Practice: the parallelogram

Geometry, grade 9 Lesson 14 of 21Practice

Practice: the parallelogram

$\displaystyle ABCD$ is a parallelogram.
Find all its angles.

Angle $\displaystyle B$ is $\displaystyle 40^{\circ}$ larger than angle $\displaystyle A$ .

\displaystyle \angle A

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\displaystyle \angle B

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\displaystyle \angle C

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\displaystyle \angle D

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Angle $\displaystyle A$ is four times angle $\displaystyle B$ .

\displaystyle \angle A

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\displaystyle \angle B

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\displaystyle \angle C

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\displaystyle \angle D

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$\displaystyle ABCD$ is a parallelogram.
Find the lengths of the sides $\displaystyle AB$ and $\displaystyle BC$ .

The perimeter of the parallelogram is $\displaystyle 36$ , and the side $\displaystyle AB$ is $\displaystyle 4$ longer than the side $\displaystyle BC$ .

\displaystyle AB

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\displaystyle BC

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The perimeter of the parallelogram is $\displaystyle 42$ , and $\displaystyle {AB}={2BC}$ .

\displaystyle AB

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\displaystyle BC

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The diagonals of parallelogram $\displaystyle ABCD$ cross at the point $\displaystyle O$ .

$\displaystyle {AC}={16}$ , $\displaystyle {BD}={12}$ and $\displaystyle {AB}={10}$ .
What is the perimeter of triangle $\displaystyle AOB$ ?

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$\displaystyle {AO}={x+3}$ and $\displaystyle {OC}={2x-1}$ .
Find $\displaystyle x$ and the length of the diagonal $\displaystyle AC$ .

\displaystyle x

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\displaystyle AC

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Is quadrilateral $\displaystyle ABCD$ sure to be a parallelogram? Give the reason.

$\displaystyle {\angle A}={\angle C}={70^{\circ}}$ and $\displaystyle {\angle B}={110^{\circ}}$ .

$\displaystyle AB\parallel DC$ and $\displaystyle {AB}={DC}={9}$ .

$\displaystyle {AB}={BC}={5}$ and $\displaystyle {CD}={DA}={8}$ .

$\displaystyle AD\parallel BC$ and $\displaystyle {\angle A}={\angle C}$ .

The points $\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ and $\displaystyle D$ are the vertices of quadrilateral $\displaystyle ABCD$ .

Prove that $\displaystyle ABCD$ is a parallelogram.

Find the length of the side $\displaystyle AD$ and the perimeter of the parallelogram.

\displaystyle AD

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perimeter

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$\displaystyle ABCD$ is a parallelogram, $\displaystyle {AB}={10}$ and $\displaystyle {AD}={6}$ .
The bisector of angle $\displaystyle A$ meets the side $\displaystyle CD$ at the point $\displaystyle E$ .

Prove: triangle $\displaystyle ADE$ is isosceles.

Find $\displaystyle DE$ and $\displaystyle EC$ .

\displaystyle DE

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\displaystyle EC

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$\displaystyle ABCD$ is a parallelogram.
The point $\displaystyle E$ is on the side $\displaystyle AD$ , and the point $\displaystyle F$ on the side $\displaystyle BC$ , so that $\displaystyle {AE}={CF}$ .
Prove: $\displaystyle EBFD$ is a parallelogram.

The diagonals of parallelogram $\displaystyle ABCD$ cross at the point $\displaystyle O$ .
A line through $\displaystyle O$ meets the side $\displaystyle AB$ at the point $\displaystyle E$ and the side $\displaystyle CD$ at the point $\displaystyle F$ .
Prove: $\displaystyle {OE}={OF}$ .

$\displaystyle ABCD$ is a parallelogram.
The bisector of angle $\displaystyle A$ meets the side $\displaystyle CD$ at the point $\displaystyle E$ , and the bisector of angle $\displaystyle C$ meets the side $\displaystyle AB$ at the point $\displaystyle F$ .
Prove: $\displaystyle AFCE$ is a parallelogram.

Formulas: geometry, grade 9

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Angles and sides in a triangle

The sum of the angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
An exterior angle $\displaystyle \delta$ equals the sum of the two interior angles not next to it
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
The larger side faces the larger angle, and the other way round
$\displaystyle a>b\ \Leftrightarrow\ \alpha>\beta$

The right triangle

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^{2}+b^{2}$ $\displaystyle =$ $\displaystyle c^{2}$
The leg $\displaystyle a$ facing a $\displaystyle 30^{\circ}$ angle is half the hypotenuse $\displaystyle c$
$\displaystyle a$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$
The median $\displaystyle m_c$ to the hypotenuse is half the hypotenuse $\displaystyle c$
$\displaystyle m_c$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$

Angles between parallel lines

Corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Co-interior angles
$\displaystyle \alpha+\beta=180^{\circ}$

Angles in a quadrilateral

The sum of the angles of a quadrilateral
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$
In a trapezoid: the two angles beside a leg
$\displaystyle \alpha+\beta=180^{\circ}$

Areas

The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a rectangle
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^{2}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a rhombus or a kite – $\displaystyle d_1$ , $\displaystyle d_2$ the diagonals
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{d_1\cdot d_2}{2}$
The ratio of the areas of similar triangles – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$