Spatial Sense · SHS1 Mathematics

Lesson notes

Learning Objectives

Indicator: 1.3.1.LI.3 - State and apply the exterior angle theorem of a triangle to solve problems and identify various properties of special triangles.

By the end of the lesson, learners can:

State the exterior angle theorem of a triangle in their own words and identify exterior angles versus interior angles in a given triangle.
Prove the exterior angle theorem using the angle sum property of a triangle and the linear pair property of angles.
Apply the exterior angle theorem to find the measure of unknown interior and exterior angles in a variety of triangle problems.
Identify fundamental properties of special triangles (scalene, isosceles, equilateral, right-angled) and use these properties to solve angle problems.
Solve word problems involving exterior angles and special triangle properties in practical situations.

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Curriculum details

Strand: Geometry Around Us (Strand 3)
Sub-strand: Spatial Sense (3.1)
Content standard: 1.3.1.CS.1 - Demonstrate a conceptual understanding of spatial sense with respect to angles, parallel lines, transversal and polygons, and apply their properties to solve everyday life problems. 1.3.1.LO.1 Draw and describe angles of various measures; solve problems on the Pythagorean theorem, parallel lines, perpendicular lines and transversal; use the exterior angle theorem of a triangle and calculate the sums of interior and exterior angles of polygons.
Indicator: 1.3.1.LI.3 - State and apply the exterior angle theorem of a triangle to solve problems and identify various properties of special triangles.
Suggested placement: Semester 1, Week 20 (Week 20 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Curriculum reference: NaCCA curriculum document, p. 93

Exemplars (from the NaCCA curriculum)

Using talk for Learning Strategy, task learners in convenient groups to discuss the Exterior Angle Theorem of a triangle and establish the formula to solve problems on triangles.
Activity 1: Exterior Angle Theorem According to the exterior angle theorem, the exterior angle that results from stretching a triangle's side is equal to the sum of the dimensions of the triangle's two opposed interior angles. The theorem can be used to find the measure of an unknown angle in a triangle.
For instance: Three internal angles in a triangle always add up to 180 degrees. This theorem is applied to each of the outer angles, which total si. As they constitute a linear pair of angles, take note that an exterior angle is supplementary to the neighbouring interior angle. Exterior angles are those that are created between a polygon's side and its extended neighbouring side.

Triangle with exterior angle e and remote interior angles a and b, with the rule angle e = angle a + angle b. — A triangle with one side extended to make an exterior angle. The two interior angles remote from it are labelled a and b and captioned "Two opposite interior angles", the third interior angle is c, the exterior angle is e, and a box below reads "angle e = angle a + angle b".

Activity 2: Proof of the exterior angle theorem. We can verify the exterior angle theorem with the known properties of a triangle. Consider a Δ ABC. The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1 c= 180 - (a+b) ----- Equation 2 (rewriting equation 1) e = 180 - c----- Equation 3 (linear pair of angles) Substituting the value of c in equation 3, we get e = 180 - [180 - (a + b)] e = 180 - 180 + (a + b) e = a + b Hence verified.

Triangle marked 30 degrees, x and 88 degrees with exterior angle y, worked to x = 92 and y = 58 degrees. — A worked "Example 3" on exterior angles, with a triangle whose angles are marked 30 degrees, x and 88 degrees and whose exterior angle is y. The solution beneath solves x plus 88 equals 180 to get x equals 92, then x plus y plus 30 equals 180 to get y equals 58 degrees.

 Example 3: Use the properties of exterior angles of a triangle to determine the measure of the angles represented by x and y.
Solution x + 88º = 180º x + 88º − 𝟖𝟖º = 180º − 𝟖𝟖º x = 92
x + y + 30º = 180º 92 + y + 30º = 180º y + 122º = 180º 𝑦 + 122º − 𝟏𝟐𝟐º = 180º − 𝟏𝟐𝟐º 𝑦 = 58º
Teaching and Learning Resources:
- * Mathematical sets. * Technology tools such as computers, mobile phones, etc.
- * Computer software applications like GeoGebra. * Tape measure, carpenters square, compass, clock face, etc.
Assessment (1.3.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.