Angles & parallel lines · Ages 12–13
Angles of a triangle
Angles add to 180°; exterior = the two far interior angles.
How to do it
Method
Angles add to 180°; exterior = the two far interior angles
∠ACD = 60° + 50° = 110° (ext. ∠ of △)
Another way
The interior angle first
180° − 60° − 50° = 70°, 180° − 70° = 110°
In full
The three angles of a triangle add up to 180° (reason: ∠ sum of △). Why? Draw a line through the top parallel to the base: two alternate angles bring all three angles onto one straight line. Extend a side to get an exterior angle: it equals the two interior angles not next to it added together (reason: ext. ∠ of △). Say ∠A = 60° and ∠B = 50°. One way: the exterior angle ∠ACD = 60° + 50° = 110°. Another: first ∠ACB = 180° − 60° − 50° = 70°, then the straight line, 180° − 70° = 110°. The same answer; the first is a step shorter.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
One side of the triangle is extended. x =
Show the answer and working
Answer: 139°
x is an exterior angle, equal to the two interior angles not next to it: x = 49° + 90° = 139° (ext. ∠ of △).
Example 2
BC is extended to D, as shown. ∠ACB = ° (∠ sum of △), ∠ACD = ° (adj. ∠s on st. line).
Fill in each blank
Show the answer and working
Answer: 74, 106
∠ACB = 180° − 47° − 59° = 74°, ∠ACD = 180° − 74° = 106°. Or straight from the exterior angle: 47° + 59° = 106° (ext. ∠ of △).
Example 3
Which reason finds x in the picture most directly?
- A. adj. ∠s on st. line
- B. alt. ∠s, AB // CD
- C. ∠ sum of △
- D. ext. ∠ of △
Choose the right answer
Show the answer and working
Answer: C. ∠ sum of △
An interior x: the three angles add to 180° (∠ sum of △); an exterior x: it equals the two far interior angles added (ext. ∠ of △). Here x = 49°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.One side of the triangle is extended. x =
Answer:
2.Which reason finds x in the picture most directly?
- A. ext. ∠ of △
- B. ∠ sum of △
- C. adj. ∠s on st. line
- D. alt. ∠s, AB // CD
Choose the right answer
Answer:
3.Two angles of the triangle are marked. x =
Answer:
4.One side of the triangle is extended, and the exterior angle is marked. x =
Answer:
5.In △ABC, ∠A = x°, ∠B = 2x° and ∠C = (x + 88)°. ∠C =
Answer:
All the answers
- 1. 97°
- 2. B. ∠ sum of △
- 3. 48°
- 4. 83°
- 5. 111°
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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