Geometry

A corner where two lines meet is a vertex 顶点. Two lines are parallel 平行 if they never meet, and perpendicular 垂直 if they meet at a right angle. Angles are named by their size:

Angles by size: acute (under $90^\circ$), right ($90^\circ$, shown by a square), obtuse ($90^\circ$–$180^\circ$) and reflex ($180^\circ$–$360^\circ$).

We name an angle with three letters, e.g. angle $ABC$ is the angle at $B$.

Learn these basic facts. Each one is a valid reason in the exam.

Angles around a point add up to 360 degrees

Angles on a straight line add up to 180 degrees

• Angles at a point add up to $360^{\circ}$.
• Angles on a straight line add up to $180^{\circ}$.
• Vertically opposite angles 对顶角 (made by two crossing lines) are equal.

Worked example. Three angles on a straight line are $x$, $50^{\circ}$ and $70^{\circ}$. Find $x$.

When a line crosses two parallel lines:

• Corresponding angles 同位角 (in matching positions, an "F" shape) are equal.
• Alternate angles 内错角 (opposite sides of the crossing line, a "Z" shape) are equal.
• Co-interior angles 同旁内角 (a "C" shape) add up to $180^{\circ}$; we say they are supplementary 互补.

Worked example. A straight line crosses two parallel lines. One angle is $110^{\circ}$. The co-interior angle $y$ satisfies $110 + y = 180$, so $y = 70^{\circ}$.

Corresponding (F) and alternate (Z) angles are equal; co-interior (C) angles add to $180^\circ$.

A triangle 三角形 has three sides and angles that add up to $180^{\circ}$.

The three angles of a triangle add up to 180 degrees

Worked example. A triangle has angles $x$, $2x$ and $90^{\circ}$. Find $x$.

A quadrilateral 四边形 has four sides and angles that add up to $360^{\circ}$.

A honeycomb tessellates the plane with regular hexagons.

A polygon 多边形 is a shape with straight sides. A regular polygon 正多边形 has all sides and all angles equal.

For a polygon with $n$ sides:

The interior angle 内角 and the exterior angle 外角 at each corner add up to $180^{\circ}$.

Worked example. Find each interior angle of a regular hexagon.

The exterior angle is $\dfrac{360^{\circ}}{6} = 60^{\circ}$, so each interior angle is $180^{\circ} - 60^{\circ} = 120^{\circ}$.

At each corner the interior and exterior angles add to $180^\circ$; a regular hexagon has $60^\circ$ exterior and $120^\circ$ interior angles.

The Taj Mahal has a clear line of symmetry down its centre.

• A shape has line symmetry 轴对称 if a mirror line splits it into two matching halves.
• A shape has rotational symmetry 旋转对称 if it fits onto itself as you turn it. The order is how many times it fits in one full turn.

For solids (Extended), a flat slice that splits the solid into mirror halves is a plane of symmetry 对称面, and a line you can spin it around is an axis of symmetry 对称轴.

The main parts of a circle: a chord joins two points, while a tangent touches at just one.

A sector is a slice between two radii, a segment is cut off by a chord, an arc is part of the circumference, and a semicircle is half the circle.

Use these to find unknown angles, always giving the reason.

For both levels:

• The angle in a semicircle is $90^{\circ}$.
• The angle between a tangent and a radius is $90^{\circ}$.

Two theorems for both levels: the angle in a semicircle is $90^\circ$, and a tangent meets a radius at $90^\circ$.

Extended (theorems I):

• The angle at the centre is twice the angle at the circumference (standing on the same arc).
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral 圆内接四边形 add up to $180^{\circ}$.
• The alternate segment theorem 弦切角定理: the angle between a tangent and a chord equals the angle in the other segment.

Extended (theorems II): equal chords are the same distance from the centre; the perpendicular bisector of a chord passes through the centre; two tangents from the same outside point are equal in length.

Worked example. A, B, C are on a circle. The angle at the circumference $ABC$ is $40^{\circ}$. Find the angle $AOC$ at the centre $O$.

The angle at the centre is twice the angle at the circumference: $2 \times 40^{\circ} = 80^{\circ}$.

The angle at the centre ($80^\circ$) is twice the angle at the circumference ($40^\circ$) standing on the same arc.

Two shapes are similar 相似 if one is an enlargement of the other: same angles, and all sides multiplied by the same scale factor 比例因子 $k$. (Shapes that are exactly the same size and shape are congruent 全等.)

Nesting dolls are similar shapes: each one is the same shape as the others, just multiplied by a scale factor

For similar shapes and solids:

Worked example. Two similar solids have lengths in the ratio $2 : 3$. The smaller has volume $40\text{ cm}^{3}$. Find the volume of the larger.

The volume ratio is $2^{3} : 3^{3} = 8 : 27$. So the larger volume is $40 \times \dfrac{27}{8} = 135\text{ cm}^{3}$.

A bearing 方位角 gives a direction as a three-figure angle, measured clockwise 顺时针 from north, from $000^{\circ}$ to $360^{\circ}$. So due east is $090^{\circ}$ and due south is $180^{\circ}$.

Worked example. The bearing of $B$ from $A$ is $025^{\circ}$. Find the bearing of $A$ from $B$.

The return (back) bearing differs by $180^{\circ}$: $025^{\circ} + 180^{\circ} = 205^{\circ}$.

A bearing is measured clockwise from north as three figures; the back bearing differs by $180^\circ$.

• To construct 作图 a triangle from three given sides, draw the base with a ruler, then use a pair of compasses 圆规 to mark each other side as an arc. Leave the construction arcs showing.
• A net 展开图 is a flat shape that folds up into a solid. You can use a net to work out surface areas.
• A scale drawing 比例图 shows a real object smaller (or larger) by a fixed scale, such as $1\text{ cm}$ to $5\text{ m}$.

Unfolding a solid gives its net. Add the areas of the pieces to get the surface area: a cylinder is $2\pi r^{2} + 2\pi r h$.

Common solids 几何体 and their parts:

A flat side of a solid is a face 面, two faces meet at an edge 棱, and the whole outside is its surface 表面.

• Angles on a straight line add to $180°$, around a point to $360°$, and in a triangle to $180°$. Give a reason for every step of an angle question.
• Learn the circle theorems: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is $90°$; angles in the same segment are equal; opposite angles of a cyclic quadrilateral add to $180°$.
• The exterior angles of any polygon add to $360°$, and each interior angle + its exterior angle = $180°$.
• Bearings are measured clockwise from north and always written with three figures (e.g. $072°$).