Triangles

Question 1

median-what

Answer 1

connects vertex to midpoint of opposite side

Question 2

median-how

Answer 2

perpendicular bisect a side. connect where they intersect to opposite vertex

Question 3

midsegment-what

Answer 3

connects midpoints of 2 sides of a triangle

Question 4

altitude-how

Answer 4

swing arc with point on vertex that hits opposite side twice. then swing another arc from each new point. connect newest point to vertex.

Question 5

midsegment-how

Answer 5

perpendicular bisect two sides. connect the midpoints

Question 6

altitude-what

Answer 6

perpendicular segment from vertex to opposite side or line containing opposite side

Question 7

Centroid

Answer 7

Medians

Question 8

Circumcenter

Answer 8

Perpendicular bisector

Question 9

Incenter

Answer 9

Angle bisector

Question 10

Orthocenter

Answer 10

Altitudes

Question 11

acute

Answer 11

all angles smaller than 90

Question 12

obtuse

Answer 12

one angle larger than 90

Question 13

right

Answer 13

one right(90 degrees) angle

Question 14

angle bisector-what

Answer 14

cuts an angle into two exactly

Question 15

angle bisector-how

Answer 15

swing an arc from vertex that intersects both sides of the angle. rom each of those two intersection points, draw smaller arcs in the interior of the angle that intersect each other. draw a straight line from the vertex to the intersection of these smaller arcs to create the bisector.

Question 16

perpendicular bisector-how

Answer 16

swing an arc from one endpoint with radius the length of segment. repeat on other side. connect two intersecetion points

Question 17

perpendicular bisector-what

Answer 17

divides a side into two equal segemnts

Question 18

scalene

Answer 18

no side is the same as another

Question 19

isosceles

Answer 19

two sides are the same

Question 20

equilateral

Answer 20

all sides are the same

Question 21

Conjecture 9-converse of perpendicular bisector

Answer 21

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

Question 22

Conjecture 10-shortest distanmce

Answer 22

Shortest Distance - The shortest distance from a point to a line is measured along the perpendicular segment from the point to the line.

Question 23

Conjecture 11-angle bisector

Answer 23

Angle Bisector - If a point is on the bisector of an angle, then the point is equidistant from both sides of the angle.

Question 24

Conjecture 12-centroid existence

Answer 24

Centroid Existence - The three medians of a triangle are concurrent at a point called the centroid.

Question 25

Conjecture 13-centroid location

Answer 25

Centroid Location - For any given triangle, the centroid is on the interior of the triangle.

Question 26

Conjecture 14- centroid distance

Answer 26

Centroid Distance -The centroid divides the median into two segments in such a way that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint.

Question 27

Conjecture 15-center of gravity

Answer 27

Center of Gravity - The centroid of a triangle is the center of gravity of the triangular region.

Question 28

Conjecture 16- orthocenter existence

Answer 28

Orthocenter Existence - The three altitudes of a triangle are concurrent at a point called the orthocenter.

Question 29

Conjecture 17- orthocenter location

Answer 29

Orthocenter Location: If a triangle is acute, then the orthocenter is in the interior region of the triangle. If a triangle is right, then the orthocenter is the vertex of the right angle. If a triangle is obtuse, then the orthocenter is on the outside of the triangle.

Question 30

Conjecture 18- orthocenter distance

Answer 30

Orthocenter Distance - The product of the parts into which the orthocenter divides an altitude is the equivalent for all three perpendiculars.

Question 31

Conjecture 19- circumcenter existence

Answer 31

Circumcenter Existence - The three perpendicular bisectors of a triangle are concurrent at a point called the circumcenter.

Question 32

Conjecture 20- circumcenter location

Answer 32

Circumcenter Location: If a triangle is acute, then the circumcenter is on the interior region of the triangle. If the triangle is right, then the circumcenter is the midpoint of the hypotenuse. If the triangle is obtuse, then the circumcenter is on the outside of the triangle.

Question 33

Conjecture 21-circumcenter distance

Answer 33

Circumcenter Distance - The circumcenter is equidistant from the vertices of a triangle.

Question 34

Conjecture 22- incenter existence

Answer 34

Incenter Existence - The three angles bisectors of a triangle are concurrent at a point called the incenter.

Question 35

Conjecture 23- incenter location

Answer 35

Incenter Location - For any given triangle, the incenter is on the interior of the triangle.

Question 36

Conjecture 21- incenter distance

Answer 36

Incenter Distance - The incenter is equidistant from the sides of a triangle.

Question 37

Conjecture 22- equilateral triangle

Answer 37

Equilateral Triangle - For an equilateral triangle, all 4 points of concurrency are the same point.

Question 38

Conjecture 23- isoscles triangle

Answer 38

Isosceles Triangle - For an isosceles triangle, the points of concurrency are all collinear (from vertex angle, it is the Euler Line)

Question 39

Conjecture 24- euler line

Answer 39

Euler Line - The circumcenter, centroid, and orthocenter are collinear in any triangle that is not equilateral.

Question 40

Conjecture 25- euler segment

Answer 40

Euler Segment - The centroid divides the Euler Segment into 2 parts so that the distance from the circumcenter to the centroid is half of the distance from the centroid to the orthocenter.

Question 41

Conjecture 26- triangle sum

Answer 41

Triangle Sum - The sum of the angles in every triangle is 180 degrees.

Question 42

Conjecture 27- third angle

Answer 42

Third Angle - If 2 angles of 1 triangle are equal in measure to 2 angles of another triangle, then the third angle of each triangle must be congruent.

Question 43

Conjecture 28- isosceles triangle

Answer 43

Isosceles Triangle - If a triangle is isosceles, then the base angles must be congruent.

Question 44

Conjecture 29- converse of isoscles triangle

Answer 44

Converse of Isosceles Triangle - If a triangle has 2 congruent angles, then the triangle is isosceles.

Question 45

Conjecture 30- triangle inequality

Answer 45

Triangle Inequality - The sum of the length of any 2 sides of a triangle is greater than the length of the third side.

Question 46

Conjecture 3- converse of triangle inequality

Answer 46

Converse of the Triangle Inequality - If 3 positive real numbers exist such that each is less than the sum of the other 2, then there exists a triangle with these numbers as its side lengths.

Question 47

Conjecture 32- side angle inequality

Answer 47

Side-Angle Inequality - In a triangle, if one side is longer than the other side, then the side angle opposite the larger side is larger than the angle opposite the shorter side.

Question 48

Conjecture 33- triangle exterior angle

Answer 48

Triangle Exterior Angle - The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.