triangles

Types of triangles
[]

Triangles classified in terms of angle and lengths....

**Congruence**

 * Key stage 3 specification examples:**

Parallel Lines and Similar and Congruent Triangles
Theorems 6.[|1], [|2], [|3], [|4], [|5], [|6], [|7], [|8], [|9], [|10], [|11] **[|Theorem 6.1]**: If two parallel lines are transected by a third, the alternate interior angles are the same [|size]. **[|Theorem 6.2]**: If a line intersects two other lines then the following conditions are equivalent. **[|Theorem 6.3]**: The angles in a triangle always add up to 180o. **[|Theorem 6.4]**: If two lines are crossed by a third, then the following conditions are equivalent. [|top] Why SSA is not congruent: [|ssa not congruent] Why AAA not congruent: [|aaa does not work] **[|Theorem 6.5]**: Two line segments are congruent if and only if they have the same length. **[|Theorem 6.6]**: Two angles are congruent if and only if they have the same size. **[|Theorem 6.7]**: (SSS) Two triangles are congruent if and only if their corresponding sides all have the same lengths.[|T4] **[|Theorem 6.8]**: (SAS) Two triangles are congruent if and only if two sides and the angles between them in one triangle are congruent to the two sides and the angle between them in the other triangle. [|T4] **[|Theorem 6.9]**: Two triangles are similar if and only if the ratios of their corresponding sides are all the same. **[|Theorem 6.10]**: (ASA) Two triangles are congruent if and only if two angles and the side between them in one triangle are congruent to two angles and the side between them in a second triangle, then the triangles are congruent.[|2] **[|Theorem 6.11]**: (AAS) Two triangles are congruent if and only if two angles and the side next to one of them in one triangle are congruent to two angles and the corresponding side in a second triangle, then the triangles are congruent.
 * a) The alternate interior angles are the same size
 * b) The corresponding angles are the same size
 * c) The opposite interior angles are supplementary.
 * a) The alternate interior angles are the same size
 * b) The corresponding angles are the same size
 * c) The opposite interior angles are supplementary.
 * d) The two lines are parallel.[|1]

[|Properties of triangles]


 * [|Types of triangles]**
 * [|definition of regular polygons and exterior angles]**


 * angles and parallel lines**

Angles of a triangle equal 180
 * [[image:http://www.mathsisfun.com/geometry/images/parallel-angle-pairs.gif width="183" height="215"]] ||  ||   ||   ||
 * [|Corresponding Angles] are equal, or || //a = e// ||
 * [|Alternate Interior Angles] are equal, or || //c = f// ||
 * [|Alternate Exterior Angles] are equal, or || //b = g// ||
 * [|Consecutive Interior Angles] add up to 180° || //d + f = 180°// ||
 * [|Consecutive Interior Angles] add up to 180° || //d + f = 180°// ||

Proof
This is a proof that the angles in a triangle equal 180°:

The top line (that touches the top of the triangle) is running parallel to the base of the triangle. So: And you can easily see that A + C + B does a **complete rotation** from one side of the straight line to the other, or **180°**
 * angles A are the same
 * angles B are the same