# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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The phrase “angle of inclination” is also used to describe the angle between two planes.

Reference: Angle of Depression, Angle of Elevation, Plane.

ANGLE SUM OF A TRIANGLE

This is a geometry theorem, which states that the three angles of a triangle add up to 180°. Alternatively, this can be expressed as follows: The three angles of a triangle are supplementary.

30 ANGLE SUM OF A TRIANGLE

This geometry theorem can be demonstrated by the following experiment: Draw a triangle on a sheet of paper and using scissors, carefully cut it out. Tear off each of the three angles A, B, and C and rearrange them as shown in figure a and you will discover that they form a straight line, which is an angle of 180°.

Example. Figure b shows the gable end of a house with an angle at the vertex of 136°. Find the angle that the roof makes with the horizontal, marked by x in figure b.

Solution. The rooflines are symmetrical, and the triangle in figure b is therefore isosceles. The two equal base angles are marked as x. Write

2x + 136° = 180° Sum of angles of a triangle is 180°

2x = 180° — 136° Subtracting 136° from both sides of equation

2x = 44°

x = 22° Dividing both sides of equation by 2

The roof makes an angle of 22° with the horizontal.

Example. In figure c, find the size of angle y if angle ABC is 90°.

ANGLES AT A POINT 31

Solution. Write

Angle ACB = 53° Sum of adjacent angles is 180° y + 53° + 90° = 180° Sum of angles of triangle ABC = 180°

y = 37° Subtracting 53° and 90° from both sides of equation References: Adjacent Angles, Equations, Geometry Theorems, Supplementary Angles, Vertex.

ANGLES AT A POINT

When two or more angles meet at a point and together make up a full turn, we say they are angles at a point. In figure a, the three angles 100°, 95°, and x are angles at a point. When two or more angles meet at a point and together make up a full turn, the sum of the angles is 360°, since there are 360° in a full turn. Angles that add up to 360° are called conjugate angles. With reference to figure a, we write

x + 100° + 95° = 360° Sum of angles at a point = 360° jc = 165°

Example. A family share a birthday cake, so that Jo has a slice with an angle of 121°, Sarah’s has an angle of 37°, and Andy’s slice has an angle of 162°. What angle does Luke’s slice have?

Sarah

32 ANGLES AT THE CENTER AND CIRCUMFERENCE OF A CIRCLE

Solution. A view of the top of the cake is drawn in figure b, with Luke’s slice having an angle x. The figure is not drawn to scale. Write

121° + 37° + 162° + x = 360° Sum of angles at a point = 360°

320° + jc = 360° jc =40°

Luke’s slice has an angle of 40°.

Reference: Geometry Theorems.

ANGLES AT THE CENTER AND CIRCUMFERENCE OF A CIRCLE

This statement refers to a circle geometry theorem. The angle subtended at the center O of a circle by the arc AB is equal to twice the angle subtended by the same arc at any point C on the circumference of the same circle.

C

(a)

This means that angle AOB = twice angle ACB.

Example. If O is the center of the circle and angle AOB = 104°, find the size of angle ACB (figure b).

ANGLES AT THE CENTER AND CIRCUMFERENCE OF A CIRCLE

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Solution. Write

Angle ACB = 104° 2 Angle at center is twice angle at circumference

Angle ACB = 52°

A special case of this theorem occurs when the arc AB is greater than a semicircle; the diagram looks quite different, as shown in figure c. Nevertheless, the theorem is still true, but the angles are larger:

Reflex angle AOB = twice obtuse angle ACB

Example. In figure d, angle ADC = 64° and O is the center of the circle. Find the sizes of angle AOC and angle ABC, denoted by x and y, respectively.

(d)

Solution. Write

x = twice 64° Angle at center is twice angle at circumference jc = 128°

and

Reflex angle AOC = 360° — 128 Sum of angles at a point = 360° = 232°

y = 232° -j- 2 y = 116°

Angle at center is twice angle at circumference

References: Geometry Theorems, Circle Geometry Theorems, Obtuse Angle, Reflex Angle, Semicircle, Subtend.

34 ANGLES ON THE SAME ARC

ANGLES ON THE SAME ARC

This entry refers to a circle geometry theorem, and states that all angles subtended at the circumference of a circle by the arc AB are equal in size. In figure a, there are two angles on the circumference of the circle, one at the point R and one at Q, that are subtended by the arc AB. Therefore, this theorem states the following:

The two angles ARB and AQB are equal in size.

The theorem is not limited to just two angles, but is true for any number of angles at the circumference of the circle, provided they are all subtended by the arc AB.

Example. William is having shooting practice at soccer training and his coach has drawn a large circle on the gymnasium floor. The straight line AS in figure b represents the goal, and players are shooting from anywhere on the major arc of the circle trying to score. The coach is at point C and William is at point W. If angle AWB = 34°, find angle ACB, which is the coach’s shooting angle.

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