Practice: angles

Geometry, grade 8 Lesson 3 of 21Practice

Practice: angles

1

In the figure, $\displaystyle AB\parallel CD$ , and the line $\displaystyle EF$ cuts them.
Find $\displaystyle \alpha$ , $\displaystyle \beta$ and $\displaystyle \gamma$ , and give the reasons.

\displaystyle \alpha

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\displaystyle \beta

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\displaystyle \gamma

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2

A line cuts two parallel lines.
Two co-interior angles are $\displaystyle 3x+20$ and $\displaystyle 2x-15$ degrees.
Find $\displaystyle x$ and the size of the two angles.

\displaystyle x

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\displaystyle 3x+20

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\displaystyle 2x-15

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3

In the figure, $\displaystyle AB\parallel CD$ .
The point $\displaystyle P$ is on $\displaystyle AB$ , and the points $\displaystyle Q$ and $\displaystyle R$ are on $\displaystyle CD$ .
$\displaystyle {\angle APQ}={63^{\circ}}$ and $\displaystyle {\angle RPB}={53^{\circ}}$ .
Find the three angles of triangle $\displaystyle PQR$ , and give the reasons.

\displaystyle \angle PQR

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\displaystyle \angle PRQ

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\displaystyle \angle QPR

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4

In the figure, $\displaystyle AB\parallel CD$ .
$\displaystyle {\angle BME}={27^{\circ}}$ and $\displaystyle {\angle MEN}={56^{\circ}}$ .
Find $\displaystyle \angle END$ , and give the reasons.

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5

In triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={2x}$ and $\displaystyle {\angle B}={3x}$ degrees.
The exterior angle at the vertex $\displaystyle C$ is $\displaystyle 110^{\circ}$ .
Find $\displaystyle x$ and the three angles of the triangle.

\displaystyle x

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\displaystyle \angle A

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\displaystyle \angle B

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\displaystyle \angle C

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6

In triangle $\displaystyle ABC$ , $\displaystyle {\angle B}={40^{\circ}}$ .
The exterior angle at the vertex $\displaystyle C$ is twice the angle $\displaystyle A$ .
Find $\displaystyle \angle A$ and $\displaystyle \angle ACB$ .

\displaystyle \angle A

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\displaystyle \angle ACB

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7

The angles of a triangle are in the ratio $\displaystyle 2:3:4$ .
Find the angles of the triangle.
Find the exterior angle at the vertex of the largest angle.

The smallest angle

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The middle angle

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The largest angle

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The exterior angle

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8

At one vertex of a triangle there is an exterior angle of $\displaystyle 80^{\circ}$ , and at another vertex an exterior angle of $\displaystyle 90^{\circ}$ .
Is there such a triangle? Explain.

Formulas: geometry, grade 8

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
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$