Angles & parallel lines

Skills and methods in this topic, easy to hard

1. 1Angles on a line and at a point180° on a line, 360° round a point

   Methodx + 65° = 180° (adj. ∠s on st. line), x = 115°

   Another way: Write an equationx + x + 100° = 360°, x = 130°

2. 2Vertically opposite anglesOpposite angles equal; angles side by side add to 180°

   MethodOne angle 70°: the opposite one 70°, the next one 110°

   Another way: By angles on a linea + b = 180°, b + c = 180°, so a = c

3. 3Angles with an unknownRelation first, then the equation; asked for the angle, put x back

   Method3x − 20 = 2x + 10 (vert. opp. ∠s), x = 30

   Another way: Check by putting x back2 × 30 + 10 = 70, 3 × 30 − 20 = 70

4. 4Corresponding, alternate and interior anglesCorresponding F, alternate Z, interior C

   Method∠AEF and ∠EFD: opposite sides, both between the lines, so alternate

   Another way: Ask two questionsSame side? Between the lines?

5. 5Telling parallel linesCorr. equal, alt. equal, or int. adding to 180°

   Method∠AEF = ∠EFD = 65°, so AB // CD (alt. ∠s equal)

   Another way: Turn it into another relation70° + 110° = 180°, or 180° − 110° = 70°

6. 6Angles with parallel linesParallel: corr. and alt. equal, int. add to 180°

   Method∠EFD = ∠AEF = 65° (alt. ∠s, AB // CD)

   Another way: In two stepsVert. opp. ∠s first, then corr. ∠s

7. 7Bends and reflex anglesA parallel line through the bend: two alternate angles

   Methodx = 40° + 35° = 75°, reflex angle = 360° − 75° = 285°

   Another way: Extend one sideAlternate angle 35°, exterior angle = 40° + 35° = 75°

8. 8Angles of a triangleAngles add to 180°; exterior = the two far interior angles

   Method∠ACD = 60° + 50° = 110° (ext. ∠ of △)

   Another way: The interior angle first180° − 60° − 50° = 70°, 180° − 70° = 110°

9. 9Isosceles trianglesEqual sides face equal angles; given one angle, top or base?

   MethodTop angle 40°: base angle = (180° − 40°) ÷ 2 = 70°

   Another way: Write an equation2x + 40° = 180°, x = 70°

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Angles on a line and at a point →

   The four angles go round one point, and the three marked x are equal. x =

   ▶Show the answer and working

   Answer: 40°

   x + x + x + 240° = 360° (∠s at a pt.), 3x = 360° − 240° = 120°, x = 40°.

2. Example 2Telling parallel lines →

   The line EF cuts AB and CD, as shown, and ∠AEF = 135°. If AB // CD, ∠EFD = ° and ∠EFC = °.

   Fill in each blank

   ▶Show the answer and working

   Answer: 135, 45

   ∠EFD = ∠AEF = 135° (alt. ∠s, AB // CD); ∠EFC = 180° − 135° = 45° (int. ∠s, AB // CD).

3. Example 3Isosceles triangles →

   Two sides of the triangle are equal, as marked. x =

   ▶Show the answer and working

   Answer: 34°

   The two base angles are equal (base ∠s, isos. △): x = (180° − 112°) ÷ 2 = 34° (∠ sum of △).