# Congruent triangles and triangle properties · Grade 8

> Congruent triangles and triangle properties, Grade 8: The exterior angle and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Congruent triangles and triangle properties (Grade 8) you practise 7 skills: The congruence conditions · Corresponding sides and angles · The triangle inequality · The median of a triangle · The exterior angle · The isosceles triangle · Equal angles opposite equal sides.

Two triangles are congruent if one can be placed on top of the other so that they match exactly. To know that they are congruent, three matching facts are enough: three sides (SSS), two sides and the angle between them (SAS), or a side and the two angles next to it (ASA).

In congruent triangles, the matching sides are equal and the matching angles are equal. The order of the letters in the triangles' names shows what matches what.

In every triangle, any two sides added together are longer than the third side. A median joins a vertex to the middle of the side opposite it, and cuts the triangle into two triangles with equal areas.

An exterior angle of a triangle is equal to the sum of the two interior angles that are not next to it. In an isosceles triangle, the base angles are equal, and the height to the base cuts the base in half. It works the other way too: if two angles of a triangle are equal, the sides opposite them are equal.

**Tip:** Watch the order of the letters in a congruence: the first letter matches the first letter, the second the second, and the third the third.

## What you learn

### The congruence conditions

- Example: (Two triangles. In each triangle three parts are marked, and each marked part is equal to the matching part in the other triangle.) In the two triangles, the equal parts are marked: BC = EF , AC = DF and AB = DE . Which rule shows that the triangles are congruent?
- Options: ASA · Not enough information · SAS · SSS
- Answer: SSS
- Solution: The three sides of one triangle are equal to the three sides of the other triangle. So the triangles are congruent by SSS.

### Corresponding sides and angles

- Example: The triangles are congruent: ΔABC ≅ ΔFED . Which side is equal to side AB ?
- Options: DF · EF · DE
- Answer: EF
- Solution: When two triangles are congruent, the letters in the same place in their names match each other. AB = FE , that is, AB = EF . The side is EF .

### The triangle inequality

- Example: Is there a triangle with sides of 4, 20 and 4 cm?
- Options: Yes · No
- Answer: No
- Solution: Check the two shorter sides against the longest side: 4 + 4 = 8 8 is less than 20, so there is no such triangle. The answer is no.

### The median of a triangle

- Example: (A triangle, and the line from the top vertex to the middle of the side opposite it. The two parts of that side are marked as equal.) In triangle ABC , the line AM is the median to side BC , and BC = 6 cm. What is the length of BM ?
- Answer: 3 cm
- Solution: The median goes to the middle of the side, so BM = MC : BM = 6 ÷ 2 = 3 The length is 3 cm.

### The exterior angle

- Example: (A triangle with one side extended past a vertex to a new point. The exterior angle this makes is marked with an arc.) In triangle ABC , side BC is extended past vertex C to point D . ∢A = 30° and ∢B = 100° . What size is the exterior angle ∢ACD ?
- Answer: 130 °
- Solution: An exterior angle of a triangle is equal to the sum of the two interior angles that are not next to it. ∢ACD = 30° + 100° = 130°

### The 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 apex angle is ∢A = 160° . What size is the base angle ∢B ?
- Answer: 10 °
- Solution: The two base angles are equal, and together with the apex angle they make 180°: ∢B = (180° − 160°) ÷ 2 = 10°

### Equal angles opposite equal sides

- Example: (A triangle, with the names of its vertices.) In triangle ABC , the angles are ∢A = 85° and ∢B = 85° . Is the triangle isosceles?
- Options: Yes · No
- Answer: Yes
- Solution: The third angle: ∢C = 180° − 85° − 85° = 10° There are two equal angles: ∢A = ∢B . A triangle with two equal angles is isosceles. The answer is yes.

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 8 transition maths curriculum (2026–27).

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