4.1.2 Angle Relationships in Triangles Key Objectives
• Find the measure of interior and exterior angles of triangles. • Apply theorems about the interior and exterior angles of triangles. Key Terms • An auxiliary line is a line that is added to a figure to aid in a proof. • A corollary is a theorem whose proof follows directly from another theorem.
• The interior of a figure is the set of all points inside the figure. • The exterior of a figure is the set of all points outside the figure. • An interior angle is formed by two sides of a polygon. • An exterior angle is formed by one side of a polygon and the extension of an adjacent side. • Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Theorems, Postulates, Corollaries, and Properties • Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. • Corollary The acute angles of a right triangle are complementary. • Corollary The measure of each angle of an equiangular triangle is 60°. • Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. • Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Example 1 Triangle Sum Theorem
The Triangle Sum Theorem states that the sum of the interior angles of any triangle is 180°. This conclusion can be visualized by manipulating a triangle’s angles. Begin with a triangle of any type or size and cut the angles (corners) off so that each piece of the triangle contains one vertex. Each piece represents one of the triangle’s interior angles. Then, place the three angles so that they meet at their vertices. The three angles always form a straight angle, no matter what type or size of triangle is used. The measure of a straight angle is always 180°. Therefore, the sum of the angle measures of a triangle is always 180°.
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