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Triangle postulate: The sum of the angles of a triangle is two right angles. Playfair's axiom: Given a straight line and a point not on the line, exactly one straight line may be drawn through the point parallel to the given line. Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also. [3]
This allows the determination of the measure of the third angle of any triangle, given the measure of two angles. [33] An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle.
Given triangle sides b and c and angle γ there are sometimes two solutions for a. The theorem is used in solution of triangles , i.e., to find (see Figure 3): the third side of a triangle if two sides and the angle between them is known: c = a 2 + b 2 − 2 a b cos γ ; {\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}\,;}
AAS (angle-angle-side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°.
If b ≥ c, then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ ′ = 180° − γ.
One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ABC is similar to the reflection of CAD, the triangle DAC in the lower panel. Taking the ratio ...
The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line segments). [3] Only one of these angles contains the third side of the triangle in its interior, and this angle is called an interior angle of the triangle. [4]
In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles. [6] The vertex opposite the base is called the apex. [7]