E2.2 Solve problems involving angle properties, including the properties of intersecting and parallel lines and of polygons.
Skill: Solving Problems Involving Angle Properties
Angles can be measured indirectly (calculated) by applying angle properties.
If a larger angle is composed of two smaller angles, only two of the angles are needed to calculate the third.
Angle properties can be used to determine unknown angles.
- A straight angle measures 180°; this is used to determine the measure of a supplementary angle.
- A right angle measures 90°; this is used to determine the measure of a complementary angle.
- The interior angles of triangles sum to 180°, the interior angles of quadrilaterals sum to 360°, the interior angles of pentagons sum to 540°, and the interior angles of n-sided polygons sum to (n − 2) × 180. The angle properties of a polygon can be used to determine the measure of a missing angle. A straight angle measures 180°; knowing this, it is possible to determine the measure of additional angles.
Note
- The aim of this expectation is not to memorize these angle theorems or the terms, but to use spatial reasoning and known angles to determine unknown angles.
- Smaller angles may be added together to determine a larger angle. This is the additivity principle of measurement.
- If two shapes are similar, their corresponding angles are equal (see E1.3). Recognizing similarity between shapes (for example, by ensuring that the corresponding side lengths of a shape are proportional) can help to identify their corresponding angles.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Properties of Intersecting and Parallel Lines
A transversal line intersects two or more other lines.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
The properties above can be used to determine unknown angles when a line (transversal) intersects two parallel lines:
Image Two parallel lines are crossed by an intersecting line. The
angles are measured with the upper line as follows, counterclockwise. Angle " a ", angle " b ", angle " c ", angle
" d ". The angles are measured with the lower line as follows, counterclockwise. Angle " e ", angle " f ", angle "
g ", angle " h ".- Alternate interior angles are equal, so \(\ {\angle}c\ = {\angle}e \) and \(\ {\angle}d\ = {\angle} f \) (Z-pattern).
- Opposite angles are equal, so \(\ {\angle}b\ ={\angle}d\), \(\ {\angle}a\ = {\angle}c\), \(\ {\angle}f\ = {\angle}h\) and \(\ {\angle}e\ = {\angle}g\) (angles formed by two lines intersecting).
- Alternate exterior angles are equal, so \(\ {\angle}b\ = {\angle}h\), and \(\ {\angle}a\ = {\angle}g\).
- Corresponding angles are equal, so \(\ {\angle}b\ = {\angle}f\), \(\ {\angle}c\ = {\angle}g\), \(\ {\angle}a\ = {\angle}e\), and \(\ {\angle}d\ = {\angle}h\) (F-pattern).
- Co-interior angles sum to 180° and \(\ {\angle}d\ + {\angle}e\ = 180°\) (C-pattern).
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Properties of Polygons
The sum of the interior angles of a triangle is 180°; the sum of the interior angles of a quadrilateral is 360°; the sum of the interior angles of a pentagon is 540°; the sum of the interior angles of a polygon with n sides is (n - 2) × 180. The properties of the angles of a polygon can be used to find the measure of a missing angle.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
The interior angle is formed by the intersection of two adjacent sides within a polygon. A polygon with n sides has n interior angles.
The sum of the measures (in degrees) of the interior angles of any polygon with n sides is 180° × (n - 2). The expression (n - 2) represents the number of triangles that can be drawn inside a polygon.
Note: The formula for quickly finding the sum of the interior angles of a polygon is given here for teachers only. Intermediate students are not required or expected to know and use this concept.
\(1 \ \mathrm{triangle} \times 180^{\circ} = 180^{\circ}\)
\(6 \ \mathrm{triangles} \times 180^{\circ} = 1080^{\circ}\)
Note
When students are asked to construct a polygon and display the angle measurements, they should check to see if the sum of the interior angles is correct. A table containing the sum of the angles of various polygons can be posted on the wall and students can refer to it.
Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
The exterior angle is formed by one of the sides of a polygon and the extension of one of its adjacent sides.
In a polygon, the exterior angle and the adjacent interior angle are supplementary, that is, their sum is always 180°.
The sum of the exterior angles of a polygon is always 360°.
Examples of Polygons
Hexagon
Image A hexagon with all segments extended on the outside, clockwise. The angles resulting from this
extension are 60 degrees. The angles inside the hexagon are 120 degrees. A hexagon with all segments extended on the
outside, counterclockwise. The angles resulting from this extension are 60 degrees. The angles inside the hexagon are
120 degrees.
\(60^{\circ} + 60^{\circ} + 60^{\circ} + 60^{\circ} + 60^{\circ} + 60^{\circ} = 360^{\circ}\)
Rectangle
Image A rectangle with all segments extended on the outside, clockwise. The angles resulting from this extension are 90 degrees.
A hexagon with all segments extended on the outside, counterclockwise. The resulting angles are 60 degrees. The angles inside the hexagon are 120 degrees.
\(90^{\circ} + 90^{\circ} + 90^{\circ} + 90^{\circ} = 360^{\circ}\)
Triangle
Image A triangle with all segments extended on the outside, clockwise. The angles that result from this extension are 120 degrees, 150 degrees and 90 degrees. The angles inside the triangle are 60 degrees, 90 degrees and 30 degrees.
A triangle with all segments extended on the outside, counterclockwise. The angles that result from this extension are 120 degrees, 90 degrees, and 150 degrees. The angles inside the triangle are 60 degrees, 90 degrees, and 30 degrees.
\(90^{\circ} + 120^{\circ} + 150^{\circ} = 360^{\circ}\)
Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).