iteatf o«f /Ac following:
a) 4a2 x 2a2
b) xy
c) 3^2xy
d) S = nr2
e) a = —
2"
f) J?
h) cos a = 4 cos . —3 cos —
3 3
i) 2 cos a = a j)x3-3x-a = 0 k) + V3
m)
n) V«s o)4'2
FOCUS B
Angles and Triangles
I. Angles
In plane geometry an angle is a figure which is formed by two straight lines which meet at a point. The lines of an angle are called the -sides. The point where they meet is called the vertex. When the sides of an angle are perpendicular to each other, they form a right angle. A right angle has ninety degrees. An angle of less than 90° is an acute angle, and an angle of more than 90* but less than 180* is an obtuse angle. An angle of more than 180* is a reflex angle.
Classification of angles according to their magnitude:
zero acute right obtuse straight reflex
(a = 0°)   (0*<a = <90*)   (a = 90°)   (90° < a = <180°)  (a = 180°) (180° < a = <360°)
/.   What kind of angle does a clock make at:
a) two o'clock?
b) three o'clock?
c) four o'clock?
d) twenty to ten?
e) twelve minutes past seven?
f) twenty-nine minutes past twelve?
2. Name the kinds of angles shown in these figures: a) b)
3. Are the following statements true or false?
a) The exterior angles of a triangle are always obtuse.
b) Only two angles of a triangle can be obtuse.
c) The smallest angle of a triangle is opposite the shortest side.
d) The point where the sides of an angle meet is called the vertex.
e) A triangle with two obtuse angles is called an obtuse triangle.
4. Describe the lines and angles in the following figures;
a) b)
5. Look at this and describe the figure:
E
F
6. Look at the figure above and read the following:
11. A triangle
A triangle is a three-sided figure. The three sides of a triangle meet at points called vertices (sg. vertex), the vertex at the top of a triangle may be called the apex, and the line at the bottom may be called the base. Triangles are often named according to their angles. A right-angled triangle has one right angle and two acute angles. An obtuse triangle has one obtuse and two acute angles. An acute triangle has three acute angles.
Classification of triangles according to their angles:
acute-angled right-angled obtuse-angled
(all the interior angles        (one interior angle is (one interior angle
are acute) a right angle) is obtuse)
Classification of triangles according to their sides:
scalene equilateral isosceles
(a A with no sides (a A with all sides (a A with two sides
of equal length) of equal length) of equal length)
Z AGH = Z EGB They are vertically opposite angles. (Vert. opp. Z s)
Z AGH = Z CHF They are corresponding angles. (Corr. Z s)
Z AGH = Z GHD They are alternate angles. (Alt. Z s)
Z AGH + Z AGE = 180°    They are adjacent angles on a straight line. (Adj. Z s) Z AGH + Z CHG = 180°    They are interior angles on the same side of the transverse
line (Int. Z s)
Z AGH = Z EAG + Z AEG The exterior angle of a triangle equals the sum of the interior
opposite angles (Ext. Z of A)
• Now make similar statements about EGB:
• Find other angles in the diagram which are equal and say why. If EB is equal and parallel to CF, compare A CHF and A EGB. Give reasons for what you say.
L Fill in the missing expressions:
a) If each of the angles in a triangle is equal to 60°, the triangle is called...............
b) If two angles of a triangle are equal to 45°, the triangle is called a............triangle.
c) If two lines meet at an angle of 90°, they are...................to each other.
d) Each triangle has three points, or...............
2.  Look and read, and fill in the missing expressions; A straight line drawn from any
...............of a triangle to the mid-point of
the............................is known as a
median of a triangle. The..............
where they meet is called the centroid, or the centre of gravity of the triangle.
III. A right-angled triangle 1. Look and read:
In a right-angled triangle the side c opposite the right angle is called the hypotenuse. A, B, C are vertices opposite the sides a, b, c. The line leading from the vertex C perpendicular to the hypotenuse is called an altitude of the triangle.
Complete this statement of Pythagoras's Theorem: The square of the.........is equal to the sum of the..............
Label the three sides of A ABC in relation to CAB, using the words hypotenuse, opposite and adjacent.
Complete the following:
opposite
a) tangent = -
adjacent
b) sine =
c) cosine =
d) tan CAB =
e) sin CAB =
f) cos CAB =
g) cotangent =
IV. Congruence, similarity and symmetry
4.
1. Read this:
Two triangles are congruent if the following parts are equal:
• two sides and the included angle; or
• a right angle, hypotenuse and side; or
• two angles and a corresponding side; or
• all three sides.
Two triangles are similar if they have their corresponding angles equal.
These two triangles are symmetrical, they are on either side of an axis of symmetry (or centre line).
2. Describe each triangle, and find any relationships between the triangles (i.e. symmetry,
similarity or congruence): a)    „ b) k c)
d)
S)
i)
FOCUS C
ARTICLES
I. Use of the indefinite article (sg.: a, an; pi.: 0) /. Instead of the number "one":
... for an object to be an element of Ar\B, it must be . ... look at the partial derivatives off at a point p ... In a paper on "Spaces of Type H<c + C"I learned...