Spatial Sense · SHS1 Mathematics

Lesson notes

Learning Objectives

Indicator: 1.3.1.LI.2 - Solve problems that involve parallel lines, perpendicular lines transversal, and pairs of angles formed between them.

By the end of the lesson, learners can:

Sort and classify given pairs of lines as parallel, perpendicular, or neither, with justification based on their defining properties.
Identify and name the six pairs of angles formed when a transversal intersects two lines: corresponding angles, vertically opposite angles, alternate interior angles, alternate exterior angles, interior angles on the same side of the transversal, and exterior angles on the same side of the transversal.
State and apply the relationships between angles formed by parallel lines and a transversal (equal angles for corresponding, alternate interior, and alternate exterior; supplementary angles for interior and exterior angles on the same side).
Solve for unknown angle measures in diagrams involving parallel lines cut by a transversal, using algebraic equations where necessary.
Verify that the angle relationships hold only when the lines are parallel, and apply these relationships to solve real-life problems such as finding unknown widths, slopes, or distances.

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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.2 - Solve problems that involve parallel lines, perpendicular lines transversal, and pairs of angles formed between them.
Suggested placement: Semester 1, Week 19 (Week 19 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. 91

Exemplars (from the NaCCA curriculum)

Experiential Learning, Group discussions, think-pair-share activities, Talk for Learning (use of blended experiences that include both group and individual learning experiences).
Experiential Learning: In convenient groups, engage learners to sort a set of lines as perpendicular, parallel or neither, and justify it.
Activity 1:

Three diagrams captioned Parallel lines, Neither and Perpendicular lines. — A three-cell table of line diagrams. The first cell shows two lines that never meet and is captioned "Parallel lines", the second shows two lines crossing obliquely and is captioned "Neither", and the third shows two lines crossing at a right angle and is captioned "Perpendicular lines".

Parallel Neither Perpendicular lines lines
In a collaborative group activity, engage learners to illustrate and describe complementary and supplementary angles. Then, learners should go on to identify, in a set of angles, adjacent angles that are not complementary or supplementary.
Activity 2 Identifying and naming pairs of angles.

Table of complementary angles, illustrated 52 and 38 degrees, and supplementary angles, illustrated 128 and 52. — A table headed "Type of Angles", "Description" and "Example". The first row reads "Complementary Angles - Angles that add up to 90" and shows two red angles marked 52 and 38 degrees making a right angle; the second reads "Supplementary Angles - Angles that add up to 180" and shows angles marked 128 and 52 degrees on a straight line.

In mixed-ability group activities, engage learners to identify and name pairs of angles formed by parallel lines and a transversal, including corresponding angles, vertically opposite angles, alternate interior angles, and alternate exterior angles, interior angles on the same side of transversal and exterior angles on the same side of the transversal.
Example

Parallel lines a and b cut by transversal l, with the eight angles labelled 1 to 8 in coloured sectors. — Two horizontal parallel lines, a and b, cut by a transversal l. The eight angles formed are labelled angle 1 to angle 4 at the upper intersection and angle 5 to angle 8 at the lower, each drawn as a coloured sector.

In mixed-ability group activities, Engage learners to explain and illustrate the relationships between angles formed by parallel lines and a transversal.
Using think-pair-share activities, learners should explain, using examples, why the angle relationships do not apply when the lines are not parallel.
Using think-pair-square activities, learners should determine if lines or planes are perpendicular or parallel. E.g., A wall perpendicular to the floor, and describe the strategy used.
Using think-pair-share activities, learners should determine the measures of angles involving parallel lines and a transversal, using angle relationships.
Example: Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'.

Parallel lines a and b cut by transversal t, with angles 60 degrees and 8x minus 4 marked. — Two horizontal parallel lines, a above and b below, cut by a sloping transversal t. The angle at the upper intersection is marked 60 degrees and the angle at the lower is marked 8x minus 4.

Solution: The given parallel lines are cut by a transversal; therefore, the marked angles in the figure are the alternate interior angles, which are equal in measure. This means, 8x - 4º = 60°, and 8x = 64º, x = 8º. Therefore, the value of x = 8º.
In convenient groups, engage learners to solve a contextual problem that involves angles formed by parallel lines and a transversal (including perpendicular transversal).
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.2). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.