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Angles 3 – Circles/Euclidean Geometry
1.
Rotate the triangle ABC about O. What do you
notice? What do you notice about angle ABC?
2.
Rotate the diagram180° about 0 so that R R' and
S S'.
Show the
following.
a. R = R’
b. S = S’
c. R = S
B
A
C
O
3.
Find the marked angles in the diagrams.
54
e
c
4.
Prove that a + d = 90°
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
Euclidean Geometry
A study lamp has freely turning hinges at A, B, C, D, E,
F, G and H. ABCD and DEFG are parallelograms. Only
AB is rigidly fixed in position. ED is vertical.
1. If BAD = 46° find the sizes of the following. Give
reasons for your answer. a ADC b DCB c ABC.
2. Explain why FG is vertical.
3. BAD is changed to 58° by rotating the arm AD about
A. a Is ED still vertical? Explain.
b Is FG still vertical? Explain.
c Explain why the angle of the lamp remains at 21°
to the horizontal. Why is this design feature useful?
4. EDG is changed by rotating DG about D.
Does the lamp angle remain the same? Explain your answer.
6. Name the sets of parallel lines. Give reasons.
7.
Find the size of the marked angles for the following regular polygons. Give reasons. (O is
the centre of each polygon.)
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
8.
a
b
The diagram shows part of a regular polygon with n
sides (an n-gon).
Write down the size of the following angles. Give
reasons.
(i) y
(ii) z in terms of n
If y = 20°, how many sides does the polygon have?
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
2.
Measure the angles x and y and make a comment about the results.
3.
Predict, without measuring, the values of the marked angles. The diagrams are not
accurately drawn.
4.
Measure the angles x and y and make a comment about the results.
5.
Predict the size of the following marked angles.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
6.
Find the marked angles. Give reasons for your answers.
7.
Use the fact that angles in a semi-circle (angle off the diameter) equal 90 degrees e.g. PQR to
find the marked angles.
R
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
8. A sailor at Z can see two
landmarks X and Y 1 000 metres
apart. He measures XZY and finds
it is 44°.
a Explain why Z could be at any
place on the circle.
b When Z is on the mediator of
XY, calculate the value of b.
c Make a scale drawing (using
your answer in b) of the circle on
which Z is positioned.
Z
A ship's captain sailing along a coastline looks at a
chart and notices the following diagram. How should the
captain use the chart to ensure that he avoids the rocks?
9.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
Cyclic quadrilaterals and tangents
A, B, C and D all lie on a circle centre 0. ABCD is called a cyclic
quadrilateral.
QUESTIONS
Explain why
a. DOB (reflex) = 2x and DOB=2y. Why does 2x+2y= 360°?
b. Explain why x + y =180°
c. Will the result in b be true for all choices of x when 0 < x < 180°?
d. Does KCD = x? Explain.
1.
Calculate the marked angles. Give reasons.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
3. Which of these Venn diagrams are correct? Explain your answers. C = {cyclic
quadrilaterals}.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
4. Choose one word from the list to complete these statements. (Try drawing the quadrilaterals
first.)
a. A cyclic parallelogram is a
Parallelogram
b. A cyclic rhombus is a
Trapezium
c. A cyclic trapezium with at least one right angle is a
Square
Rectangle
Kite
Rhombus
5.
In which of the following is ABCD cyclic? Carefully explain your answers.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
10.
XY.YZ and XZ are tangents.
a. Find the marked angles.
Give reasons
b. Is AXYZ isosceles
11.
ABCDE is a regular
pentagon. Find the marked angles. Give
reasons.
12.
ABCDEF is a regular
hexagon. XY is a tangent. Find the
marked angles. Give reasons.
13.
AB is parallel to DC.
Find the marked angles. Give reasons.
14.
AB is a tangent. Find the marked
angles. Give reasons.
15.
Find the marked angles. Give
reasons.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
16.*
AT is a tangent. Find the marked angles. Give reasons.
17.* AT and DT are tangents. Find the marked angles.
Give reasons.
Proofs
1.
2.
a.
Prove x =18°.
If a + b = c, prove c = 90°.
3.
4.
Prove y = 20°.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Prove z = 60°.
Angles 3 – Circles/Euclidean Geometry
W
A
5.
A
Prove WX//AY
6.
7.
Prove ABC is isosceles.
Prove the triangle is equilateral.
Prove AOC = 2x + 2y .
8.
9.
XY is a tangent. Prove QRP = z.
10.
PT and QT are tangents. Prove POQT is a cyclic
quadrilateral.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
11.
Prove BDC =180° - 2q.
12.
Prove y = a + b.
13.
Prove y = z.
14.
Prove a = b.
15.
Prove that the angles of the triangle add up to 180°.
16.
The North Star is directly above the North
Pole. d is the latitude of a boat. a = angle of North
Star from the horizon.
a.
Prove a = d.
b.
What use is the result in (a) for
navigation?
c*.
How is the longitude of a boat's position
at sea in the Northern Hemisphere
17*.
a.
b.
AT is a tangent and x = y.
Prove that ABC is right-angled
Prove that AB is a diameter.
Making Sense with Mathematics – Murray Britt and Peter Hughes
Angles 3 – Circles/Euclidean Geometry
18.
CT is a tangent, AB is a diameter and x = y. Prove that CTB = 90°.
Making Sense with Mathematics – Murray Britt and Peter Hughes