Grade 8

Parallel lines and polygon angles: practice problems

Angle problems are about choosing the right property. Look at the figure and decide whether parallel lines, triangles or polygons are involved.

Basic, Standard, Advanced: Grade 8 · Term 2

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Parallel lines

Vertical angles are equal. When a line crosses two parallel lines, corresponding angles and alternate angles are equal.

Bent lines

For a bent line between parallel lines $\ell$ and $m$ , the angle at the bend equals the sum of the angles at $\ell$ and $m$ . Draw a line through the bend parallel to $\ell$ and use alternate angles.

Angles of a triangle

The angles of a triangle add up to $180^\circ$ , and an exterior angle equals the sum of the two opposite interior angles. The five tip angles of a star add up to $180^\circ$ , which follows from this property.

Polygons

Formulas

\text{interior angle sum of an } n\text{-gon} = 180^\circ \times (n - 2),\qquad \text{exterior angle sum} = 360^\circ

A regular polygon with interior angles of $140^\circ$ has exterior angles of $40^\circ$ , so it has $360 \div 40 = 9$ sides.

Q1BasicGr 8 · Term 2Parallel lines and polygon angles

Find the size of one exterior angle of a regular 10-gon.

Answer

36^\circ

Solution

The exterior angles add up to $360^\circ$ :
$360^\circ \div 10 = 36^\circ$

Q2StandardGr 8 · Term 2Parallel lines and polygon angles

A regular polygon has interior angles of $108^\circ$ . How many sides does it have?

Answer

5

Solution

One exterior angle is
$180^\circ - 108^\circ = 72^\circ$
The exterior angles add up to $360^\circ$ :
$360 \div 72 = 5$

Q3AdvancedGr 8 · Term 2Parallel lines and polygon angles

In the figure, $\ell \parallel m$ . Find the size of $\angle x$ .

Answer

\angle x = 105^\circ

Solution

Drawing parallel lines through each bend and using alternate angles, the angles opening to the right add up to the same as those opening to the left.
$36^\circ + 111^\circ = \angle x + 42^\circ$
Therefore
$\angle x = 36^\circ + 111^\circ - 42^\circ = 105^\circ$

Q1BasicGr 8 · Term 2Parallel lines and polygon angles

Find the size of $\angle x$ in the figure.

Q2StandardGr 8 · Term 2Parallel lines and polygon angles

In the 6-gon ABCDEF, $\angle \mathrm{A} = 126^\circ,\ \angle \mathrm{B} = 133^\circ,\ \angle \mathrm{C} = 128^\circ,\ \angle \mathrm{D} = 100^\circ,\ \angle \mathrm{E} = 138^\circ$ . Find $\angle \mathrm{F}$ .

Q3AdvancedGr 8 · Term 2Parallel lines and polygon angles

Find the size of $\angle x$ in the figure.

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Parallel lines and polygon angles: practice problems

Parallel lines

Angles of a triangle

Polygons

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Parallel lines

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Related topics

Triangles and quadrilaterals

The inscribed angle theorem