Isosceles Triangles & Points of Concurrency

Geometry

Chapter 4-6 notes

Isosceles Triangles & Points of Concurrency

Lesson 4-6 OBJECTIVE: SWBAT apply the isosceles triangle theorems and be able to identify the various points of concurrency.

An _______________________ has two congruent sides.

The angle formed by these sides is

called the ________________.

The other two angles are called the ___________________.

Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles

opposite those sides are congruent.

Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides

opposite those angles are congruent.

Practice Problems

2. Find x.

An _______________________ has three congruent sides.

The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles.

1.

A triangle is equilateral if and only if it is equiangular. 2.

Each angle of an equilateral triangle is 60?

.

Practice Problems

Find the value of x.

Medians, Altitudes & Points of Concurrency

A segment from a vertex of a triangle to the midpoint of the opposite side of the triangle is called

the _____________ of that triangle.

A segment from a vertex of a triangle perpendicular to the line that contains the opposite side of the triangle is called the ___________ of that triangle. XA is the altitude of triangle XYZ

Every triangle has _________ medians and ________ altitudes. The medians of a triangle are always __________ the triangle. However, an altitude of a triangle can be ___________, ____________, or part of the triangle.

Practice Problems

For each triangle below, draw the median from A, the altitude from B, and the perpendicular bisector of AB.

Points of concurrency related to triangles

The term "concurrent" simply means "meeting or intersecting at a point." Therefore, "points of

concurrency" refers to the points where segments of a triangle meet.

The centroid of a triangle is the point where the medians meet.

The incenter of a triangle is the point where the angle bisectors meet.

The orthocenter of a triangle is the point where the altitudes meet.

The circumcenter of a triangle is the point where the perpendicular bisectors meet.

Reminder: The 4 points of concurrency are... 1. Centroid (medians) 2. Incenter (angle bisectors) 3. Orthocenter (altitudes) 4. Circumcenter (perpendicular bisectors)

The orthocenter and the circumcenter can be located inside, on the border of, or outside the triangle. The centroid and the incenter must be inside the triangle.

Practice Problems

Name each blue point as one of the following: 1. Centroid 2. Incenter 3. Orthocenter 4. Circumcenter

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