Lines, angles, triangles and quadrilaterals
| English | Chinese | Pinyin |
|---|---|---|
| parallel | 平行 | píng xíng |
| perpendicular | 垂直 | chuí zhí |
| acute | 锐角 | ruì jiǎo |
| obtuse | 钝角 | dùn jiǎo |
| reflex | 优角 | yōu jiǎo |
| vertex | 顶点 | dǐng diǎn |
| equilateral | 等边 | děng biān |
| isosceles | 等腰 | děng yāo |
| scalene | 不等边 | bù děng biān |
| quadrilateral | 四边形 | sì biān xíng |
| parallelogram | 平行四边形 | píng xíng sì biān xíng |
| rhombus | 菱形 | líng xíng |
| trapezium | 梯形 | tī xíng |
The shapes around you
- Look at any room: the door is a rectangle, a table might be circular, roof trusses are triangles.
- Geometry is the mathematics of shape — and it starts with naming the basics precisely.
Shape and angle lab
Classify angle facts by the diagram feature that creates them.
Lines and angles
- Two lines are parallel 平行 if they never meet; perpendicular 垂直 if they meet at exactly $90^{\circ}$.
- Angles are named by size:
| Name | Size |
|---|---|
| acute 锐角 | less than $90^{\circ}$ |
| right | exactly $90^{\circ}$ |
| obtuse 钝角 | between $90^{\circ}$ and $180^{\circ}$ |
| reflex 优角 | between $180^{\circ}$ and $360^{\circ}$ |
- Name an angle with three letters: angle $ABC$ is the angle at vertex 顶点 $B$.
Reflex angles are easy to misread. If an angle is $250^{\circ}$, that's reflex — not $110^{\circ}$. Always check: the smaller angle and the reflex add to $360^{\circ}$.
Angles by size: acute (under 90°), right (90°, shown by a square), obtuse (90°–180°) and reflex (180°–360°)
The Taj Mahal has a clear line of symmetry down its centre
An angle of 40° is called:
Less than 90° is acute.
An angle of 250° is an obtuse angle.
250° is between 180° and 360°, so it is a reflex angle, not obtuse.
Triangles
- A triangle has three sides; its angles always add up to $180^{\circ}$.
- Equilateral 等边: all sides equal, all angles $60^{\circ}$.
- Isosceles 等腰: two equal sides, two equal angles.
- Scalene 不等边: all sides and angles different.
Angles $x$, $2x$, and $90^{\circ}$: $\;x + 2x + 90 = 180 \Rightarrow 3x = 90 \Rightarrow x = 30^{\circ}$.
The angle at the centre ($80^\circ$) is twice the angle at the circumference ($40^\circ$) on the same arc
A triangle has angles x, 2x and 90°. Find x (in degrees).
x + 2x + 90 = 180, so 3x = 90 and x = 30°.
An isosceles triangle has ______ equal sides.
An isosceles triangle has exactly two equal sides (and two equal angles).
Quadrilaterals 四边形
- A quadrilateral has four sides; its angles always add up to $360^{\circ}$.
- Key types: square (4 equal sides, 4 right angles), rectangle (4 right angles), parallelogram 平行四边形 (opposite sides parallel), rhombus 菱形 (4 equal sides), kite (2 pairs of adjacent equal sides), trapezium 梯形 (exactly 1 pair of parallel sides).
Like number families, shapes nest inside each other: every square is a rectangle, but not every rectangle is a square.
The four angles of a quadrilateral add up to how many degrees?
A quadrilateral can be split into two triangles: 2 × 180 = 360°.
Match each quadrilateral to its property.
A square has 4 equal sides and 4 right angles; a trapezium has one parallel pair; a kite has two adjacent equal pairs.
Why triangles are special
- Triangles are the only polygon that is always rigid — you can't change the shape without bending a side.
- That's why bridges, roof trusses, and crane booms are built from triangles.
Two theorems for both levels: the angle in a semicircle is $90^\circ$, and a tangent meets a radius at $90^\circ$
You've got it
- acute $<90^{\circ}$, right $=90^{\circ}$, obtuse $90$–$180^{\circ}$, reflex $180$–$360^{\circ}$
- triangle angles $= 180^{\circ}$; quadrilateral angles $= 360^{\circ}$
- name an angle by three letters — the middle letter is the vertex