Angle Properties of Triangles
The following states some angle properties of triangles. IMPT: When solving problems, do quote the bracket as stated beside each rule when using the property! 1. The sum of the 3 angles of a triangle is = 180°. (∠ sum of triangle) 2. The exterior angle of a triangle is equal to the sum of the interior opposite angles. (ext. ∠ of triangle) 3. An isoceles triangle has 2 equal angles opposide the 2 equal sides. (base ∠s of isos. triangle) 4. An equilateral triangle has 3 equal sides and 3 equal angles, each equal to 60°. (∠ of equi. triangle) 5. A triangle can also be grouped according to the types of angles it contains as shown below: (a) Acute-angled triangle where all three angles are acute. (b) Right-angled triangle where there is one right angle. (c) Obtuse-angled triange where there is one obtuse angle.
Angle Properties of Quadrilaterals
The sum of all the angles in a quadrilateral is 360°. When quoting this property, do write "(∠ sum of quad.)".
Angle Properties of Polygons
1. A polygon is a closed plane figure with three or more straight lines e.g. triangle (3 sides), quadrilateral (4 sides), pentagon (5 sides). 2. Polygons are named according to the number of sides they have. Besides the three examples stated above, - 6 sides: hexagon - 7 sides: heptagon - 8 sides: octagon - 9 sides: nonagon - 10 sides: decagon 3. In a regular polygon, the sides are all equal in length and the interior angles are equal.
Triangles
The following text are some examples on how triangles are constructed when given certain information. Do try them out yourself! Example 1: To construct a triangle given the lengths of all its three sides, use a ruler and a pair of compass. Construct triangle ABC in which AB = 12 cm, BC = 8 cm and AC = 9 cm. Steps for construction: 1. Draw a line segment AB 12 cm long. 2. With A as centre, draw an arc of radius 9 cm using a pair of compasses. 3. With B as centre and radius 8 cm, draw an arc to cut the first arc at C. 4. Join AC and BC to obtain triangle ABC. Example 2: To construct a triangle given the lengths of two sides and an included angle, use a pair of compasses, a protractor and a ruler. Construct triangle XYZ in which XY = 7.6 cm, ∠XYZ = 135° and YZ = 5.6 cm. Steps for construction: 1. Draw a line segment XY 7.6 cm long. 2. Using a protractor, draw a ray with Y as endpoint and ∠XYZ = 135°. 3. With Y as centre, draw an arc of radius 5.6 cm to cut the ray at Z. 4. Join XZ to obtain triangle XYZ. Example 3: To construct a triangle given two angles and an included length, use a pair of compasses, a protractor and a ruler. Construct triangle PQR such that PQ = 10 cm, ∠PQR = 65° and ∠QPR = 46°. Steps for construction: 1. Draw a line segment PQ 10 cm long. 2. Using a protractor, draw a ray with endpoint P and ∠RPQ = 46°. 3. Using a protractor, draw a ray with endpoint Q and ∠RQP = 65°. 4. Produce the two rays to meet at R to obtain triangle PQR. Do watch the following video to see how constructing triangles can be done:
Quadrilaterals
The following text are some examples on how quadrilaterals are constructed when given certain information. Do try them out yourself, using a pair of compasses, a ruler and a protractor. Example 1: Construct a parallelogram ABCD such that AB = 7 cm, AD = 5.4 cm and ∠BAD = 58°. Steps for construction: 1. Draw a line AB 7 cm long. 2. Draw a ray starting at A with ∠BAD = 58°. 3. With A as centre and radius 5.4 cm, draw an arc to cut the produced arm of A at D. 4. With B as centre and radius 5.4 cm, draw an arc. 5. With D as centre and radius 7 cm, draw an arc to cut the arc at (4) at C. Join BC, CD and AD to obtain the parallelogram ABCD. Example 2: Construct a quadrilateral ABCD such that AB = 5 cm, AD = 6.5 cm, CD = 8.5 cm, ∠BAD = 95° and ∠ABC = 120°. Steps for construction: 1. Draw line segment AB 5 cm long. 2. Draw a ray beginning at A with ∠BAD = 95°. 3. With centre A and radius 6.5 cm, draw an arc to cut the ray at D. 4. Draw a ray beginning at B with ∠ABC = 120°. 5. With centre D and radius 8.5 cm, draw an arc to cut the ray at (4) at C. 6. Join C to D to obtain quadrilateral ABCD.
