Year 8 Angle relationships and properties of geometrical figures: free practice test with answers

Sample angle relationships and properties of geometrical figures questions with answers

Try each one before you open the answer.

1. Reviewing angles

   Find the complement and the supplement of an angle of $38^\circ$.

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   Key idea: Complementary angles add to $90^\circ$; supplementary angles add to $180^\circ$.

   1. Complement: $90^\circ - 38^\circ = 52^\circ$
   2. Supplement: $180^\circ - 38^\circ = 142^\circ$

   Answer: Complement $52^\circ$, supplement $142^\circ$

2. Transversal lines and parallel lines

   Two parallel lines are cut by a transversal, forming eight angles. One of the angles is $72^\circ$. Find the sizes of all eight angles.

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   Key idea: With parallel lines, corresponding, alternate and vertically opposite angles are equal, and angles on a straight line add to $180^\circ$.

   1. At each crossing point, the angle vertically opposite the $72^\circ$ angle is $72^\circ$ and the other two angles are $180^\circ - 72^\circ = 108^\circ$
   2. The lines are parallel, so the corresponding angles at the other crossing point are the same: two of $72^\circ$ and two of $108^\circ$
   3. Altogether four angles are $72^\circ$ and four are $108^\circ$

   Answer: Four angles are $72^\circ$ and four angles are $108^\circ$

3. Triangles

   Two angles of a triangle are $47^\circ$ and $68^\circ$.

   (a) Find the third angle.

   (b) Is the triangle acute, right or obtuse?

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   Key idea: The angles of a triangle add to $180^\circ$.

   1. (a) $180^\circ - 47^\circ - 68^\circ = 65^\circ$
   2. (b) All three angles ($47^\circ$, $68^\circ$, $65^\circ$) are less than $90^\circ$, so the triangle is acute

   Answer: (a) $65^\circ$ (b) Acute

4. Quadrilaterals

   A quadrilateral has three known interior angles of $112°$, $75°$ and $96°$. Find the fourth interior angle.

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   Key idea: The interior angles of any quadrilateral add to $360°$.

   1. Write the angle sum equation:\

   $$112 + 75 + 96 + x = 360$$

   2. Add the three known angles:\

   $$283 + x = 360$$

   3. Solve for $x$:\

   $$x = 360 - 283 = 77$$

   Answer: The fourth angle is $77°$.

5. Polygons

   Find the angle sum of an octagon.

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   Key idea: The angle sum of a polygon with $n$ sides is $S = (n - 2) \times 180^\circ$.

   1. An octagon has $n = 8$ sides
   2. $S = (8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ$

   Answer: $1080^\circ$

6. Three-dimensional solids and Euler's formula

   A pentagonal prism has two pentagonal faces joined by rectangles.

   (a) Find the number of faces ($F$), vertices ($V$) and edges ($E$).

   (b) Show that Euler's formula $F + V - E = 2$ holds.

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   Key idea: A prism with an $n$-sided base has $n + 2$ faces, $2n$ vertices and $3n$ edges.

   1. (a) Faces: $2$ pentagons $+ 5$ rectangles $= 7$
   2. (a) Vertices: $5$ on each pentagon, so $10$
   3. (a) Edges: $5$ on each pentagon plus $5$ joining them, so $15$
   4. (b) $F + V - E = 7 + 10 - 15 = 2$ ✓

   Answer: (a) $F = 7$, $V = 10$, $E = 15$ (b) $7 + 10 - 15 = 2$

7. Three-dimensional coordinate systems

   Point $P$ has coordinates $(3, 5, 2)$. Write down its $x$, $y$ and $z$ coordinates.

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   Answer: $x = 3$, $y = 5$, $z = 2$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 8 properties of quadrilaterals worksheet