几何

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:

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 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 $y = 70^{\circ}$.

Worked example: two steps with reasons. Lines $PQ$ and $RS$ are parallel, with $P$ and $R$ on the same side. A transversal makes angle $PXY = 65^{\circ}$ at $X$ on $PQ$, where $Y$ is the point where the transversal meets $RS$. Find angle $XYS$ and the obtuse angle $XYR$.

Angle $XYS = 65^{\circ}$ (alternate angles are equal). Angle $XYR = 180^{\circ} - 65^{\circ} = 115^{\circ}$ (angles on a straight line add to $180^{\circ}$).

Each step gets its own reason, written with the correct name. "Because of the Z" or "parallel lines" alone does not earn the mark.

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

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 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 $180^{\circ} - 60^{\circ} = 120^{\circ}$.

Worked example: find the number of sides. Each interior angle of a regular polygon is $156^{\circ}$. How many sides does it have?

Each exterior angle is $180^{\circ} - 156^{\circ} = 24^{\circ}$, and the exterior angles add to $360^{\circ}$, so there are $360 \div 24 = 15$ sides.

Worked example: an irregular polygon. A pentagon has angles $100^{\circ}$, $110^{\circ}$, $120^{\circ}$, $x$ and $2x$. Find $x$.

The interior angles of a pentagon add to $(5 - 2) \times 180^{\circ} = 540^{\circ}$, so

• 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.

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 $90^{\circ}$.

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: $2 \times 40^{\circ} = 80^{\circ}$.

Worked example (both levels). $AB$ is a diameter of a circle and $C$ is a point on the circle. Angle $CAB = 35^{\circ}$. Find angle $ABC$.

Angle $ACB = 90^{\circ}$ (angle in a semicircle). Then angle $ABC = 180^{\circ} - 90^{\circ} - 35^{\circ} = 55^{\circ}$ (angles in a triangle add to $180^{\circ}$).

Worked example (both levels). $PT$ is a tangent to a circle with centre $O$, touching it at $T$. Angle $TOP = 62^{\circ}$. Find angle $OPT$.

Angle $OTP = 90^{\circ}$ (tangent meets radius at $90^{\circ}$), so angle $OPT = 180^{\circ} - 90^{\circ} - 62^{\circ} = 28^{\circ}$ (angles in a triangle).

Worked example (Extended). $ABCD$ is a cyclic quadrilateral with angle $DAB = 105^{\circ}$. Find angle $BCD$.

Angle $BCD = 180^{\circ} - 105^{\circ} = 75^{\circ}$ (opposite angles of a cyclic quadrilateral add to $180^{\circ}$).

Worked example (Extended: several steps). Two tangents from a point $P$ touch a circle with centre $O$ at $A$ and $B$. Angle $APB = 50^{\circ}$, and $C$ is a point on the major arc. Find angle $ACB$.

• Angles $OAP$ and $OBP$ are $90^{\circ}$ (tangent meets radius at $90^{\circ}$).
• Angle $AOB = 360^{\circ} - 90^{\circ} - 90^{\circ} - 50^{\circ} = 130^{\circ}$ (angles in a quadrilateral add to $360^{\circ}$).
• Angle $ACB = 130^{\circ} \div 2 = 65^{\circ}$ (angle at the centre is twice the angle at the circumference).

Worked example (Extended: alternate segment). A tangent touches a circle at $A$. The angle between the tangent and the chord $AB$ is $58^{\circ}$. $C$ is a point on the circle in the other segment. Then angle $ACB = 58^{\circ}$ (alternate segment theorem).

Common mistakes. Giving an angle with no reason, or with an informal reason such as "it looks like $90^{\circ}$"; using the angle-at-the-centre rule when the two angles do not stand on the same arc; and assuming a line is a diameter because it looks like one (the question must say so, or it must pass through $O$).

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.)

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 $40 \times \dfrac{27}{8} = 135\text{ cm}^{3}$.

Worked example: lengths (both levels). In triangle $ABC$, $D$ is on $AB$ and $E$ is on $AC$, with $DE$ parallel to $BC$. $AD = 4\text{ cm}$, $AB = 10\text{ cm}$ and $DE = 6\text{ cm}$. Find $BC$.

Triangles $ADE$ and $ABC$ are similar. (Extended reason: angle $A$ is shared, and angles $ADE$ and $ABC$ are equal because they are corresponding angles.) The scale factor is $\dfrac{AB}{AD} = \dfrac{10}{4} = 2.5$, so

Worked example: area to length (Extended). Two similar shapes have areas $20\text{ cm}^{2}$ and $45\text{ cm}^{2}$. The smaller is $8\text{ cm}$ tall. How tall is the larger?

The area ratio is $20 : 45 = 4 : 9$, so the length ratio is $\sqrt{4} : \sqrt{9} = 2 : 3$. The larger is $8 \times \dfrac{3}{2} = 12\text{ cm}$ tall.

Common mistakes. Multiplying an area by the length scale factor $k$ (use $k^{2}$ for area and $k^{3}$ for volume), and matching the wrong pairs of sides. Match sides that sit opposite equal angles.

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 $180^{\circ}$: $025^{\circ} + 180^{\circ} = 205^{\circ}$.

• 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 $5\text{ m}$.

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 a convex polygon, one at each vertex, 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. $072°$).