Triangles : a three-sided polygon Polygon: a closed figure in a plane that is made of segments, called sides, that intersect only at their endpoints, called vertices. Acute triangle: All angles are acute. Obtuse triangle: One angle is obtuse. Right triangle: One angle is right. Equiangular triangle: acute triangle in which all angles are equal Scalene triangle: No two sides are congruent Isosceles triangle: At least two sides are congruent Equilateral triangle: all the sides are congruent.

Angles in triangles Theorem 4.1: The sum of the measures of the angles of a triangle is 180° Theorem 4.2: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Exterior angle: angle formed by one side of a triangle and the extension of another side. Exterior angle Remote interior angles: the interior angles not adjacent to a given exterior angle.

Theorem 4.3: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles Corollary 4.1: The acute angles of a right triangle are complementary Corollary 4.2: There can be at most one right or obtuse angle in a triangle. Congruent triangles are triangles that are the same size and shape. Congruent triangles have the corresponding six parts (three angles, three sides) congruent. Definition: Two triangles are congruent if and only if their corresponding parts are congruent. Congruence of triangles is reflexive, symmetric, and transitive.

Proving Triangles Congruent SSS: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Given  STU with vertices S(0,5), T(0,0) and U(-2,0) and  XYZ with vertices X(4,8), Y(4,3) and Z(6,3), determine if the triangles are congruent. Postulate 4-1 Side-Side-Side

SAS: If two sides and the included angle of a triangle are congruent to two sides and the included angle in another triangle, then the triangles are congruent. Postulate 4-2Side-Angle-Side Write a proof for the following: Given: X is the midpoint of BD X is the midpoint of AC Prove:  DXC   BXA A BC D X

ASA: If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Postulate 4-3 Angle-Side-Angle Write a two-column proof Given VR  RS UT  SU RS  US Prove VR  TU V R S UT

AAS: If two angles and a non-included side of a triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent. Postulate 4-5 Angle-Angle-Side Some of the measurements of  ABC and  DEF are given. Determine if the triangles are congruent. A 4 cm30° C B E F D 4 cm 2.5 cm 30°

SSS: The three sides of one triangles are congruent with the three sides of the other. SAS: Two sides and the included angle of one triangle are congruent with the two sides and the included angle of the other. ASA:Two angles and the included side of one triangle are congruent with the two angles and the included side of the other. AAS:Two angles and the nonincluded side of one triangle are congruent with the two angles and the nonincluded side of the other. Qualifications of congruent triangles: All six parts of one triangle must be congruent with all six part of the other triangle.

Isosceles Triangles Theorem 4.6: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4.7: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollary 4.3: A triangle is equilateral if an only if it is equiangular. Corollary 4.4: Each angle of an equilateral triangle measures 60°

Qualifications of congruent triangles: SSS: The three sides of one triangles are congruent with the three sides of the other. SAS: Two sides and the included angle of one triangle are congruent with the two sides and the included angle of the other. ASA:Two angles and the included side of one triangle are congruent with the two angles and the included side of the other. AAS:Two angles and the nonincluded side of one triangle are congruent with the two angles and the nonincluded side of the other. All six parts of one triangle must be congruent with all six part of the other triangle.

Isosceles Triangles Theorem 4.6: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Theorem 4.7: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollary 4.3: A triangle is equilateral if an only if it is equiangular. Corollary 4.4: Each angle of an equilateral triangle measures 60°