Review: the whole course

Geometry, grade 9 Lesson 21 of 21Practice

Review: the whole course

In triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={70^{\circ}}$ and $\displaystyle {\angle B}={55^{\circ}}$ .
Give the reason for every answer.

Which sides of the triangle are equal?

Which side of the triangle is the longest?

In triangle $\displaystyle ABC$ , angle $\displaystyle C$ is right, $\displaystyle {\angle A}={30^{\circ}}$ , and the hypotenuse $\displaystyle AB$ is $\displaystyle 14$ .

Find $\displaystyle BC$ .

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Find $\displaystyle AC$ .

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A line crosses two lines, $\displaystyle a$ and $\displaystyle b$ .
One of the angles it makes with $\displaystyle a$ is $\displaystyle 108^{\circ}$ .
Is $\displaystyle a\parallel b$ ? Give the reason.

The co-interior angle to it, which the line makes with $\displaystyle b$ , is $\displaystyle 72^{\circ}$ .

The corresponding angle to it, which the line makes with $\displaystyle b$ , is $\displaystyle 72^{\circ}$ .

$\displaystyle ABCD$ is a kite: $\displaystyle {AB}={AD}$ and $\displaystyle {CB}={CD}$ .
$\displaystyle {\angle BAD}={70^{\circ}}$ and $\displaystyle {\angle BCD}={50^{\circ}}$ .

Find $\displaystyle \angle ABC$ .

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Find $\displaystyle \angle BAC$ .

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$\displaystyle ABCD$ is a rhombus, and $\displaystyle {\angle BAD}={64^{\circ}}$ .
Its diagonals meet at the point $\displaystyle O$ .

Find $\displaystyle \angle ABC$ .

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Find $\displaystyle \angle ABD$ .

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Find $\displaystyle \angle AOB$ .

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The quadrilateral $\displaystyle ABCD$ is drawn in the coordinate system.

Which quadrilateral is $\displaystyle ABCD$ ? Give the reason by the slopes of its sides.

Find the area of $\displaystyle ABCD$ .

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Find the length of the side $\displaystyle AD$ .

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In triangle $\displaystyle ABC$ , angle $\displaystyle C$ is right, $\displaystyle {\angle A}={30^{\circ}}$ , and $\displaystyle {AB}={12}$ .
The segment $\displaystyle CM$ is the median to the hypotenuse.

Find $\displaystyle CM$ .

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Find $\displaystyle BC$ .

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What kind of triangle is $\displaystyle CMB$ ? Give the reason.

In triangle $\displaystyle ABC$ , the point $\displaystyle D$ is on $\displaystyle AB$ , the point $\displaystyle E$ is on $\displaystyle AC$ , and $\displaystyle DE\parallel BC$ .
The area of triangle $\displaystyle ADE$ is $\displaystyle 12$ , and the area of the trapezoid $\displaystyle DBCE$ is $\displaystyle 15$ .

Find the area of triangle $\displaystyle ABC$ .

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Find the ratio of similarity of the triangles $\displaystyle ADE$ and $\displaystyle ABC$ .

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$\displaystyle {AB}={9}$ .
Find $\displaystyle AD$ .

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$\displaystyle ABCD$ is a parallelogram.
The bisector of angle $\displaystyle B$ meets the side $\displaystyle AD$ at the point $\displaystyle E$ .
Prove: $\displaystyle {AB}={AE}$ .

$\displaystyle ABCD$ is an isosceles trapezoid: $\displaystyle AB\parallel DC$ and $\displaystyle {AD}={BC}$ .
Its diagonals meet at the point $\displaystyle O$ .
Prove: $\displaystyle {OA}={OB}$ .

In triangle $\displaystyle ABC$ , the segments $\displaystyle BE$ and $\displaystyle CF$ are heights: $\displaystyle E$ is on $\displaystyle AC$ , and $\displaystyle F$ on $\displaystyle AB$ .
The point $\displaystyle M$ is the midpoint of $\displaystyle BC$ .
Prove: $\displaystyle {ME}={MF}$ .

$\displaystyle ABCD$ is a trapezoid: $\displaystyle AB\parallel DC$ .
Its diagonals meet at the point $\displaystyle O$ .

a. Prove: $\displaystyle \triangle AOB\sim\triangle COD$ .

$\displaystyle {AB}={2DC}$ , and the area of triangle $\displaystyle COD$ is $\displaystyle 5$ .
Find the area of triangle $\displaystyle AOB$ .

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Formulas: geometry, grade 9

Open as a page

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
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$