Congruent triangles and triangle properties · Year 9
In Congruent triangles and triangle properties (Year 9) you practise 6 skills: The congruence conditions · Corresponding sides and angles · The triangle inequality · 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
For example: In the two triangles, the equal parts are marked: BC = EF , AC = DF and AB = DE . Which rule shows that the triangles are congruent?
- ASA
- Not enough information
- SAS
- SSS
Answer and solution
The answer: SSS
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
For example: The triangles are congruent: ΔABC ≅ ΔFED . Which side is equal to side AB ?
- DF
- EF
- DE
Answer and solution
The answer: EF
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
For example: Is there a triangle with sides of 4, 20 and 4 cm?
- Yes
- No
Answer and solution
The answer: No
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 exterior angle
For example: 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 and solution
The answer: 130 °
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
For example: 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 and solution
The answer: 10 °
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
For example: In triangle ABC , the angles are ∢A = 85° and ∢B = 85° . Is the triangle isosceles?
- Yes
- No
Answer and solution
The answer: Yes
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, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).