Review: geometry, grade 7

Geometry, grade 7 Lesson 18 of 18Practice

Review: geometry, grade 7

Angles

Here are five angles: $\displaystyle 35^\circ$ , $\displaystyle 90^\circ$ , $\displaystyle 128^\circ$ , $\displaystyle 180^\circ$ , $\displaystyle 89^\circ$ .
Write beside each one whether it is acute, right, obtuse or straight.

The lines $\displaystyle AB$ and $\displaystyle CD$ cross at the point $\displaystyle O$ .
$\displaystyle {\angle BOD}={70^\circ}$ .
Find $\displaystyle \angle AOC$ , $\displaystyle \angle BOC$ and $\displaystyle \angle AOD$ , and write the reason for each.

\displaystyle \angle AOC

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\displaystyle \angle BOC

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\displaystyle \angle AOD

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Find, in degrees:

\displaystyle {\angle ABC}={130^\circ}

, and the ray

\displaystyle BD

bisects it. How many degrees is

\displaystyle \angle ABD

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Three angles around a point:

\displaystyle 120^\circ

\displaystyle 95^\circ

, and the third?

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Triangles and quadrilaterals

Find the missing angle, in degrees.

Two angles of a triangle are

\displaystyle 47^\circ

and

\displaystyle 68^\circ

. What is the third?

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In an isosceles triangle the apex angle is

\displaystyle 40^\circ

. What is each base angle?

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In a right-angled triangle one acute angle is

\displaystyle 34^\circ

. What is the other acute angle?

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Three angles of a quadrilateral are

\displaystyle 85^\circ

\displaystyle 110^\circ

and

\displaystyle 95^\circ

. What is the fourth?

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Is there a triangle with two obtuse angles?
And is there a triangle with two right angles?
Explain.

Congruent shapes

Which of the shapes 2, 3, 4, 5 are congruent to shape 1?
For each congruent shape, write which transformation takes shape 1 onto it: a translation, a reflection or a rotation.

The triangles $\displaystyle ABC$ and $\displaystyle DEF$ are congruent: $\displaystyle A$ matches $\displaystyle D$ , $\displaystyle B$ matches $\displaystyle E$ , and $\displaystyle C$ matches $\displaystyle F$ .
$\displaystyle {AB}={5}$ cm, $\displaystyle {BC}={7}$ cm, $\displaystyle {\angle B}={50^\circ}$ .
How long are $\displaystyle DE$ and $\displaystyle EF$ , and how many degrees is $\displaystyle \angle E$ ? Explain.

\displaystyle DE

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\displaystyle EF

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\displaystyle \angle E

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Perimeter and area

A garden is a rectangle 12 m long and 7 m wide.

How many metres of fence does it take to go round it?

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What is its area, in m²?

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Find the area, in cm².

A parallelogram: a side of 9 cm, and the height to it 4 cm

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A triangle: a side of 10 cm, and the height to it 6 cm

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A trapezoid: the bases 8 cm and 5 cm, and the height 4 cm

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Here is a plan of a room’s floor. The measurements are in metres.

What is the floor’s area, in m²?

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What is its perimeter, in metres?

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A square’s perimeter is 36 cm.
What is its area, in cm²?

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The cube and the box

A box is 9 cm long, 5 cm wide and 4 cm high.

What is its volume, in cm³?

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What is its surface area, in cm²?

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A swimming pool is 25 m long, 10 m wide and 2 m deep.
How many litres of water does it take to fill it?

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A cube’s surface area is 54 cm².
What is its volume? Explain.

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Draw a net of a cube that is not the cross.
Imagine folding it, to check that it makes a cube.

Formulas: geometry, grade 7

Open as a page

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