JOURNAL CHAPTER

5Sara Wunderlich

Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each. 

Perpendicular bisector

Perpendicular bisector- line that intersects a segment, forming a 90° angle, at the segment’s midpoint. Basically, It is when you bisect and it crosses a segment exactly at a right angle

Perpendicular bisector theorem-If a segment is bisected by a perpendicular line, then any point on the perpendicular bisector is equidistant of the endpoints of the segment.

Converse-if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Perpendicular bisector theoremGiven: L is the perpendicular bisector of AB. Prove: AYX = BYX Since L is the perpendicular bisector of AB, L AB and Y is the midpoint of AB. By the def. of perp, <AYX and <BYX are right angles and <AYX = <BYX. BY the definition of midpoint, AY= BY. And by the reflexive property, XY = XB so, AYX = BYX by SAS.

EXAMPLE 1

example2

Example 3 constructing it

Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each. 

Angle bisector- is the line that divides the angle into two equal parts, or congruent angles.

Angle bisector theorem: if a point is on the bisector of an angle, then it is equidistant from the 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 the bisector.

Angle Bisector

Example 2

Constructing it, example 3

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

When three or more lines intersect in a single point they are said to be concurrent.

In a triangle, the three medians, three perpendicular bisectors, three angle bisectors, and three altitudes are each concurrent.

CIRCUMCENTER: the point of congruency, where three or more perpendicular lines intersect.

Concurrency of Perpendicular Bisectors of a Triangle Theorem   The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

Example 1

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

Incenter – the point where the three angle bisectors meet, making it a point of concurrency.

the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. 

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

The median of a triangle is a line segment that goes from a vertex to the midpoint of the opposite side.

Centroid- the point where the medians of the triangle intersect, it is always inside the triangle.

Concurrency of Medians of a Triangle Theorem   The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

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. 

Altitude

Altitude of a triangle- is a line segment from a vertex to the opposite side so that the line segment is perpendicular to the side. Since a triangle has 3 vertices, it has 3 altitudes. It can be inside outside or on triangle.

Orthocenter – the point of intersection of the three altitudes, making the concurrent.

Concurrency of Altitudes of a Triangle Theorem   The lines containing the altitudes of a triangle are concurrent.

Describe what a Midsegment is. Explain the Midsegment theorem. Give at least 3 examples. 

Midsegment

Midsegment - is a line segment whose endpoints are the midpoints of a triangle. Every triangle has three Midsegment.Midsegment theorem:A Midsegment of a triangle is parallel to a side of the triangle and is half as long.

Example 1

Example 2

Example 3

Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples.In any triangle the side opposite to the biggest angle will always have the longest length. The side opposite to the smallest angle will have smallest length. 

longer and shorter sides of a triangle

Example 1 Example 2

Example 3

Describe the exterior angle inequality. Give at least 3 examples.

 

Exterior Angle inequality

Describe the triangle inequality. Give at least 3 examples.

In any triangle, any 2 sides of a triangle must add up to more than third side.AB+BC>ACBC+AC>ABAC+BC>BC 

Triangle inequality

Example 1

Example 2

Example 3

Describe how to write an indirect proof. Give at least 3 examples.

To write and indirect proof:1: assume that what you are trying to prove is false.2: Try to prove using steps of a proof.3: when you come to a point of contradiction, you have proof that the theory is true 

Indirect proof

Example 1A scalene triangle cannot have two congruent angles.A scalene triangle can have 2 congruent angles Base angle theorem, it would be isosceles isosceles cannot be scalene by def. of isosceles a scalene triangle cannot have 2 congruent angles.

Example 2A triangle cannot have two obtuse anglesA triangle can have two obtuse angles <1,<2 ,<1+<2+<3 has to equal 180 <1 and <2 do not equal 180 therefore a triangle cannot have 2 obtuse angles.

Example 3An isosceles triangle cannot have a base < that is a right angle.An isosceles triangle can have a base < that is a right angle.<1+<2+<3 have to equal 180<1+<2 = 180, so therefore an isosceles triangle cannot have a base < that is right.

Describe the hinge theorem and its converse. Give at least 3 examples.

Hinge Theorem - If 2 sides of 2 triangles are congruent and the angle between them is not congruent, then the triangle with the larger angle will have a longer third side. 

Hinge Theorem

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

a a1

c

b

c1

b1

a is congruent to a1 and b to b1

Example 1

Example 2

Example 3