 1), and AC and BD it is in its diagonals. The Theorem says that the diagonal AC of the rhombus is the angle bisector to each of the two angles DAB and BCD, while the diagonal BD is the edge bisector to each of the two angles ABC and ADC.Let us think about the triangles ABC and also ADC (Figure 2). The sides BC and DC are of equal length by the condition. The sides ad and ABare that equal length by the condition too. The diagonal AC is the typical side of these triangles. Therefore, the triangles ABC and ADC room congruent in accordance through the postulate 3 (SSS) the the class Congruence tests for triangles, i beg your pardon is under the topic triangle of the section Geometry in this site. number 1. To the to organize 1 number 2. Come the proof of the organize 1 Hence, the angles ACB and also ACD room congruent as the corresponding angles of the congruent triangles ABC and ADC.The angle CAB and CAD space congruent too by the exact same reason. So, we have proved that the diagonal line AC of the rhombus ABCD is the edge bisector to the angles DAB and also BCD. Similarly, we can prove the the diagonal line BD that the rhombus ABCD is the angle bisector to the angle ADC and also ABC. The evidence is completed.NoteIn the food of proving the organize 1 we obtained that the angle ACB and ACD to be congruent, and also the angle CAB and CAD were congruent too.From the other side, the angle ACD and CAB space congruent together the alternative interior angles at the parallel present DC and abdominal and the transverse AC;the angle CAD and also ACB are congruent together the alternating interior angle at the parallel lines ad and BC and also the transverse AC.It way that all 4 acute angle in the figure 2 (LACB, LACD, LCAB and also LCAD) are congruent.This agrees through the reality that the opposite angle DAB and also DCB of the parallelogram ABCD room congruent and the diagonal AC is the angle bisector come the angle DAB and DCB. This is also consistent with the fact that the triangles ADC and also ABC are congruent isosceles triangles having actually congruent angles at their base AC. another proof come the to organize 1Let the square ABCD be the rhombus (Figure 3) v the diagonalsAC and BD intersecting in ~ the point P. Think about the triangle BCD and also DAB. Because all the sides of a rhombus room of equal length, these triangles room isosceles.The diagonals AC and BD bisect each other as the diagonals that the parallel in accordance with the lesson nature of diagonals that parallelograms (under the present topic Parallelograms that the section Geometry in this site). Therefore, the directly segment CP, i m sorry is the part of the diagonal line AC, is the average in the triangle BCD. figure 3. One more proof to the organize 1 In an isosceles triangle, the median corresponds with the edge bisector in accordance through the lesson An altitude a median and also an edge bisector in the isosceles triangle, i m sorry is under the topic triangle of the section Geometry in this site. It means that the diagonal line AC is the angle bisector for the angle BCD.In order come prove that the diagonal BD is the edge bisector for the angle ADC and ABC, think about the isosceles triangle ADC and also ABC and apply similar arguments reflecting that the segment DP is the median and also coincides through the edge bisector come the angle ADC in the triangle ADC.This completes an additional proof come the to organize 1.Theorem 2If in a parallel the 2 diagonals space the edge bisectors, climate the parallelogram is a rhombus. ProofWe are provided the parallel ABCD (Figure 4), in i beg your pardon the two diagonals AC and also BD space the edge bisectors. We must prove the the parallelogram is the rhombus, in various other words, that all 4 sides that the parallelogram have the exact same length.Consider the triangle ABD and CBD produced by the parallelogram sides and also the diagonal BD. The angles ADB and CDB are congruent due to the fact that the diagonal line BD is the edge bisector to the angle ADC by the condition. figure 4. Come the to organize 2 The angle ABD and also CBD are congruent since the diagonal line BD is the angle bisector to the angle abc by the condition.So, the triangle ABD and also CBD have the typical side BD and the two congruent angles surrounding to the common side. Therefore, this triangles room congruent in accordance through the postulate 2 (ASA) the the great Congruence tests because that triangles, i beg your pardon is under the topic triangle of the section Geometry in this site. It indicates that the segments ad and DC room congruent as the matching sides of the congruent triangles ABD and also CBD: ad = DC.From the various other side, the opposite political parties of the parallel ABCD are congruent in accordance with the lesson nature of the sides of a parallelogram, i m sorry is under the topic triangles of the ar Geometry in this site: advertisement = BC and ab = DC. This chain the the debates leads to the conclusion the all political parties of the parallel ABCD have the very same length.Hence, the parallelogram ABCD is a rhombus. The theorem 2 is proved. Organize 3If in a parallelogram the diagonal line bisects an inner angle, then this diagonal line bisects the opposite internal angle too, and also the parallel is a rhombus. ProofLet assume that us are offered the parallelogram ABCD (Figure 5),in i m sorry the diagonal AC bisects the edge DAB. We need to prove that then the diagonal AC bisects the edge DCB too, and also the parallel is a rhombus, in other words, that all four sides of the parallelogram have the exact same length. Consider the triangle ADC and also ABC developed by the parallelogramsides and the diagonal line AC. The angle DAC and BAC room congruent since the diagonal line AC is the edge bisector to the edge DAB through the condition: LDAC = LBAC. number 5. To the to organize 3 from the various other side, the angles BAC and also ACD space congruent as the alternative interior angle at the parallel straight lines abdominal and DC and the transverse AC (see the lesson Parallel lines under the object Angles, complementary, supplementary angle of the ar Geometry in this site): LBAC = LACD.Therefore, the angle DAC and also ACD space congruent: LDAC = LACD. (1)Similarly, the angles DAC and ACB are congruent together the alternate interior angle at the parallel right lines advertisement and BC and the transverse AC : LDAC = LACB.Therefore, the angle ACD and also ACB space congruent as they both room congruent to the angle DAC: LCAB = LACB. The last congruency way that the diagonal line AC bisects the angle ABC, and this proves the very first part of the theorem 3 statement.Next, the congruency (1) over implies that the triangle ADC is one isosceles triangle due to the fact that it has congruent angles LDAC and also LACD (see the lesson Isosceles triangle under the topic triangles of the section Geometry in this site). Hence, the sides ad and DC of this triangle room congruent. Hence the parallelogram ABCD has two consecutive sides congruent. It implies that the parallelogram is a rhombus, due to the fact that the opposite political parties of a parallelogram room congruent. The organize 3 is totally proved. An overview 1. In a rhombus, the diagonals are the edge bisectors. 2. If in a parallelogram the two diagonals space the edge bisectors, then the parallelogram is a rhombus. 3. If in a parallel the diagonal bisects an internal angle, climate this diagonal bisects the opposite inner angle too, and also the parallel is a rhombus. My other lessons top top rhombis in this website are - Diagonals that a rhombus space perpendicular- The length of diagonals of a rhombus- A one inscribed in the rhombus- how TO solve difficulties on the rhombus sides and also diagonals measures - Examples,- The size of diagonals the a rhombus - properties OF RHOMBISTo navigate over all topics/lessons the the digital Geometry Textbook usage this file/link GEOMETRY - YOUR online TEXTBOOK.
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