Congruent triangles · Ages 13–14
Properties of isosceles triangles
Base angles equal; the bisector, median and height from the top are one line.
How to do it
Method
Base angles equal; the bisector, median and height from the top are one line
AD ⊥ BC, BC = 10 cm: BD = 10 ÷ 2 = 5 cm (prop. of isos. △)
Another way
By congruent triangles
△ABD ≅ △ACD (RHS), so BD = CD
In full
An isosceles triangle ABC with AB = AC has equal base angles, ∠B = ∠C (reason: base ∠s, isos. △). And its three lines are one: the bisector of the top angle, the median to the base and the height to the base are the same line (reason: prop. of isos. △). So knowing AD is one of them tells you it is the other two: if AD ⊥ BC, then BD = DC and ∠BAD = ∠CAD. With BC = 10 cm, BD = 10 ÷ 2 = 5 cm; with ∠BAC = 50°, ∠BAD = 50° ÷ 2 = 25°. Why? Another way is to prove it by congruence: AB = AC (given), AD = AD (common side), ∠ADB = ∠ADC = 90°, so △ABD ≅ △ACD (RHS), and the corresponding parts give BD = CD and ∠BAD = ∠CAD. An equilateral triangle has three equal sides and three angles of 60°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
An isosceles triangle's top angle is 100°, and its base angles are 100°.
True or false?
Show the answer and working
Answer: False
The two base angles are equal and make 180° with the top: each = (180° − 100°) ÷ 2 = 40°.
Example 2
In the picture, AB = AC and AD bisects ∠BAC. x =
Show the answer and working
Answer: 32°
The bisector of the top angle is at right angles to BC (prop. of isos. △), ∠ADB = 90°: x = 180° − 90° − 58° = 32° (∠ sum of △).
Example 3
In the isosceles triangle ABC, AB = AC = 6 cm, AD ⊥ BC with D on BC, and BD = 3 cm. The perimeter of △ABC =
Show the answer and working
Answer: 18 cm
The height to the base is also its median (prop. of isos. △): DC = BD = 3 cm, BC = 6 cm; perimeter = 6 + 6 + 6 = 18 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In the picture, AB = AC. x =
Answer:
2.In the picture, AB = AC, D is the midpoint of BC and ∠BAC = 50°. ∠BAD = °, ∠ADB = °, ∠ABD = °.
Fill in each blank
Answer:
3.An isosceles triangle's top angle is 122°, and its base angles are 29°.
True or false?
Answer:
4.△ABC is equilateral with sides of 14 cm, and AD ⊥ BC. BD = cm, ∠BAD = °.
Fill in each blank
Answer:
5.In the picture, AB = AC and AD bisects ∠BAC. x =
Answer:
All the answers
- 1. 54°
- 2. 25, 90, 65
- 3. True
- 4. 7, 30
- 5. 53°
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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