All Conjectures

Question 1

Line Segment Congruence :

Answer 1

Two line segments are congruent if and only if they have the same length or measure.

Question 2

Angle Congruence :

Answer 2

Two angles are congruent if and only if they have the same measure.

Question 3

Polygon Congruence :

Answer 3

Two polygons are congruent if and only if their corresponding sides and angles are congruent.

Question 4

Linear Pair :

Answer 4

If two angles form a linear pair then they are supplementary.

Question 5

Vertical Angle :

Answer 5

If two angles are vertical angles then they are congruent.

Question 6

Lines :

Answer 6

If two parallel lines are cut by a transversal then the corresponding angles are congruent

Question 7

Converse of Parallel Lines :

Answer 7

If two lines are cut by a transversal to form pairs of congruent corresponding angles then the lines are parallel.

Question 8

Perpendicular Bisector :

Answer 8

If a point is on the perpendicular bisector of a line segment then the point is equidistant from the endpoints of the segment.

Question 9

Converse of Perpendicular Bisector :

Answer 9

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

Question 10

Shortest Distance :

Answer 10

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

Question 11

Angle Bisector :

Answer 11

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

Question 12

Centroid Existence :

Answer 12

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

Question 13

Centroid Location :

Answer 13

For any given triangle the centroid is on the interior of the triangle.

Question 14

Centroid Distance :

Answer 14

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 15

Center of Gravity :

Answer 15

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

Question 16

Orthocenter Existence :

Answer 16

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

Question 17

Orthocenter Location:

Answer 17

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 18

Orthocenter Distance :

Answer 18

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

Question 19

Circumcenter Existence :

Answer 19

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

Question 20

Circumcenter Location:

Answer 20

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 21

Circumcenter Distance :

Answer 21

The circumcenter is equidistant from the vertices of a triangle.

Question 22

Incenter Existence :

Answer 22

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

Question 23

Incenter Location :

Answer 23

For any given triangle the incenter is on the interior of the triangle.

Question 24

Incenter Distance :

Answer 24

The incenter is equidistant from the sides of a triangle.

Question 25

Equilateral Triangle :

Answer 25

For an equilateral triangle all 4 points of concurrency are the same point.

Question 26

Isosceles Triangle :

Answer 26

For an isosceles triangle the points of concurrency are all collinear (from vertex angle

Question 27

Euler Line :

Answer 27

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

Question 28

Euler Segment :

Answer 28

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 29

Triangle Sum :

Answer 29

The sum of the angles in every triangle is 80 degrees.

Question 30

Third Angle :

Answer 30

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

Question 31

Isosceles Triangle :

Answer 31

If a triangle is isosceles then the base angles must be congruent.

Question 32

Converse of Isosceles Triangle :

Answer 32

If a triangle has 2 congruent angles then the triangle is isosceles.

Question 33

Triangle Inequality :

Answer 33

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

Question 34

Converse of the Triangle Inequality :

Answer 34

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 35

Side:Angle Inequality :

Answer 35

In a triangle if one side is longer than the other side

Question 36

Triangle Exterior Angle :

Answer 36

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

Question 37

Triangle Side:Side:Side :

Answer 37

If the three sides of one triangle are congruent to the three sides of another trianglethen the triangles are congruent.

Question 38

Triangle Side:Angle:Side :

Answer 38

If two sides and their included angle of a triangle are congruent to another two sides and their included angle of a triangle then the two triangles are congruent.

Question 39

Triangle Angle:Side:Angle :

Answer 39

If two angles and their included side of a triangle are congruent to another two angles and their included side of a triangle then the two triangles are congruent.

Question 40

Triangle Side:Angle:Angle :

Answer 40

If two angles and a non:included side of a triangle are congruent to the corresponding angles and side of a different triangle then the triangles are congruent.

Question 41

Hypotenuse:Leg :

Answer 41

If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of another triangle then the triangles are congruent.

Question 42

Vertex Angle Bisector :

Answer 42

In an isosceles triangle the bisector of a vertex angle is also the median to the base and the altitude.

Question 43

Equilateral/Equiangular Triangle :

Answer 43

A triangle is equilateral if and only if it is equiangular.