Parallel lines, triangle angles and the triangle inequality
Week 12 · Parallel lines and the triangle: finding angles, angle sum, exterior angle, triangle inequality · 3 lessons
1
Angles with parallel lines: finding angles and showing lines are parallel
Mathematics Prep 1 Term 1, Unit 4, Lessons 2–3
GoalUse equal and supplementary angles to find angles and to prove two lines are parallel.
New words
parallel lines · lines in one plane that never meetconverse · the statement turned around: if the angles are equal, then the lines are parallelprove · to show with reasons that a statement is truereason · the rule that tells why a step is true
Explanation
When a transversal cuts two parallel lines three facts hold: corresponding angles are equal, alternate angles are equal, and same-side interior angles add up to 180°. The converse is also true: if a transversal makes equal corresponding angles, or equal alternate angles, or same-side interior angles that add up to 180°, then the two lines are parallel. In the figure the arrows show parallel lines and only two sizes appear: 65° and 115°, which add up to 180°. In every answer write the reason, for example: corresponding angles, parallel lines.
Worked examples
Corresponding. The lines are parallel and one angle is 70°. The angle in the same position at the other crossing is x. Corresponding angles are equal, so x = 70°.
Alternate. The lines are parallel and one interior angle is 125°. The alternate angle on the other side of the transversal is y. Alternate angles are equal, so y = 125°.
Same-side interior. The lines are parallel and one interior angle is 105°. The interior angle on the same side is z, and 105° + z = 180°, so z = 75°.
With an equation. The lines are parallel and the corresponding angles are (3x + 10)° and (2x + 40)°. They are equal: 3x + 10 = 2x + 40, so x = 30. Each angle is 3 × 30 + 10 = 100°. Check: 2 × 30 + 40 = 100.
Proving parallel. A transversal makes two same-side interior angles of 100° and 80°. Their sum is 180°, so the lines are parallel. If the angles were 100° and 85°, the sum 185° would show they are not parallel.
Class activity
Parallel lines on paper. With a grown-up, use the two edges of a ruler to draw two parallel lines and cross them with a line. Measure the eight angles with a protractor and check that you only get two different sizes.
Practice
1
The lines are parallel and the top right angle is 50°. Find the bottom left angle x and give the reason.
The angle at the top right of the lower crossing is 50° (corresponding angles), and x is vertically opposite to it, so x = 50°.
2
The lines are parallel. Find y.
Same-side interior angles add up to 180°: y = 180° − 70° = 110°.
3
A transversal makes alternate angles of 72° and 72°. Are the lines parallel? Why?
Yes. Equal alternate angles show the lines are parallel.
4
The lines are parallel and the alternate angles are (2x + 15)° and (3x − 5)°. Find x and the angles.
2x + 15 = 3x − 5, so x = 20. The angles are 2 × 20 + 15 = 55° each.
5
Rana says: “For parallel lines the same-side interior angles are equal, so if one is 70° the other is 70°.” What is wrong?
Same-side interior angles add up to 180°, so the other is 110°. It is the alternate angles that are equal.
