Download Angles and triangles

11-B
Angles and triangles
KEY CONCEPTS
Interior angles
In the triangle at right, a + b + c = 180°, where a, b and c represent the
three interior angles of the triangle.
b
a
c
The sum of the interior angles of a triangle is 180 °.
This rule can be used to find unknown values of interior angles of
triangles.
Calculate the value of the pronumeral in the triangle below.
a
65o
33o
WRITE
1
The sum of the three interior angles is 180°.
Write this as an equation.
2
Simplify by adding 33° and 65°.
3
Solve the equation for a; that is, subtract 98°
from 180°.
33° + 65° + a = 180°
98° + a = 180°
a = 180° − 98°
= 82°
Calculate the value of the pronumeral in the triangle below.
70°
b
WRITE
330
1
The triangle is an isosceles triangle, so the two base
angles are equal. The sum of the interior angles of a
triangle is 180°.
2
To find the value of b, subtract 140° from 180°.
Maths XPRESS 7
180° = b + 70° + 70°
= b + 140°
b = 180° − 140°
= 40°
Exterior angles
If one edge of a triangle continues past the vertex, the angle
formed is called an exterior angle.
B
The interior angle and the exterior angle next
(or adjacent) to it are called supplementary
angles because they form a straight line.
Supplementary angles sum to 180°. For the
triangle at right:
A
∠ACB + ∠BCD = 180°
∠ACB = 180° − ∠BCD
∠BCD = 180° − ∠ACB
Exterior
angle
C
D
B
Determine the value of the exterior angle
∠BCD in the diagram at right.
70o
50o
A
1
The sum of the supplementary angles, ∠ACB and
∠BCD, is 180°.
2
∠ACB is 60°. Subtract this value from 180° to
determine the value of ∠BCD.
60o
C
D
WRITE
∠ACB + ∠BCD = 180°
The sum of the interior angles of a triangle is 180°, and there are
180° in a straight angle. Therefore, we can establish the rule at right:
In Example 3, ∠BCD = 50° + 70° = 120°.
∠BCD = 180° − ∠ACB
= 180° − 60°
= 120°
An exterior angle of a
triangle is equal to the sum
of the two opposite interior
angles.
LEARNING EXPERIENCE
Puzzles with shapes
Equipment: coloured cardboard, ruler, protractor, scissors
1 Using coloured cardboard, draw and cut out:
a one equilateral triangle with edge lengths equal to 4 cm and interior angles equal to 60°
b one obtuse-angled, isosceles triangle with a 120° angle and two edges equal to 4 cm
c four scalene, right-angled triangles with a hypotenuse of 4 cm and other interior angles of 30° and 60°.
2 Use the cardboard triangles from step 1 to make:
a a right-angled triangle using only two pieces
b a right-angled triangle using three pieces
c a right-angled triangle using four pieces
d a single triangle with an exterior angle of 120° using all pieces
e a single triangle with an exterior angle of 150° using all pieces.
3 Measure the interior angles of each construction. What do you notice about the sum of the interior angles?
Chapter 11
Shapes
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