Adjacent and vertical angles

Geometry, grade 7 Lesson 4 of 18Learning

Adjacent and vertical angles

Adjacent angles

Adjacent angles are two angles with a common vertex and a common arm, whose other two arms make one straight line. In the figure, $\displaystyle \alpha$ and $\displaystyle \beta$ are adjacent:

The common vertex is $\displaystyle B$ , and the common arm is $\displaystyle BD$ .
The other arms, $\displaystyle BA$ and $\displaystyle BC$ , make the straight line $\displaystyle AC$ .
Together, $\displaystyle \alpha$ and $\displaystyle \beta$ fill the straight angle $\displaystyle \angle ABC$ , so their sum is $\displaystyle 180^{\circ}$ :

\displaystyle \alpha+\beta

\displaystyle =

\displaystyle 180^{\circ}

For example: when $\displaystyle {\beta}={65^{\circ}}$ , the angle adjacent to it is $\displaystyle {\alpha}={180^{\circ}-65^{\circ}}={115^{\circ}}$ .

Vertical angles

Vertical angles are two angles whose arms are the extensions of each other’s arms, and they are equal. When two straight lines cross, they make four angles around the crossing point $\displaystyle O$ :

$\displaystyle \alpha$ and $\displaystyle \gamma$ are vertical angles, and so are $\displaystyle \beta$ and $\displaystyle \delta$ .
Vertical angles are equal:

\displaystyle \alpha=\gamma

The reason: $\displaystyle \alpha$ is adjacent to $\displaystyle \beta$ , so $\displaystyle {\alpha}={180^{\circ}-\beta}$ . $\displaystyle \gamma$ is adjacent to $\displaystyle \beta$ too, so $\displaystyle {\gamma}={180^{\circ}-\beta}$ . Both are the same number.
For example: when $\displaystyle {\alpha}={80^{\circ}}$ , $\displaystyle {\gamma}={80^{\circ}}$ as well, and $\displaystyle {\beta}={\delta}={180^{\circ}-80^{\circ}}={100^{\circ}}$ .

Open scissors have vertical angles too: the angle between the blades equals the angle between the handles.

Angles around a point

Angles that fill the whole turn around a point, without covering each other, make a full turn together, so their sum is $\displaystyle 360^{\circ}$ :

Here: $\displaystyle {105^{\circ}+100^{\circ}+65^{\circ}+90^{\circ}}={360^{\circ}}$ .
A pizza works the same way: the angles of all the slices, at the centre, make $\displaystyle 360^{\circ}$ together.
The rule holds for any number of angles. With four angles:

\displaystyle \alpha+\beta+\gamma+\delta

\displaystyle =

\displaystyle 360^{\circ}

Exercise 1

In the figure, the points $\displaystyle A$ , $\displaystyle B$ and $\displaystyle C$ are on one straight line.
$\displaystyle {\angle DBC}={47^{\circ}}$ .
Calculate $\displaystyle \alpha$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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Exercise 2

In the figure, the lines $\displaystyle AB$ and $\displaystyle CD$ cross at the point $\displaystyle O$ .
$\displaystyle {\angle COB}={58^{\circ}}$ .
Write the size of each of the other three angles around $\displaystyle O$ .
Give the reason for each answer.

\displaystyle \angle BOD

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\displaystyle \angle DOA

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\displaystyle \angle AOC

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Exercise 3

Are $\displaystyle \angle ABD$ and $\displaystyle \angle DBC$ adjacent angles? Explain.

Exercise 4

Three rays start at the point $\displaystyle O$ .
$\displaystyle {\angle AOB}={110^{\circ}}$ and $\displaystyle {\angle BOC}={135^{\circ}}$ .
Calculate $\displaystyle \alpha$ .

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Exercise 5

Two adjacent angles are equal.
Calculate the size of each of them.

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Exercise 6

A round pizza is cut into 8 equal slices, by cuts through the centre.
Calculate the angle of each slice at the centre.

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Exercise 7

In the figure, three straight lines cross at the point $\displaystyle O$ .
$\displaystyle {\angle AOC}={50^{\circ}}$ and $\displaystyle {\angle COE}={60^{\circ}}$ .
Calculate $\displaystyle \angle EOB$ , $\displaystyle \angle BOD$ , $\displaystyle \angle DOF$ and $\displaystyle \angle FOA$ .
Give the reason for each answer.

\displaystyle \angle EOB

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\displaystyle \angle BOD

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\displaystyle \angle DOF

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\displaystyle \angle FOA

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Formulas: geometry, grade 7

Angles

The sum of angles – the ray $\displaystyle BD$ runs inside $\displaystyle \angle ABC$
$\displaystyle \angle ABC$ $\displaystyle =$ $\displaystyle \angle ABD+\angle DBC$
The difference of angles
$\displaystyle \angle DBC$ $\displaystyle =$ $\displaystyle \angle ABC-\angle ABD$
The bisector $\displaystyle BD$ cuts $\displaystyle \angle ABC$ into two equal angles
$\displaystyle \angle ABD=\angle DBC=\frac{\angle ABC}{2}$
Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
The angles around a point
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$

The angles of a triangle and a quadrilateral

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
The two acute angles of a right triangle
$\displaystyle \alpha+\beta=90^{\circ}$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$
The angles of a quadrilateral
$\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ}$

The perimeter and area of a rectangle and a square

The area of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The perimeter of a rectangle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\cdot(a+b)$
The area of a square – $\displaystyle a$ the side
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^2$
The perimeter of a square
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4\cdot a$

The areas of polygons

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 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 right triangle – $\displaystyle a$ , $\displaystyle b$ the legs
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot b}{2}$ $\displaystyle \angle C=90^{\circ}$
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 box and the cube

The surface area of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle 2(ab+bc+ac)$
The surface area of a cube – $\displaystyle a$ the edge
$\displaystyle S$ $\displaystyle =$ $\displaystyle 6a^2$
The volume of a box
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\cdot b\cdot c$
The volume of a cube
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$