Practice: the angles of a triangle and a quadrilateral

Geometry, grade 7 Lesson 9 of 18Practice

Practice: the angles of a triangle and a quadrilateral

1

In triangle $\displaystyle ABC$ : $\displaystyle {\angle A}={47^{\circ}}$ , $\displaystyle {\angle B}={68^{\circ}}$ .
Calculate $\displaystyle \angle C$ .

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2

In the drawing, the side $\displaystyle BC$ of triangle $\displaystyle ABC$ is extended to the point $\displaystyle D$ .
$\displaystyle {\angle BAC}={60^{\circ}}$ , $\displaystyle {\angle ACD}={125^{\circ}}$ .
Find $\displaystyle \angle ACB$ and $\displaystyle \angle B$ , and explain.

\displaystyle \angle ACB

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\displaystyle \angle B

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3

In the drawing, the segments $\displaystyle AD$ and $\displaystyle BE$ cross at the point $\displaystyle O$ .
$\displaystyle {\angle A}={70^{\circ}}$ , $\displaystyle {\angle B}={50^{\circ}}$ , $\displaystyle {\angle D}={65^{\circ}}$ .
Find $\displaystyle \angle AOB$ , $\displaystyle \angle DOE$ and $\displaystyle \angle E$ , and explain.

\displaystyle \angle AOB

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\displaystyle \angle DOE

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\displaystyle \angle E

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4

The apex angle of an isosceles triangle is $\displaystyle 36^{\circ}$ .
Calculate a base angle.

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5

A base angle of an isosceles triangle is $\displaystyle 54^{\circ}$ .
Find the apex angle.

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6

In a right-angled triangle, one acute angle is twice the other acute angle.
Find the two acute angles.

Smaller

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Larger

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7

Triangle $\displaystyle ABC$ is equilateral, and the side $\displaystyle BC$ is extended to the point $\displaystyle D$ .
Find $\displaystyle \angle ACD$ , and explain.

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8

In triangle $\displaystyle ABC$ : $\displaystyle {AB}={AC}$ .
The side $\displaystyle CB$ is extended beyond $\displaystyle B$ to the point $\displaystyle D$ , and $\displaystyle {\angle ABD}={115^{\circ}}$ .
Find the three angles of triangle $\displaystyle ABC$ , and explain.

\displaystyle \angle B

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\displaystyle \angle C

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\displaystyle \angle A

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9

Three angles of a quadrilateral are $\displaystyle 85^{\circ}$ , $\displaystyle 95^{\circ}$ and $\displaystyle 100^{\circ}$ .
Calculate the fourth angle.

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10

In quadrilateral $\displaystyle ABCD$ : $\displaystyle {\angle A}={\angle C}$ .
$\displaystyle {\angle B}={110^{\circ}}$ , $\displaystyle {\angle D}={90^{\circ}}$ .
Find $\displaystyle \angle A$ .

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11

$\displaystyle ABCD$ is a rectangle, and $\displaystyle AC$ is one of its diagonals.
$\displaystyle {\angle BAC}={35^{\circ}}$ .
Find $\displaystyle \angle ACB$ and $\displaystyle \angle CAD$ , and explain.

\displaystyle \angle ACB

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\displaystyle \angle CAD

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12

True or false?
Explain: if true – say why; if false – show why not.

a

An obtuse-angled triangle can be isosceles.

b

A right-angled triangle can be equilateral.

c

Every equilateral triangle is acute-angled.

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
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$