5.1 Angle Relationships in a 5.1 Angle Relationships in a TriangleTriangle

Triangles can be classified by the measure of Triangles can be classified by the measure of their angles. These classifications include their angles. These classifications include acute triangles, , obtuse trianglesobtuse triangles, , right trianglesright triangles, and , and equiangular trianglesequiangular triangles..

The longest side of a triangle is opposite the The longest side of a triangle is opposite the largest interior angle and the shortest side of a largest interior angle and the shortest side of a triangle is opposite the smallest interior angle.triangle is opposite the smallest interior angle.

The measure of an exterior angle of a triangle is The measure of an exterior angle of a triangle is equal to the sum of the measures of the two equal to the sum of the measures of the two remote interior angles of a triangle.remote interior angles of a triangle.

The The Exterior Angle Inequality Theorem Exterior Angle Inequality Theorem states that states that the measure of an exterior angle of a triangle is the measure of an exterior angle of a triangle is greater than the measure of either of its remote greater than the measure of either of its remote interior angles.interior angles.

Lesson 5.2 Lesson 5.2 Side Relationships of a TriangleSide Relationships of a Triangle Triangles can be classified by the Triangles can be classified by the

lengths of their sides. These lengths of their sides. These classifications include classifications include scalene scalene trianglestriangles, , isosceles trianglesisosceles triangles, and , and equilateral trianglesequilateral triangles..

The The Triangle Inequality Theorem Triangle Inequality Theorem states states that the sum of the lengths of any two that the sum of the lengths of any two sides of a triangle is greater than the sides of a triangle is greater than the length of the third side of the triangle.length of the third side of the triangle.

Lesson 5.3Lesson 5.3Points of ConcurrencyPoints of Concurrency

An An angle bisector angle bisector is a line segment, or ray that is a line segment, or ray that divides an angle into two smaller angles of equal divides an angle into two smaller angles of equal measures.measures.

Concurrent lines Concurrent lines are three or more lines that are three or more lines that intersect at the same point.intersect at the same point.

The The incenter incenter of a triangle is the point at which the of a triangle is the point at which the three angle bisectors intersect.three angle bisectors intersect.

A A segment bisector segment bisector is a line, segment, or ray that is a line, segment, or ray that divides a segment into two smaller segments of divides a segment into two smaller segments of equal length.equal length.

The The circumcenter circumcenter of a triangle is the point at of a triangle is the point at which the three perpendicular bisectors intersect.which the three perpendicular bisectors intersect.

Lesson 5.3 Cont’dLesson 5.3 Cont’d

A A median median of a triangle is line segment that of a triangle is line segment that connects a vertex to the midpoint of the connects a vertex to the midpoint of the side opposite the vertex.side opposite the vertex.

The The centroid centroid of a triangle is the point at of a triangle is the point at which the three medians intersectwhich the three medians intersect

An An altitude altitude of a triangle is a perpendicular of a triangle is a perpendicular line segment that is drawn from a vertex line segment that is drawn from a vertex to the opposite side.to the opposite side.

The The orthocenter orthocenter of a triangle is the point of a triangle is the point at which the three altitudes intersect.at which the three altitudes intersect.

Lesson 5.4 Lesson 5.4 Direct and Indirect ProofsDirect and Indirect Proofs

A A two-column two-column formal proof is a way of formal proof is a way of writing a proof such that each step is listed writing a proof such that each step is listed in one column and the reason for each in one column and the reason for each step is listed in the other column.step is listed in the other column.

A A proof of contradiction proof of contradiction begins with a begins with a negation of the conclusion, meaning that negation of the conclusion, meaning that you assume the opposite of the you assume the opposite of the conclusion. When a contradiction is conclusion. When a contradiction is developed, then the conclusion must be developed, then the conclusion must be true.true.

Lesson 5.5Lesson 5.5Proving Triangles Congruent: SSS and Proving Triangles Congruent: SSS and

SASSAS If two triangles are similar, then the ratios of the If two triangles are similar, then the ratios of the

lengths of the corresponding sides are proportional lengths of the corresponding sides are proportional and the measures of the corresponding angles are and the measures of the corresponding angles are equal.equal.

If two triangles are congruent, then the triangles If two triangles are congruent, then the triangles are similar and the ratios of the lengths of the are similar and the ratios of the lengths of the corresponding sides are equal to 1.corresponding sides are equal to 1.

The The Side-Side-Side Congruence Theorem Side-Side-Side Congruence Theorem states states that if all corresponding sides of two triangles are that if all corresponding sides of two triangles are congruent, then the triangles are congruent.congruent, then the triangles are congruent.

The The Side-Angle-Side Congruence Theorem Side-Angle-Side Congruence Theorem states states that if two sides and the included angle of one that if two sides and the included angle of one triangle are congruent to two sides and the triangle are congruent to two sides and the included angle of a second triangle, the then included angle of a second triangle, the then triangles are congruent triangles are congruent

Lesson 5.6Lesson 5.6Proving Triangles Congruent: ASA and Proving Triangles Congruent: ASA and

AASAAS The The Angle-Side-Angle Congruence Postulate Angle-Side-Angle Congruence Postulate

states if two angles of one triangle are states if two angles of one triangle are congruent to two angles of another congruent to two angles of another triangle, then the triangles are congruent.triangle, then the triangles are congruent.

The The Angle-Angle-Side Congruence Theorem Angle-Angle-Side Congruence Theorem states if two angles of one triangle are states if two angles of one triangle are congruent to two angles of another triangle congruent to two angles of another triangle and two corresponding non-included sides and two corresponding non-included sides are congruent, then the triangles are are congruent, then the triangles are congruent.congruent.

Lesson 5.7 Lesson 5.7 Proving Triangles Congruent: Proving Triangles Congruent:

HLHL The The Hypotenuse-Leg Congruence Hypotenuse-Leg Congruence

Theorem Theorem states if the hypotenuse states if the hypotenuse and a leg of a right triangle are and a leg of a right triangle are congruent to the hypotenuse and leg congruent to the hypotenuse and leg of another triangle, then the triangles of another triangle, then the triangles are congruent.are congruent.