Review: geometry, grade 8

Geometry, grade 8 Lesson 21 of 21Practice

Review: geometry, grade 8

The lines $\displaystyle AB$ and $\displaystyle CD$ are parallel, and the line $\displaystyle EF$ cuts them at the points $\displaystyle P$ and $\displaystyle Q$ .
Given: $\displaystyle {\angle APE}={65^\circ}$ .
Find $\displaystyle \angle BPQ$ , $\displaystyle \angle CQP$ and $\displaystyle \angle DQP$ . Give a reason for each angle.

\displaystyle \angle BPQ

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\displaystyle \angle CQP

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\displaystyle \angle DQP

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In triangle $\displaystyle ABC$ , the exterior angle at the vertex $\displaystyle C$ is $\displaystyle 130^\circ$ .
The angle $\displaystyle A$ is $\displaystyle 30^\circ$ larger than the angle $\displaystyle B$ .
Find the three angles of the triangle. Let $\displaystyle {\angle B}={x}$ , write an equation and solve it.

\displaystyle \angle A

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\displaystyle \angle B

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\displaystyle \angle C

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A rectangular sign is $\displaystyle 48$ cm wide and $\displaystyle 20$ cm high.
How long is the sign’s diagonal?

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These are the side lengths of triangles. Which of them is right-angled? Explain.

$\displaystyle 9$ , $\displaystyle 40$ , $\displaystyle 41$

$\displaystyle 6$ , $\displaystyle 7$ , $\displaystyle 9$

Triangle $\displaystyle A'B'C'$ is the image of triangle $\displaystyle ABC$ in a reflection in a line.
Given: $\displaystyle {AB}={7}$ , $\displaystyle {BC}={5}$ , $\displaystyle {\angle A}={35^\circ}$ .

How long are $\displaystyle A'B'$ and $\displaystyle B'C'$ , and how large is $\displaystyle \angle A'$ ? Explain.

\displaystyle A'B'

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\displaystyle B'C'

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\displaystyle \angle A'

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Are the triangles congruent? If so, write the congruence, with the letters in the right order.

The segments $\displaystyle AB$ and $\displaystyle CD$ cross at the point $\displaystyle O$ .
Given: $\displaystyle {AO}={OB}$ and $\displaystyle {CO}={OD}$ .
Prove: $\displaystyle {AC}={BD}$ .

Answer with the triangle inequality.

Is there a triangle with sides $\displaystyle 3$ , $\displaystyle 5$ and $\displaystyle 9$ ? Explain.

Two sides of a triangle are $\displaystyle 4$ and $\displaystyle 9$ , and the third side is a whole number.
What can the third side be? Write every possibility.

In triangle $\displaystyle ABC$ , given: $\displaystyle {AB}={AC}={25}$ and $\displaystyle {BC}={14}$ .
$\displaystyle AD$ is the median to the side $\displaystyle BC$ .

Explain why $\displaystyle AD\perp BC$ .

Work out $\displaystyle AD$ .

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In the isosceles triangle $\displaystyle ABC$ ( $\displaystyle {AB}={AC}$ ), the exterior angle at the apex $\displaystyle A$ is $\displaystyle 110^\circ$ .
Find the base angles $\displaystyle \angle B$ and $\displaystyle \angle C$ . Write an equation and solve it.

In triangle $\displaystyle ABC$ , the point $\displaystyle D$ is on the side $\displaystyle AB$ , the point $\displaystyle E$ is on the side $\displaystyle AC$ , and $\displaystyle DE\parallel BC$ .
Given: $\displaystyle {AD}={6}$ , $\displaystyle {DB}={3}$ , $\displaystyle {DE}={8}$ .
Find $\displaystyle BC$ . Write a proportion and solve it.

Formulas: geometry, grade 8

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Angles between parallel lines

Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
Between parallel lines: corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Between parallel lines: co-interior angles
$\displaystyle \gamma+\delta=180^{\circ}$

The angles of a triangle

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
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
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
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$

Pythagoras' theorem

In a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
Finding a leg – from the hypotenuse and the other leg
$\displaystyle a^2$ $\displaystyle =$ $\displaystyle c^2-b^2$

The sides of a triangle

The triangle inequality – two sides together are longer than the third
$\displaystyle a+b>c$
The third side – between the difference of the other two and their sum
$\displaystyle a-b<c<a+b$ $\displaystyle a\ge b$

Similar triangles

In similar triangles $\displaystyle \triangle ABC\sim\triangle DEF$ the corresponding sides are in one ratio – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{DE}{AB}=\frac{EF}{BC}=\frac{FD}{CA}=k$
The ratio of the perimeters – the perimeter grows $\displaystyle k$ times too
$\displaystyle P_{DEF}$ $\displaystyle =$ $\displaystyle k\cdot P_{ABC}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$