L4.6 Application of Triangle Theorems

L4.6 Application of Triangle Theorems

Geometry

Triangle Theorems

• The sum of the measures of the interior angles of a triangle is 180∘.

• The base angles of an isosceles triangle are congruent.

• If a triangle has two congruent angles then it is isosceles.

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

• The segment connecting two midpoints of a triangle is both parallel to and one half the length of the third side of the triangle.

Triangle Theorems

• The three medians of a triangle meet at a point called the centroid. The centroid divides each median in a 2:1 ratio with the larger segment being the one from the vertex to the centroid.

• The three angle bisectors of a triangle meet at a point called the incenter.

• The three altitudes of a triangle meet at a point called the orthocenter.

• The three perpendicular bisectors of a triangle meet at a point called the circumcenter.

Fill in the Blanks

The three medians of a triangle meet at a point called the centroid. The centroid divides each median in a 2:1 ratio with the larger segment being the one from the vertex to the centroid.
•Point P is the centroid of ΔABC. Find the length of 

 $\overline{AP}$  

Fill in the Blanks

The sum of the measures of the interior angles of a triangle is 180∘.

Solve for x

Prove that two of the medians of an isosceles triangle are congruent

• Hint: prove that ΔDBC≅ΔECB and then show that the medians are congruent because they are corresponding parts of the triangles.

Fill in the Blanks

Point P is the centroid of ΔABC and 
BD=18. Find the length of

 $\overline{BP}$  

Fill in the Blanks

Solve for x.

Open Ended

Name the four points of concurrency for a triangle.

Multiple Choice

Which of these is called the center of gravity.

• orthocenter

• incenter

• circumcenter

• centroid

Multiple Choice

What is the point of concurrency?

• Orthocenter

• Circumcenter

• Incenter

• Centroid

Multiple Choice

What type of special segments are being used?

• Perpendicular Bisectors

• Angle Bisectors

• Medians

• Altitudes

Multiple Choice

What type of special segments are being used?

• Altitudes

• Perpendicular Bisectors

• Medians

• Angle Bisectors

Multiple Choice

What is the point of concurrency?

• Orthocenter

• Centroid

• Circumcenter

• Incenter

Multiple Choice

If D is the orthocenter, what type of segments are drawn?

• medians

• altitudes

• angle bisectors

• perpendicular bisectors

Multiple Choice

Name the point of concurrency shown.

• Circumcenter
• Incenter
• Supercenter
• Neither

Multiple Choice

The figure is an example of a(n) ...

• altitude

• perpendicular bisector

• midsegment

• angle bisector

Multiple Choice

The figure is an example of a(n) ...

• angle bisector

• midsegment

• altitude

• median