Angle Geometry
EXERCISE 1
ORAL EXAMPLES: Identify the following angles by a, b, c etc.Q
P c d
b
h e
g f
R
Note: For, O is called the vertex of the angle and OA, OB are called the arms of the angle. The size of the angle is measured by the amount of turning it takes to get from one arm to the other.
For example: / e.g. is c; is f.
1. 2.
3. 4.
5. 6.
a full turn.
a half turn, which forms a straight angle.
a quarter turn, which forms a right angle. /
These diagrams show how one full turn is made up of 2 straight angles and 4 right angles.
WRITTEN EXAMPLES:1. Copy down and complete this table.
Number
of Turns / Number of
Straight Angles / Number of
Right Angles / Number of
Degrees
e.g. 1 / 2 / 4 / 360
a. /
b. /
c. / 1
d. / 2
e. /
f. / 3
Note: Unless stated otherwise angles will be measured in degrees.
Note:
An acute angle is smaller than a right angle.
An obtuse angle is greater than a right angle but less than a straight angle.
A reflex angle is greater than a straight angle but less than a full turn.
2. / Copy down and complete these statements.a.
b.
c.
d.
e. / An acute angle is less than degrees.
A right angle is equal to degrees.
An obtuse angle is between degrees and degrees.
A straight angle is equal to degrees.
A reflex angle is between degrees and degrees.
3. / Copy down and complete this table by listing these angle sizes in the correct column.
20o, 200o, 100o, 300o, 50o, 150o, / Acute / Obtuse / Reflex
250o, 350o, 75o, 175o, 275o, 85o,
185o, 285o, 95o, 295o, 55o, 355o,
4. / For each of the angles drawn below;
i.
ii. / state whether it is acute, obtuse or reflex.
estimate its size choosing from 30, 60, 120, 150, 210, 240, 300, 330.
Note: 1. Two angles which add up to 90are said to be complementary. (e.g. 40and 50).
Two angles which add up to 180are said to be supplementary. (e.g. 40and 140).
2. 50o is the complement of 40o (and vice-versa).
140o is the supplement of 40o (and vice-versa).
5. / Copy down and complete this table.Angle / Complement / Supplement / Angle / Complement / Supplement
a. / 30o / f. / 20o
b. / 60o / g. / 55o
c. / 45o / h. / 160o
d. / 25o / i. / 125o
e. / 75o / j. / o
6. / Copy down and complete the table below from the diagram. (The first one has been done for you - all points are equally spaced around the circle centre O). /
Description of Turn / Name of Angle / Size of Angle for Anticlockwise Turn / Size of Angle for Clockwise Turn
e.g. / From OA to OB / AOB (or BOA) / 45o / 315o
a. / From OA to OC
b. / From OA to OD
c. / From OA to OE
d. / From OA to OF
e. / from OA to OG
f. / From OA to OH
Note: The size of an angle is always taken to be that of the smaller turn (i.e. the acute or obtuse angle rather than the reflex angle) unless stated otherwise.
7. / Copy down and complete the table below from the diagram.
(The first one has been done for you - all points are equally spaced around the circle centre O). /
Name of Angle / Size of Angle / Name of Angle / Size of Angle
e.g. / POQ / 30o / VOQ / 150o
a.
b.
c.
d. / POR
POS
POT
POU / VOR
VOS
VOT
VOU
8.
Note: / This is a diagram of a protractor. O is called the centre of the protractor and PQ is called its base line.
The protractor has 2 scales on it. When measuring the size of POA, POB, POC, POD use the scale which starts from zero at P. When measuring the size of QOA, QOB, QOC, QOD use the scale which starts from zero at Q. /
Copy down and complete this table from the diagram above.
Name of Angle / Size of Angle / Name of Angle / Size of Angle
e.g. POA
a. POB
b. POC
c. POD / 40o / QOA
QOB
QOC
QOD / 140o
Note: / When using a protractor carry out these steps;
Step 1: Put the centre of the protractor on the vertex of the angle.
Step 2: Put the base line of the protractor on one arm of the angle.
Step 3: Choose the correct scale and read the number of degrees at the other arm of the angle.
9. / Use your protractor to measure the size of these angles.
EXERCISE 2
WRITTEN EXAMPLES: In each question
a. use your protractor to measure the size of the marked angles.
b. find the sum of these angles in each question.
c. check that the sum of the angles are within or of either or .
Copy down and complete these statements;
9. The adjacent angles at a point (such as those in questions 1 and 2) add up to .
10. The adjacent angles on a line (such as those in questions 3 and 4) add up to .
11. The angles of a triangle add up to .
12. The angles of a quadrilateral add up to .
EXERCISE 3
The previous Exercise showed how we can have the same results, but for different reasons. Hence reasons are often required for results in geometry. They are usually given in an abbreviated form as shown in the following table.
Copy down this table.
Diagram / Result / Reason/ a + b + c + d = 360 / (adj 's at a point)
/ a + b + c = 180 / (adj 's on a line)
Note: In Exercises where you have to calculate the size of angles the diagrams are not drawn to scale and so no measurements are to be taken from them.
WORKED EXAMPLE: Calculate the size of the angle marked o
/ + 85 + 125 + 90 = 360 (adj. 's at a point)+ 300 = 360
= 60
WRITTEN EXAMPLES: Calculate the size of the marked angles. Give reasons for your results.
EXERCISE 4
In a similar way it can be shown that z = 50. And so when 2 straight lines cut the "opposite" angles are equal. There are many situations in Geometry where angles are equal - these angles are said to be equal because they are vertically opposite angles. (i.e. opposite angles at the same vertex.).
Copy down this table.
WORKED EXAMPLE: Calculate the size of , y and z.
WRITTEN EXAMPLES: Calculate the size of the marked angles.
Give reasons for your results.
EXERCISE 5: Copy down this table.
WORKED EXAMPLE: Calculate the size of and y.
WRITTEN EXAMPLES: Calculate the size of the marked angles.
Give reasons for your results.
EXERCISE 6:
Note: If two lines AB and CD are cut by a third line (called a transversal) then there are
(i) 4 pairs of corresponding angles namely p and x, q and y, r and z,
s and w.
(ii) 2 pairs of alternate angles namely r and x, s and y.
(iii) 2 pairs of cointerior angles namely r and y, s and x.
ORAL EXAMPLES: In these diagrams state the pairs of a) corresponding angles.
b) alternate angles.
c) cointerior angles.
1. 2. 3.
b a p h
c d q s
f
a x
d y z
f c
g h
Note: If the 2 lines AB and CD are parallel (indicated by the arrows) then the corresponding angles are equal.
i.e. p = x, q = y, r = z, s = w.
(AB is parallel to CD is written as AB // CD.)
State the size of the marked angles - no reasons are required.
Note: Questions 8,9 and 10 could be regarded as written examples.
ii. / c = e
d = f
d + e = 180
c + f = 180 /
/ i.e. alternate angles are equal
i.e. cointerior angles are supplementary
Copy down this table.
Diagram / Result / Reason/ a = b
b = c
b + d = 180 / (corres 's PQ//RS)
(alt 's PQ//RS)
(coint 's PQ//RS) / Note: If there is only one pair of parallel lines on the diagram then there is no need to name them in the reason. If there are more than one pair of parallel lines on the diagram then the appropriate pair must be identified in the reason.
In questions 9 and 10 state
a) the pairs of corresponding angles which are equal.
b) the pairs of alternate angles which are equal.
c) the pairs of cointerior angles which are supplementary.
WRITTEN EXAMPLES: Calculate the size of the marked angles.
Give reasons for your results.
REVISION EXERCISE
1. Copy down and complete these statements.
a. i. 1 right angle = turn = degrees.
ii. 1 straight angle = turn = degrees.
b. i. The complement of 40o is .
ii. The supplement of 40o is .
c. i. If an angle is acute its supplement is .
ii. If an angle is equal to its supplement is then the angle is .
2. a. What is the maximum number of obtuse angles a triangle can have?
b. What is the maximum number of obtuse angles a quadrilateral can have?
c. What is the maximum number of reflex angles a quadrilateral can have?
d. What is the minimum number of obtuse angles a quadrilateral must have if it has no right angles?
Calculate the size of the marked angles giving reasons for your results.
TEST EXERCISE 40 marks
1. Complete these statements.
(a) Half a turn =______right angles = ______degrees.
(b) (i) The complement of is ______.
(ii) The supplement of is ______.
2. Two angles are labelled as “x” and “y” and x is an obtuse angle.
What can be said about y if (i) Answer: y is ______.
(ii) Answer: y is ______.
3. The diagram shows any quadrilateral ABCD with its opposite sides produced to cut at E and F.
(a) The name of the angle marked w is ÐEBF (or ÐFBE but we regard these names to be the same). Write down 4 different ways of naming the angle marked x.
Answers:______(4).
(b) Use your protractor to find the size of the angles marked as x, y, z, and w.
(i) x =______(ii) y =______(iii) z =______(iv) w =______
4. Calculate x, y, z, and w giving reasons for your answers.
5. Calculate p, q, and r giving reasons for your answers.
6. Calculate a, b, c, d, and e giving reasons for your answers.
EXTRA EXERCISE
Find a, b and c, if
a + b = 100 and
a + c = 145 / b. /
Find a if y + z = 250 / 2. /
Prove that .
3. / Given that PQR = 90o and PSR = 90o,
prove that b + c = a. /
4. /
Given that a + b = 180
prove that e = c. /
6. Note 1. The exterior angles of a polygon are formed by producing its sides when going around the polygon in either a clockwise or an anticlockwise direction. Thus in the diagram for (a) below the exterior angles x, y and z are formed by going around the triangle in a clockwise direction whereas in the diagram for (b) below the exterior angles p, q, r and s are formed by going around the quadrilateral in an anti-clockwise direction.
2. The interior and exterior angles at each vertex are supplementary
(i.e. , , etc).
/ a. Prove that the exterior angles of a triangle add up to 360o
(i.e.
/ b. Prove that the exterior angles of a quadrilateral add up to 360o
(i.e.
Note: You may now assume that the exterior angles of all polygons add up to 360o.
c. Consider a polygon with n sides (i.e. an n-gon).
Copy down and complete these statements.
i. The sum of its exterior angles = ? rt
ii. The sum of the interior and exterior angles at each vertex = ? rt
iii. An n-gon has n sides and so it has ? vertices.
iv. Hence the sum of all the interior and exterior angles in the n-gon = ? ? rt .
v. Hence the sum of all the interior angles in the n-gon = (? ? -? ) rt.
7. / In this question you will find the interior angle sum of polygons given that the interior angle sum of a triangle is 2 rt 's (i.e. 180o).
e.g. the interior angle sum of a quadrilateral (or 4-gon) can be found by
either drawing in one diagonal to divide
the quadrilateral into 2 triangles as shown.
Sum of interior angles of a
quadrilateral = 2 x 2
= 4 rt 's
(2 triangles each with 2 rt 's) / or taking any point inside the quadrilateral
to divide it into 4 triangles as shown.
Sum of interior angles of a
quadrilateral = 4 x 2 - 4 (= 8 - 4)
= 4 rt 's
(4 triangles each with 2 rt 's but take off the
4 rt 's at the point inside the quadrilateral)
a. / i. the interior angle sum of a pentagon (or 5-gon) can be found by
either drawing in 2 diagonals to divide
the pentagon into 3 triangles as shown.
Copy down and complete these statements.
Sum of interior angles of a
pentagon = 3 x .
= rt 's / or taking any point inside the pentagon
to divide it into 5 triangles as shown.
Copy down and complete these statements.
Sum of interior angles of a
pentagon = 5 x - .
= rt 's
ii. the interior angle sum of a hexagon (or 6-gon) can be found by
either drawing in 3 diagonals to divide
the hexagon into 4 triangles as shown.
Copy down and complete these statements.
Sum of interior angles of a
hexagon = .
= rt 's / or taking any point inside the hexagon
to divide it into 6 triangles as shown.
Copy down and complete these statements.
Sum of interior angles of a
hexagon = .
= rt 's
b. / Copy down and complete the following table to summarize and extend your results in (a).
Name of / Number / Sum of interior angles (in rt 's)
polygon / of sides / either / or
quadrilateral (or 4-gon)
pentagon (or 5-gon)
hexagon (or 6-gon)
heptagon (or 7-gon)
octagon (or 8-gon)
nonagon (or 9-gon)
decagon (or 10-gon)
20-gon
n-gon / 4
5
6
7
8
9
10
20
n / 2 x 2
3 x 2
.
.
.
.
.
.
. / 4 x 2 - 4
5 x 2 - 4
.
.
.
.
.
.
. / 4
6
.
.
.
.
.
.
Check that your results for an n-gon are true for a triangle (i.e. a 3-gon).
c. / Use your results in b. to help you find
i. the sum of the interior angles of a 50-gon.
ii. the number of sides a polygon has if the sum of its interior angles is 50 rt 's.
8. Note: 1. Three or more points are said to be collinear if they all lie on the same straight line.