5.1 Angle Relationships in a Triangle Triangles can be classified by the measure of their angles. These classifications include acute triangles, obtuse triangles, right triangles, and equiangular triangles. The longest side of a triangle is opposite the largest interior angle and the shortest side of a triangle is opposite the smallest interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of a triangle. The Exterior Angle Inequality Theorem states that the measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

Lesson 5.2 Side Relationships of a Triangle Triangles can be classified by the lengths of their sides. These classifications include scalene triangles, isosceles triangles, and equilateral triangles. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side of the triangle.

Lesson 5.3 Points of Concurrency An angle bisector is a line segment, or ray that divides an angle into two smaller angles of equal measures. Concurrent lines are three or more lines that intersect at the same point. The incenter of a triangle is the point at which the three angle bisectors intersect. A segment bisector is a line, segment, or ray that divides a segment into two smaller segments of equal length. The circumcenter of a triangle is the point at which the three perpendicular bisectors intersect.

Lesson 5.3 Cont’d A median of a triangle is line segment that connects a vertex to the midpoint of the side opposite the vertex. The centroid of a triangle is the point at which the three medians intersect An altitude of a triangle is a perpendicular line segment that is drawn from a vertex to the opposite side. The orthocenter of a triangle is the point at which the three altitudes intersect.

Lesson 5.4 Direct and Indirect Proofs A two-column formal proof is a way of writing a proof such that each step is listed in one column and the reason for each step is listed in the other column. A proof of contradiction begins with a negation of the conclusion, meaning that you assume the opposite of the conclusion. When a contradiction is developed, then the conclusion must be true.

Lesson 5.5 Proving Triangles Congruent: SSS and SAS If two triangles are similar, then the ratios of the lengths of the corresponding sides are proportional and the measures of the corresponding angles are equal. If two triangles are congruent, then the triangles are similar and the ratios of the lengths of the corresponding sides are equal to 1. The Side-Side-Side Congruence Theorem states that if all corresponding sides of two triangles are congruent, then the triangles are congruent. The Side-Angle-Side Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, the then triangles are congruent

Lesson 5.6 Proving Triangles Congruent: ASA and AAS The Angle-Side-Angle Congruence Postulate states if two angles of one triangle are congruent to two angles of another triangle, then the triangles are congruent. The Angle-Angle-Side Congruence Theorem states if two angles of one triangle are congruent to two angles of another triangle and two corresponding non-included sides are congruent, then the triangles are congruent.

Lesson 5.7 Proving Triangles Congruent: HL The Hypotenuse-Leg Congruence Theorem states if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another triangle, then the triangles are congruent.