Isosceles triangles: the converse theorems · Year 9
In Isosceles triangles: the converse theorems (Year 9) you practise 4 skills: Angles and the height in an isosceles triangle · What proves a triangle is isosceles? · Calculating with the special lines · An isosceles triangle on a coordinate grid. In an isosceles triangle the base angles are equal, and the height to the base is also the median to the base and the bisector of the apex angle. So the height cuts the base into two equal parts, and the apex angle is twice the angle between the height and an equal side.
The converse theorems work the other way round: a triangle with two equal angles is isosceles, and the sides opposite the equal angles are equal. One segment that is two of the special lines also proves the triangle isosceles: a height that is also a median, an angle bisector that is also a height, or a median that is also an angle bisector. One special line on its own is not enough, because every triangle has one. On a coordinate grid, when two vertices lie on the horizontal axis at the same distance from the origin, the vertical axis is the perpendicular bisector of the segment between them. Any point on the vertical axis makes an isosceles triangle with them, and Pythagoras' theorem gives the length of the equal sides.
Tip: To show that a triangle is isosceles, look for two equal angles, or for one segment that is two special lines at once.
What you learn
Angles and the height in an isosceles triangle
For example: In isosceles triangle ABC , the equal sides are AB = AC , and the base angle is ∢B = 30° . What size is the apex angle ∢A ?
Answer and solution
The answer: 120 °
The base angles are equal: ∢C = ∢B = 30° ∢A = 180° − 30° − 30° = 120°
What proves a triangle is isosceles?
For example: In triangle ABC , ∢B = 85°, ∢C = 85° . Is the triangle isosceles?
- No
- Yes
Answer and solution
The answer: Yes
∢A = 180° − 85° − 85° = 10° Two of the angles are equal, so the triangle is isosceles: the sides opposite the equal angles are equal.
Calculating with the special lines
For example: In isosceles triangle ABC , the equal sides are AB = AC . The segment AD is the height to the base , and BC = 6 cm. What is the length of BD ?
Answer and solution
The answer: 3 cm
The height to the base is also the median, so it cuts the base in half: BD = 6 ÷ 2 = 3 The length is 3 cm.
An isosceles triangle on a coordinate grid
For example:
The vertices of the triangle are A(−1, 0), B(1, 0), C(0, 2) . Is triangle ABC isosceles?- Yes
- No
Answer and solution
The answer: Yes
AB² = 2² = 4 AC² = 1² + 2² = 5 BC² = −1² + 2² = 5 Two sides are equal, so the triangle is isosceles.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).