Polygons · Ages 13–14
Isosceles triangles
Equal sides face equal angles; given one angle, top or base?
How to do it
Method
Equal sides face equal angles; given one angle, top or base?
Top angle 40°: base angle = (180° − 40°) ÷ 2 = 70°
Another way
Write an equation
2x + 40° = 180°, x = 70°
In full
The two angles facing the equal sides of an isosceles triangle, its base angles, are equal (reason: base ∠s, isos. △). Top angle 40°: one way, each base angle = (180° − 40°) ÷ 2 = 70°; another, let a base angle be x, so 2x + 40° = 180°, x = 70°. Base angle 70°: top angle = 180° − 2 × 70° = 40°. Careful: told only "one angle is 40°", it may be the top angle or a base angle, so think of both (an obtuse angle, though, can only be the top). With parallel lines or exterior angles, give a reason at every step. A regular polygon's sides are all equal, so one diagonal makes an isosceles triangle: in a regular hexagon ABCDEF, BA = BC, so ∠BCA = (180° − 120°) ÷ 2 = 30°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture AB = AC, ∠BAC = 70°, and BC is extended to D. ∠ACB = °, ∠ACD = °.
Fill in each blank
Show the answer and working
Answer: 55, 125
∠ACB = (180° − 70°) ÷ 2 = 55° (base ∠s, isos. △), ∠ACD = 180° − 55° = 125° (adj. ∠s on st. line).
Example 2
An isosceles triangle has a top angle of 96° and base angles of 52°.
True or false?
Show the answer and working
Answer: False
The two base angles are equal and make 180° with the top (∠ sum of △, base ∠s, isos. △): each base angle = (180° − 96°) ÷ 2 = 42°.
Example 3
In the picture CA = CB and ∠ACB = 52°. ∠BAC =
Show the answer and working
Answer: 64°
CA = CB, so the angles facing them, ∠ABC and ∠BAC, are equal (base ∠s, isos. △): ∠BAC = (180° − 52°) ÷ 2 = 64°. The equal angles are at A and B, not at B and C.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Two sides of the triangle are equal, as marked. x =
Answer:
2.In the picture AB = AC and ∠BAC = 112°. What is x?
- A. 112°
- B. 146°
- C. 68°
- D. 34°
Choose the right answer
Answer:
3.Two sides of the triangle are equal, as marked. x =
Answer:
4.Two sides of the triangle are equal and one side is extended, as shown. x =
Answer:
5.In the picture CA = CB and ∠ACB = 36°. ∠BAC =
Answer:
All the answers
- 1. 52°
- 2. D. 34°
- 3. 34°
- 4. 50°
- 5. 72°
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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