Congruent triangles · Ages 13–14
Proving a triangle isosceles
Equal angles face equal sides: ∠B = ∠C gives AB = AC.
How to do it
Method
Equal angles face equal sides: ∠B = ∠C gives AB = AC
∠C = 180° − 40° − 70° = 70° = ∠B, so AB = AC (sides opp. equal ∠s)
Another way
By congruent triangles
△ABD ≅ △ACD (AAS), AB = AC
In full
It works the other way round too: if two angles of a triangle are equal, the sides facing them are equal (reason: sides opp. equal ∠s). So to prove a triangle isosceles, you can find its angles first: ∠A = 40° and ∠B = 70° give ∠C = 180° − 40° − 70° = 70° = ∠B, so AB = AC. Mind which sides face them: ∠B faces AC and ∠C faces AB, not BC. Clues to look for: alternate or corresponding angles of parallel lines, an angle bisector, an exterior angle. Another way: by congruence. Draw AD bisecting ∠A; △ABD ≅ △ACD (AAS), so AB = AC (corr. sides, ≅△s). A triangle with all three angles equal has all three sides equal: it is equilateral.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, ∠B = ∠C. AC =
Show the answer and working
Answer: 10 cm
∠B faces AC and ∠C faces AB; ∠B = ∠C, so AC = AB = 10 cm (sides opp. equal ∠s).
Example 2
In △ABC, ∠B = ∠C = 69°. Which of these must be true?
- A. AB = BC
- B. AB = BC = CA
- C. BC = AC
- D. AB = AC
Choose the right answer
Show the answer and working
Answer: D. AB = AC
Equal angles face equal sides: AC, facing ∠B, equals AB, facing ∠C.
Example 3
In the picture BC is extended to D, ∠ACD = 80°, ∠BAC = 40° and AC = 8 cm. ∠ABC = °, BC = cm.
Fill in each blank
Show the answer and working
Answer: 40, 8
∠ABC = 80° − 40° = 40° (ext. ∠ of △), so ∠ABC = ∠BAC and BC = AC = 8 cm (sides opp. equal ∠s).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In △ABC, ∠B = ∠C = 43°. Which of these must be true?
- A. AB = BC = CA
- B. BC = AC
- C. AB = AC
- D. AB = BC
Choose the right answer
Answer:
2.In the picture, ∠B = ∠C. AC =
Answer:
3.In △ABC, ∠A = 74° and ∠B = 78°. △ABC is isosceles.
True or false?
Answer:
4.In △ABC, ∠A = 114°, ∠C = 33°, AB = 5 cm and BC = 4 cm. AC =
Answer:
5.In △ABC, BD bisects ∠ABC with D on AC; F is on AB with DF // BC, and BF = 6 cm. DF =
Answer:
All the answers
- 1. C. AB = AC
- 2. 10 cm
- 3. False
- 4. 5 cm
- 5. 6 cm
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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