Angles of a Triangle and Congruent Triangles April 24, 2008

What is a triangle? A triangle is the figured form by three segments joining three noncollinear points. Each of the three points is called a vertex of the triangle (the plural of vertex is vertices.) The segments are the sides of the triangle.

Classifying triangles by their sides We often classify triangles by the number of congruent sides it has. Scalene triangle: No sides congruent Isosceles triangle: At least two sides congruent Equilateral triangle: All sides congruent

Classifying triangles by their angles Acute: three acute angles Obtuse: one obtuse angle Right: one right angle equiangular: all angles are congruent

Theorem: The sum of the measures of the angles of a triangle is 180. Part of the proof of this theorem involves the use of an auxiliary line. An auxiliary line is a line (or segment or ray) added to a diagram to help in a proof.

Corollaries A corollary of a theorem is a statement that can be proved easily by applying a theorem. It can be easily proved from the theorem itself by simply adding a few more steps.

Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Corollary 2 Each angle of an equiangular triangle has measure 60.

Corollary 3 In a triangle, there can be at most one right angle or obtuse angle.

Corollary 4 The acute angles of a right triangle are complementary.

Interior and exterior angles When one side of a triangle is extended, an exterior angle is formed. Because an exterior angle of a triangle is always a supplement of the adjacent interior angle of the triangle, its measure is related to the meausres of the remaining angles of the triangle, called the remote interior angles

Now check out this one Notice that in both cases the measure of the exterior angle is equal to the sum of the two remote interior angles.

Another theorem The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

What does it mean to be congruent? When two figures have the same shape and size, they are called congruent. We have already discussed congruent segments (segments with equal lengths) and congruent angles (angles with equal measures).

Congruent Triangles Triangles ABC and DEF are congruent. If you mentally slide triangle ABC to the right, you can fit it exactly over triangle DEF by matching up the vertices.

Definition of congruent triangles Two triangles are congruent if and only if their vertices can be matched up so that corresponding parts (angles and sides) of the triangle are congruent.

Describing parts of a triangle Angle R is opposite line segment SQ. Line segment SQ is included between angles S and Q. Angle S is opposite line segment QR. Angle Q is included between line segments QS and QR.

Postulate 12 Side-Side-Side (SSS) Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Postulate 13 Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Postulate 14 Angle-Side-Angle Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Theorem If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

HL Theorem If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Are these two overlapping triangles congruent?