By: Sebastian Enriquez Journal Chapter 5 By: Sebastian Enriquez

Perpendicular Bisector A perpendicular bisector is what cuts a line segment into equal parts of 90 degrees. The perpendicular bisector theorem states that if a point is located on the perpendicular point of a bisector then it’s equidistant from all the endpoints in the segment. The converse of this theorem says, that any point equidistant to the endpoints is on the perpendicular bisector of the segment.

Angle Bisector An angle bisector is a the line that cuts an angle into two equal halves. The angle bisector theorem states that if a point is on the bisector of the angle then it’s equidistant to the two sides in the angle. The angle bisector theorem converse states that if a point is in the interior of an angle and it’s equidistant from the sides of that angle, then it’s lying on the bisector of the angle.

Concurrency Concurrent: it’s when three or more lines meet at a single point. The concurrency of perpendicular bisectors of a triangle states that the intersection points are all equidistant to the vertices of the triangle. A circumcenter is where the perpendicular bisectors meet.

Concurrency II The concurrency of angle bisector of a triangle states that the bisectors intersect at a point where they’re all equidistant to the sides of the triangle. An incenter is where the three angle bisectors meet.

Median A median is the segment that connects any vertex to the midpoint of the opposite side. The centroid is where the three medians meet. The concurrency of median of a triangle theorem states that the medians of a triangle intersect in a point that is two thirds of the distance from all of the vertices.

Altitude The altitude of a triangle is a segment that connects the vertex to a line that contains the opposite side to the perpendicular side. The orthocenter is the point where the three altitudes meet. The concurrency of altitudes of a triangle states that the lines that contain the altitudes are concurrent.

Midsegment A midsegment is what joins two sides in a triangle. The midsegment theorem states that the midsegment of a triangle creates another segment parallel to the base which is half as long.

Exterior Angle Inequality The Exterior Angle Inequality says that the exterior angle of a triangle is greater than the measure of either interior angles.

Triangle Inequality It says that when you add up two sides in a triangle they should be bigger than the remaining side

Indirect Proof To write an indirect proof you first have to assume the conclusion is false by negating the prove statement. Then you have to establish that the assumption in step 1 is leading to a contradiction of some fact. Then state that the assumption is false and the conclusion or prove statement is true.

Examples Let’s say John went to his brother’s house and he left his house at 5:30 and reached his brother’s house at 6:30, John exceeded the speed of 55 mph, let’s say that statement is false and he didn’t exceed that speed. He drove 80 miles at a 55mph speed. At this speed John would need 80/55 (approx)= 1 hour 27 minutes to reach his brother’s place. So this means he must’ve exceeded the 55mph because the time it took him to get to his brother’s place was 1 hour, not 1 hour 27 minutes.

Hinge Theorem The Hinge theorem states that if two sides in a triangle are congruent to another two sides in another triangle, and the included angle of the first is bigger than the included angle of the other triangle, then the third side in the first triangle is longer than the third side in the other triangle. The converse of the hinge theorem states that if two sides in a triangle are congruent to two sides in another triangle and the third sides are not congruent, then the bigger included angle is across from the longest third side.