Chapter 5 Relationships in Triangles 5.1 Bisectors, Medians, and Altitudes 5.2 Inequalities and Triangles 5.4 The Triangle Inequality 5.5 Inequalities Involving Two Triangles

Warm-up review: Draw in each line or segment on the given triangle (show all congruency markings) perpendicular bisector angle bisector altitude median sidemeasure AB BC CA anglemeasure A B C B C A Examples: Q S N R M 1 2

5.1 Bisectors, Medians, and Altitudes Objectives: To identify and use perpendicular bisectors and angle bisectors in triangles. To identify and use medians and altitudes in triangles. Let’s look at the point where these bisectors all cross!

Points of Concurrency perpendicular bisectors equidistant from each vertex PA = PB = PC angle bisectors bisect each angle of the triangle equidistant from each side PX = PY = PZ altitudes from each vertex perpendicular to the opposite side medians vertex to midpoint of opposite side B C A B C A B C A B C A When 3 or more rays, segments, or lines intersect at a point they create: *center of gravity P P X Z Y T X Z Y

For what type of is the incenter, centroid, orthocenter, & circumcenter the same point ? 2) p. 240 equilateral Points S, T, and U are midpoints of DE, EF and DF. Find x if DA = 6 and AT = 2x – 5. E U T S FD A P Q R l m n T S What is the name of point A? ________________ What is the name of point T? ________________

5.2 Inequalities and Triangles Objectives: To recognize and apply properties of inequalities to the measure of angles of a triangle AND to the relationships between angles and sides of a triangle. Exterior Angle Inequality Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. R Q P Side and Angle Inequality Relationship

Isosceles Example: Determine the relationship between the given angles. S U R V T The Triangle Inequality Objectives: We will learn how to apply the Triangle Inequality Theorem and determine the shortest distance between a point and a line. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem A C B *We can use this theorem to determine if the 3 given measures can be the sides of a triangle. Can only compare within the same triangle!

Example: Given the lengths 2, 4, 5. Can these lengths form a triangle? Use the theorem and check each combination: Yes; 2, 4, 5 can be the sides of a triangle. (we can draw a triangle with these lengths) Example: Given the lengths 6, 8, 14. Can these lengths form a triangle? No Example: P (1, 2) Q (4, -3) R (0, 5) What do we use to find the lengths of the sides? Can these coordinates form a triangle?

We can also determine the possible side lengths: A C B x 8 15 AC B 2 x 10 Example: What is the shortest distance from a point to a line? Perpendicular to it!

Identify the altitude in triangle PRH. ________ R H P Matching: 1.Angle bisectors Centroid 2.Medians Circumcenter 3.Altitudes Incenter 4.Perpendicular bisectors Orthocenter Warm-Up: Points A, B, and C are the midpoints of and. Find v and z if 4 and 1/3 C B Y A X W v + 3 YA = 6v z 4 D

5.5 Inequalities Involving Two Triangles Objectives : We will learn how to apply the SAS and SSS Inequalities. SAS Inequality - “Hinge Theorem” R Q T S H P 1.75” 1.0” Example: A C D B 6 6 What is the relationship between BC and CD?

Side-Side-Side Inequality B P R Q C A 2 1 S V R T Example: They share a common side! L N P M 15 Example: What is the relationship between ML and NP?