Angles 3 – Circles/Euclidean Geometry
BA / O / C
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
3. Find the marked angles in the diagrams.
4. Prove that
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.)
8. The diagram shows part of a regular polygon with n sides (an n-gon).
a Write down the size of the following angles. Give reasons.
(i) y (ii) z in terms of n
b If , how many sides does the polygon have?
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.
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. ÐPQR to find the marked angles.
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
9. 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?
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
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.
3. Which of these Venn diagrams are correct? Explain your answers. C = {cyclic quadrilaterals}.
4. Choose one word from the list to complete these statements. (Try drawing the quadrilaterals first.)
Parallelogram
Trapezium
Square
Rectangle
Kite
Rhombus
a. A cyclic parallelogram is a
b. A cyclic rhombus is a
c. A cyclic trapezium with at least one right angle is a
5. In which of the following is ABCD cyclic? Carefully explain your answers.
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.
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. a. If , prove c = 90°.
2. Prove . 3. Prove .
4. Prove .
5. Prove WX//AY
6. Prove DABC is isosceles.
7. Prove the triangle is equilateral.
8. Prove ÐAOC .
9. XY is a tangent. Prove QRP = z.
10. PT and QT are tangents. Prove POQT is a cyclic quadrilateral.
11. Prove .
12. Prove .
13. Prove .
14. Prove .
15. Prove that the angles of the triangle add up to .
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*. AT is a tangent and .
a. Prove that DABC is right-angled
b. Prove that AB is a diameter.
18. CT is a tangent, AB is a diameter and . Prove that CTB = 90°.
Making Sense with Mathematics – Murray Britt and Peter Hughes