Hoeken berekenen in vierhoeken

Leerdoelen

•Je kunt met behulp van de hoekensom en symmetrie-eigenschappen hoeken berekenen in vierhoeken

Hoekensom van een vierhoek

Een belangrijk feit dat je moet onthouden is dat de som van alle hoeken in een vierhoek altijd 360 graden is. Dit geldt voor alle soorten vierhoeken, of het nu een vierkant, een rechthoek of een parallellogram is.

Voorbeeld 1: Hoek C berekenen

Bekijk onderstaande vierhoek en bereken met behulp van de hoekensom\angle CCCCCCCCC.

In bovenstaande figuur zie je de volgende hoeken:\angle A=68\degree,\,\angle B=90\degree\angle A=68\degree,\,\angle B=90\degree\text{ }\angle A=68\degree,\,\angle B=90\degree\text{ }2\degree.\angle A=68\degree,\,\angle B=90\degree\text{ (dit kan je zien aan het rechthoeksteken) en }\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\left|\angle D=92\degree.\right|\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree.\angle A=68\degree,\,\angle B=90\degree\angle D=92\degree\angle A=68\degree,\,\angle B=90\degree\angle D=92\angle A=68\degree,\,\angle B=90\degree\angle D=92\angle A=68\degree,\,\angle B=90\degree\angle D=92\angle A=68\degree,\,\angle B=90\degree\angle D=92\angle A=68\degree,\,\angle B=90\degree\angle D=9\angle A=68\degree,\,\angle B=90\degree\angle D=\angle A=68\degree,\,\angle B=90\degree\angle D\angle A=68\degree,\,\angle B=90\degree\angle D=\angle A=68\degree,\,\angle B=90\degree\angle D\angle A=68\degree,\,\angle B=90\degree\angle\angle A=68\degree,\,\angle B=90\degree\angle\angle A=68\degree,\,\angle B=90\degree\angle A=68\degree,\,\angle B=90\angle A=68\degree,\,\angle B=90\angle A=68\degree,\,\angle B=90\angle A=68\degree,\,\angle B=90\angle A=68\degree,\,\angle B=9\angle A=68\degree,\,\angle B=\angle A=68\degree,\,\angle B\angle A=68\degree,\,\angle\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\,\angle A=68\degree,\angle A=68\degree,\angle A=68\degree,\angle A=68\degree\angle A=68\angle A=68\angle A=68\angle A=68\angle A=68\angle A=6\angle A=\angle A\angle(dit kan je zien aan het rechthoeksteken) en\angle D=92\degree\angle D=9\degree. Alleen\angle CCCCCis onbekend. Aangezien de hoekensom van een vierhoek gelijk is aan 360 graden, is\angle C\anglegelijk aan de hoekensom minus de andere hoeken. De som van de bekende hoeken is68+92+90=250\degree68+92+90=25068+92+90=25068+92+90=25068+92+90=25068+92+90=2568+92+90=268+92+90=68+92+9068+92+90). Dus\angle C=360-250=110\degree\angle C\text{ }=360-250=110\degree\angle C\text{ }=360--250=110\degree\angle C\text{ }=360-(-250=110\degree\angle C\text{ }=360-(=-250=110\degree\angle C\text{ }=360-(=3-250=110\degree\angle C\text{ }=360-(=36-250=110\degree\angle C\text{ }=360-(=360-250=110\degree\angle C\text{ }=360-(68+92+90)=360-250=110\degree\angle C\text{ }=360-(68+92+90)=360-250=110\angle C\text{ }=360-(68+92+90)=360-250=110\angle C\text{ }=360-(68+92+90)=360-250=110\angle C\text{ }=360-(68+92+90)=360-250=110\angle C\text{ }=360-(68+92+90)=360-250=110\text{ }\angle C\text{ }=360-(68+92+90)=360-250=110\text{ g}\angle C\text{ }=360-(68+92+90)=360-250=110\text{ gr}\angle C\text{ }=360-(68+92+90)=360-250=110\text{ gra}\angle C\text{ }=360-(68+92+90)=360-250=110\text{ grad}\angle C\text{ }=360-(68+92+90)=360-250=110\text{ grade}\angle C\text{ }=360-(68+92+90)=360-250=110\text{ graden}\angle\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{ }=360-(68+92+90)=360-250=110\text{ graden}\text{H }=360-(68+92+90)=360-250=110\text{ graden}\text{Ho }=360-(68+92+90)=360-250=110\text{ graden}\text{Hoe }=360-(68+92+90)=360-250=110\text{ graden}\text{Hoek }=360-(68+92+90)=360-250=110\text{ graden}\text{Hoek }=360-(68+92+90)=360-250=110\text{ graden}\text{Hoek C }=360-(68+92+90)=360-250=110\text{ graden}

Symmetrie in vierhoeken

Sommige vierhoeken hebben bijzondere eigenschappen zoals symmetrie. Dit kan helpen bij het berekenen van hoeken. We gaan kijken naar een specifiek voorbeeld van een vlieger en een parallellogram.

Voorbeeld 2: Vlieger GHIJ

In de vliegerwillen we de\angle JJJJJberekenen, zie onderstaande afbeelding.

We weten dat\angle J+\angle H+\angle I+\angle G=360\degree\angle J+\angle H+\angle I+\angle=360\degree\angle J+\angle H+\angle I\angle=360\degree\angle J+\angle H+\angle\angle=360\degree\angle J+\angle H\angle\angle=360\degree\angle J+\angle\angle\angle=360\degree\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=360\angle J+\angle\angle\angle=36\angle J+\angle\angle\angle=3\angle J+\angle\angle\angle=\angle J+\angle\angle\angle\angle J+\angle\angle\angle J+\angle\angle J+\angle J\angleAan het eind van deze video kan ik hoeken berekenen in vierhoeken \angleen dat\angle I=124\degree\angle I=124\angle I=124\angle I=124\angle I=124\angle I=124\angle I=12\angle I=1\angle I=\angle I\angleen\angle G=52\degree\angle G=52\angle G=52\angle G=52\angle G=52\angle G=52\angle G=5\angle G=\angle G\angle.

Daardoor kunnen we de overgebleven graden berekenen:360-124-52=184\degree360-24-52=184\degree360-(24-52=184\degree360-(124-52=184\degree360-(12452=184\degree360-(124+52=184\degree360-(124+52)=184\degree360-(124+52)=18\degree360-(124+52)=1\degree360-(124+52)=\degree360-(124+52)=2\degree360-(124+52)=20\degree360-(124+52)=200\degree360-(124+5)=200\degree360-(124+)=200\degree360-(124+7)=200\degree360-(124+76)=200\degree360-(14+76)=200\degree360-(4+76)=200\degree360-(84+76)=200\degree360-(84+76)=200360-(84+76)=200360-(84+76)=200360-(84+76)=200360-(84+76)=200\text{ }360-(84+76)=200\text{ g}.

Een vlieger is spiegelsymmetrisch. De symmetrieas loopt door de hoekenen. Daarom zijn de twee andere hoeken\angle J\angleen\angle Heven groot. Dit geeft\angle J=\angle H=\frac{184}{2}=92\degree\angle J=\angle H=\frac{184}{2}=9\degree\angle J=\angle H=\frac{184}{2}=\degree\angle J=\angle H=\frac{184}{2}=1\degree\angle J=\angle H=\frac{184}{2}=10\degree\angle J=\angle H=\frac{184}{2}=100\degree\angle J=\angle H=\frac{18}{2}=100\degree\angle J=\angle H=\frac12=100\degree\angle J=\angle H=\frac{}{2}=100\degree\angle J=\angle H=\frac22=100\degree\angle J=\angle H=\frac{20}{2}=100\degree\angle J=\angle H=\frac{200}{2}=100\degree\angle J=\angle H=\frac{200}{2}=100\angle J=\angle H=\frac{200}{2}=100\angle J=\angle H=\frac{200}{2}=100\angle J=\angle H=\frac{200}{2}=100\angle J=\angle H=\frac{200}{2}=100\angle J=\angle H=\frac{200}{2}=100\text{ }\angle J=\angle H=\frac{200}{2}=100\text{ graden}\angle J=\angle=\frac{200}{2}=100\text{ graden}\angle J==\frac{200}{2}=100\text{ graden}\angle J=\text{ }=\frac{200}{2}=100\text{ graden}\angle J=\text{ }=\frac{200}{2}=100\text{ graden}\angle J=\text{ \$\$ \textbackslash angle \$\$}=\frac{200}{2}=100\text{ graden}\angle J=\text{ \$\$ \textbackslash angle \$\$}=\frac{200}{2}=100\text{ graden}\angle J=\text{ }=\frac{200}{2}=100\text{ graden}\angle J=\text{ }=\frac{200}{2}=100\text{ graden}\angle J=\text{ H }=\frac{200}{2}=100\text{ graden}\angle J=\text{ Ho }=\frac{200}{2}=100\text{ graden}\angle J=\text{ Hoe }=\frac{200}{2}=100\text{ graden}\angle J=\text{ Hoek }=\frac{200}{2}=100\text{ graden}\angle J=\text{ Hoek }=\frac{200}{2}=100\text{ graden}\angle J=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\angle=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\angle\text{J}=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\angle\text{J }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{J }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{HJ }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{HoJ }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{HoeJ }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{HoekJ }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{Hoek J }=\text{ Hoek H }=\frac{200}{2}=100\text{ graden}\text{Hoek J }=\text{ Hoek H}=\frac{200}{2}=100\text{ graden}\text{Hoek J }=\text{ Hoek H}=\frac{200}{2}=100\text{ graden}\text{Hoek J }=\text{Hoek H}=\frac{200}{2}=100\text{ graden}.

Voorbeeld 3: Parallellogram KLMN

In een parallellogram, zoals, hebben we draaisymmetrie, zie onderstaande afbeelding.

We willen\angle Nberekenen.

In een parallellogram zijn de overstaande hoeken gelijk aan elkaar. Er geldt dus:en. Omdat, isook.

Daardoor kunnen we de overgebleven graden berekenen:360-55-55=250\degree360-5555=250\degree360-55+55=250\degree360-55+55)=250\degree360-(55+55)=250\degree360-(55+55)=250360-(55+55)=250360-(55+55)=250360-(55+55)=250360-(55+55)=250\text{ }360-(55+55)=250\text{ g}.

Deze 250 graden worden gelijk verdeeld overen. Beide hoeken zijn dus\frac{250}{2}=125\degree\frac{250}{\placeholder{}}=125\degree250=125\degree250/=125\degree.