Jose Pablo Reyes 10 – 5

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

 

Perpendicular bisector – is a line that goes through a segment cutting it into equal parts, creating 90°angles

Perpendicular bisector theorem – if a segment is bisected by a perpendicular line then any point on the perpendicular bisector is equidistant to the end point of the segment

Converse – is a point is equidistant from the endpoint of a segment, then the point lies on the perpendicular bisector of the segment

Perpendicular bisector – examples

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

Angle bisector – line/ray that goes exactly through the half of an angle

Angle bisector theorem – if an angle is bisected by a line/ray then any point on that line is equidistant from both sides of the angle

Converse – if a given point in the interior of an angle is equidistant from the sides of the angle then it is on the bisector of the angle

Angle bisector – examples

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 – three or more lines that intersect at one point

Circumcenter – is the point of congruency in a triangle which is where the points intersect.

Concurrency of perpendicular bisectors of a triangle theorem – perpendicular bisectors of a triangle intersect in a point that is equidistant from the vertices

Concurrent, circumcenter,– examples

concurrency of perpendicular bisectors of a triangle theorem – examples

Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each.

The concurrency of angle bisectors of a triangle says that the 3 bisectors are congruent

The incenter is the point of concurrency of the angle bisectors, always inside the triangle

Concurrency of angle bisector of a triangle theorem and incenter

– examples

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 segments in which the endpoints are a vertex of the triangle and the midpoint of the opposite side

Centroid – is the point of concurrency of the medians of a triangle.

The centroid of a triangle is 2/3 of the distance from each vertex to the midpoint of the opposite side

Median, Centroid of a Triangle – Examples

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 of a triangle – is a segment from one vertex to the opposite side so the segment is perpendicular to the side . There are 3 altitudes for any triangle, it can be inside, outside or in the triangle

Orthocenter – is the point where the three altitudes of a triangle intersect. In obtuse triangles the orthocenter is outside

Concurrency of altitudes of a triangle theorem – the three lines containing the altitudes of a triangle are congruent

Altitude and orthocenter – examples

Concurrency of altitudes of a triangle theorem – Examples

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

Midsegment of a triangle – any segment that joins the midpoint of two sides of the triangle

Midsegment theorem – a midsegment of a triangle is parallel to the side of the triangle and the length is half the length of that side

Midsegment theorem – examples

Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples.

 

Angle side relation in a triangle, the side opposite to the biggest angle will always have the longest length, the side opposite to the smallest angle will have the smallest length

Angle side relation – examples

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

Exterior angle inequality – the exterior angle is greater than the non-adjacent interior angles of the triangle

Triangle inequality – any two sides of a triangle must add up to more than the 3rd side

Exterior angle inequality, triangle inequality – examples

Describe how to write an indirect proof. Give at

least 3 examples.

Steps to write an indirect proof – 1. Assume that what you are trying to

prove is false2. Try to prove it using the same steps as

in normal proofs3. When you face a contradiction, you

have to prove the theory true

Indirect proofs – examples

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

Hinge theorem – if two sides of two different triangles are congruent and the angle between them is not congruent, then the triangle with the larger angle will have the longer 3rd side

Hinge theorem – examples