G1, G4 - Geometrical language, triangles and quadrilaterals

Points, lines and geometrical notation

If a square is drawn like a diamond, has its shape changed? Its position has changed, but its equal sides and right angles have not. Geometry lets you describe those facts precisely.

A point is a position with no length or width. Two distinct points determine a straight line, which continues indefinitely both ways. A line segment has two endpoints; a ray has one endpoint and continues one way. $AB$ names the segment joining $A$ and $B$, or its length in a calculation. A plane is a flat surface extending indefinitely.

A vertex is a corner; its plural is vertices. An edge of a solid is where two faces meet. A cube has 8 vertices and 12 edges: a crossing in a perspective drawing is not automatically another vertex.

Matching arrow marks show parallel sides; matching ticks show equal lengths. A small square shows a right angle. These marks give information even when the sketch is not to scale. An arrow at a line's end instead tells you that the line continues.

In $\triangle ABC$, the middle letter of $\angle ABC$ gives its vertex, $B$. Its arms are $BA$ and $BC$; $\angle CBA$ names the same angle. Lower-case side letters lie opposite the matching capitals: $a=BC$, $b=AC$, $c=AB$. This convention still works when the triangle is turned.

[DIAGRAM: asset_name: Triangle notation and geometrical marks; asset_slug: 1ma1_m45_notation; caption: The middle letter identifies an angle vertex, while lower-case side labels lie opposite matching capital vertices.; description: A triangle ABC has side a on BC, b on AC and c on AB, with angle ABC marked at B. Separate examples show matching parallel arrows, equal-length ticks and a right-angle square.]

For instance, “$AB=AC$ and $D$ lies on $BC$” tells you to mark two equal sides meeting at $A$. It does not tell you that $D$ is the midpoint. Keep given facts separate from things that merely look plausible.

Try reading the middle letter and the opposite-side convention separately.

Special triangles and plane figures

A triangle can have two useful descriptions at once: a right-angled isosceles triangle has both a right angle and two equal sides.

An equilateral triangle has three equal sides and three equal angles, each $60^\circ$. An isosceles triangle has two equal sides; the angles opposite them, its base angles, are equal. A scalene triangle has three different side lengths. Equal sides imply equal opposite angles; the congruence proof comes in the lesson on geometrical arguments.

An acute-angled triangle has all angles below $90^\circ$. A right-angled triangle has one angle of $90^\circ$; the opposite side, the hypotenuse, is longest. An obtuse-angled triangle has one angle above $90^\circ$. Two obtuse angles would already exceed a triangle's total.

The next lesson derives this result. Here it helps turn a shape definition into an angle calculation.

Change only the apex angle from $44^\circ$ to $52^\circ$. The equal-side information, and therefore the method, stays the same.

If a base angle is given instead, use it twice: base angles of $52^\circ$ leave apex angle $180^\circ-52^\circ-52^\circ=76^\circ$.

A polygon is a closed plane shape made of straight sides. Triangles have 3 sides, quadrilaterals 4, pentagons 5, hexagons 6, heptagons 7, octagons 8, nonagons 9 and decagons 10. A circle is a plane figure but not a polygon. A convex polygon has every interior angle below $180^\circ$; a concave polygon has an inward corner with a reflex interior angle.

Now change only the apex angle again.

Quadrilateral definitions and consequences

Definitions give a reliable way to recognise a quadrilateral, including when its diagonals or angles are the useful clues.

A parallelogram has both pairs of opposite sides parallel. A diagonal cuts it into two matching triangles: alternate angles agree and the diagonal is shared. This explains its equal opposite sides. Opposite angles are equal, adjacent angles total $180^\circ$, and its diagonals bisect one another, meeting at each diagonal's midpoint. They need not be equal or perpendicular.

A rectangle is a parallelogram with four right angles. Its diagonals bisect one another and are equal: the right triangles built from a length, a width and a diagonal match. Four equal sides are not required.

A rhombus is a parallelogram with four equal sides. Its diagonals bisect each other at right angles and bisect its vertex angles. To see why, let them meet at $O$ in $ABCD$. Triangles $AOB$ and $AOD$ have $AB=AD$, $OB=OD$ and common $AO$. Matching these triangles gives equal adjacent angles at $O$, so each is $90^\circ$, and equal angles at $A$. Repeat at the other vertices. The diagonals need not be equal.

A square is both a rectangle and a rhombus. It has four equal sides and four right angles, so it has both sets of diagonal properties. These quadrilateral properties are not on the formula sheet: learn them.