Geometry Journal Chapter 5          Name_________

In your own words respond to the following:

 

_____(0-10 pts.) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse.  Give 3 examples of each.

Perpendicular bisector is a line perpendicular to a segment at the segment´s midpoint.

perpendicular bisector 1.ggb    

perpendicular 2.ggb

perpendicular bisector 3.ggb

Perpendicular Bisector Theorem: if a point lies on the perpendicular bisector of a segment then it is equidistant from both of the endpoints of the segment.

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converse: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            

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_____(0-10 pts.) Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each.

Angles Bisector: is a ray that divides an angle into two congruent angles.

angle bisector 1.ggb

angle bisector 2.ggb

angle bisector 3.ggb

Angles Bisector Theorem: if a point lies on the interior of an angle and is on the angle bisector, then it is equidistant to both sides of the angle.

 

Converse: if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

_____(0-10 pts.) Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each.

Concurrent is the point where 3 or more lines intersect

The point of concurrency of the perpendicular bisector is called the circumcenter

The circumcenter is equidistance to the vertices of the triangle   

circumcenter of an obtuse triangle.ggb

circumcenter of an acute triangle.ggb

circumcenter of a right angle.ggb

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_____(0-10 pts.) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each.

Concurrency of angle bisectors of a triangle theorem: states that the angle bisectors of a triangle intersect at a point that is equidistant from the sides of a triangle.

Incenter: is a point of concurrency of the angle bisectors of a triangle, it is equidistant from the sides.

 

_____(0-10 pts.) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. Give at least 3 examples of each.

Median: is a line drawn from one vertex of a triangle to the midpoint of the opposite side.

Centroid is the center of the medians; where all the medians meet.

 

_____(0-10 pts.) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples.

Orthocenter: is where all of the altitudes meet.

Altitude: is the line from the vertex perpendicular on the opposite.

Concurrency of altitudes of a triangle theorem: states that the lines containing the altitudes of a triangle are concurrent.

cuncurrency t.ggb

_____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples.

Midsegment  is a segment that joins any two midpoints of a triangle,the midsegment is parallel to the opposite side and it is half as long as the opposite side.

Midsegment theorem states that the midsegment of a triangle must be parallel to a side of the triangle, and its length is half the size of that side.

_____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples.

Angle-Side relationships: in triangles say that if two sides of a triangle are not congruent, then the larger angle is opposite to the longer side in a triangle, the larger angle is the larger opposite longer side

If two angles of a triangle are not congruent to each other, then the longer side is opposite to the larger angle

 

_____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples.

The exterior angle is supplementary to the adjacent interior angle and it is greater than either of the non adjacent interior angles.

_____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples.

Triangle inequality: states that in every triangle the sum of the lengths of the two shorter sides must always be longer than the third sides.

8,3,10                                 7,21,23                                             10,9,20

8+3=11,10                         7+21=28,23                                    10+9=19,20

YES                                        YES                                                  NO

 

 

_____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples.

Hinge theorem states that is the two sides of two triangles are congruent but the third side is not congruent then the triangle with the longer side will have a larger included angle.

 

Converse: states that if the two sides of one angle are congruent to the two sides of another angle and its third sides are not congruent, then the larger included angle is across from the longer third side.

 

 

 

_____(0-5 pts.)   Neatness and originality bonus.

 

­­______Total points earned (120 possible)