# Isosceles triangles: the converse theorems · Grade 9

> Explanation, worked examples and free practice following the Israeli curriculum.

In Isosceles triangles: the converse theorems (Grade 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

- Example: (An isosceles triangle. The two equal sides are marked as equal.) 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: 120 °
- Solution: The base angles are equal: ∢C = ∢B = 30° ∢A = 180° − 30° − 30° = 120°

### What proves a triangle is isosceles?

- Example: In triangle ABC , ∢B = 85°, ∢C = 85° . Is the triangle isosceles?
- Options: No · Yes
- Answer: Yes
- Solution: ∢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

- Example: (An isosceles triangle, and the height from the top vertex to the base. The two equal sides are marked as equal.) 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: 3 cm
- Solution: 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

- Example: (A coordinate grid with a triangle and its three vertices) The vertices of the triangle are A(−1, 0), B(1, 0), C(0, 2) . Is triangle ABC isosceles?
- Options: Yes · No
- Answer: Yes
- Solution: AB² = 2² = 4 AC² = 1² + 2² = 5 BC² = −1² + 2² = 5 Two sides are equal, so the triangle is isosceles.

## Not on the site yet

- Proving the converse theorems

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).

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