geometry honors gifted 2008-2009 c-1 linear pair conjecture

Geometry Honors Gifted 2008-2009 Conjecture List

If 2 angles form a linear pair, then their angles sum to 180. The converse is not true; two supplementary angles need not be

linear pairs.

If two angles are vertical angles, then they are congruent. The converse is not true; two congruent angles need not be vertical

angles.

If 2 parallel lines are cut by a transversal, then corresponding angles, anterior interior angles, and alternate exterior angles are congruent.

See C-4 for converse information.

Tomás Monzón 4th Period

C-1Linear Pair Conjecture

C-2Vertical Angles

Conjecture

C-3Parallel Line Conjecture


If 2 parallel lines are cut by a transversal then the corresponding angles are congruent.


If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.


If 2 parallel lines are cut by a transversal, then alternate exterior angles are congruent.



C-3ACorresponding

Angles Conjectures

C-3BAlternate Interior Angles Conjecture

C-3CAlternate Exterior Angles Conjecture


If 2 lines are cut by a transversal so that corresponding, alternate interior angles, and angles exterior angles pairs are congruent then the lines are parallel.


C-4Converse of Parallel

Lines Conjecture


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


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

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


C-5Perpendicular

Bisector Conjecture

C-6Converse of the Perpendicular

Bisector Conjecture

C-7Shortest Distance

Conjecture


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

The three angle bisectors of a triangle are concurrent. They form the incenter of a triangle.

A converse is not applicable.

The three perpendicular bisectors of a triangle are concurrent. They form the circumcenter of triangle.



C-8Angle Bisector

Conjecture

C-9Angle Bisector Concurrency Conjecture

C-10Perpendicular

Bisectors Concurrency Conjecture


The three altitudes (or the lines containing the altitudes) of a triangle are concurrent. They form the orthocenter.

A converse is not applicable

The circumcenter of a triangle is equidistant from the vertices. The converse is true = a point in a triangle equidistant from its

vertices is its circumcenter.

The three medians of a triangle are concurrent. They form the centroid of a triangle.


The three medians of a triangle are concurrent. They form the centroid of a triangle.



C-11Altitude Concurrency

Conjecture

C-13Circumcenter

Conjecture

C-13Incenter Conjecture

C-14Median Concurrency

Conjecture


The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the

distance from the centroid to the midpoint of the opposite side. The converse is partially true = the distance from the centroid of a

triangle to the midpoint of a side is half the distance from the centroid to the vertex opposite the side.

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

The converse is true = the center of gravity of a triangle is its centroid.

The sum of the measures of the angles in every triangle is 180 ° . The converse is true = 180 ° is the sum of the angle measures in

every triangle.


C-15Centroid Conjecture

C-16Center of Gravity

Conjecture

C-17Triangle Sum Conjecture


If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle is congruent

to the third angle in the second triangle. The converse is true; simply take the above statement in the

perspective of the “second” triangle. .

If a triangle is isosceles, then its base angle are congruent. See C-20 for converse information.

If a triangle has two congruent angles, then it is isosceles.


C-18Third Angle Conjecture

C-19Isosceles Triangle

Conjecture

C-20Converse of the

Isosceles Triangle Conjecture


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

The converse is true = the length of one side of triangle is greater than the sum of the two other sides.

In a triangle,the largest side is opposite the largest angle and the smallest side is opposite the smallest angle.

The converse is true = the largest angle is opposite the largest side and the smallest angle is opposite the smallest side.

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

The converse is true = the sum of the remote angles to an exterior angle in a triangle are equal to one anoter.


C-21Triangle Inequality

Conjecture

C-22Side-Angle Inequality

Conjecture

C-23Triangle Exterior Angle Conjecture


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

The converse is not true; two congruent triangles need not be congruent by the same method or reason; this applies to

Conjectures C-24 – C-27.

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the

triangles are congruent.

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the



C-24SSS (Side-Side-Side)

Congruence Conjecture

C-25SAS (Side-Angle-Side) Congruence

Conjecture

C-26ASA (Angle-Side-

Angle) Congruence Conjecture


If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the


In an isosceles triangle, the bisector of the vertex angle is also a perpendicualr bisector and a median.

The converse is somewhat true, the median and perpendicular bisectors of the base of an isosceles triangle are both the bisector

of the isosceles triangle's vertex angle.

Every equilateral triangle is equiangular and conversely, every equiangular triangle is equilateral.


C-27SAA (Side-Angle-

Angle) Congruence Conjecture

C-28Vertex Angle Bisector

Conjecture

C-29Equilateral/

Equiangular Triangle Conjecture


The sum of the measures of the four angles of any quadrilateral is 360 degrees.

The sum of the measures of the five angles of any pentagon is 540 degrees.

The sum of the measures of the n interior angles of an n-gon is 180( n -2).

The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid

to the midpoint of the opposite side.The converse is partially true = the distance from the centroid of a triangle to the midpoint of a side is half the distance from the centroid to the vertex opposite the side.

The centroid of a triangle is the center of gravi


C-30Quadrilateral Sum

Conjecture

C-31Pentagon Sum

Conjecture

C-32Polygon Sum Conjecture


For any polygon, the sum of the measures of a set of exterior angles is 360 degrees.

You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas: 180(n-2)/n and 180 – 360/n.

The non vertex angles of a kite are congruent.


C-33Exterior Angles Sum

Conjecture

C-34Equiangular Polygon

Conjecture

C-35Kite Angles Conjecture


The diagonals of a kite are perpendicular.

The diagonal connecting the vertex angles of a kite is the perpendicular bisector of the other diagonal.

The vertex angles of a kite are bisected by a diagonal. The diagonal here is also the single line of symmetry of the kite.


C-36Kite Diagonals

Conjecture

C-37Kite Diagonal

Bisector Conjecture

C-38Kite Angle Bisector

Conjecture

C-39Trapezoid

Consecutive Angles Conjecture

C-40Isosceles Trapezoid

Conjecture

C-41Isosceles Trapezoid

Diagonals Conjecture


The consecutive angles between the bases of a trapezoid are supplementary.

The base angles of an isosceles trapezoid are congruent.

The diagonals of an isosceles trapezoid are congruent.


C-42Three Midsegments

Cojnecture

C-43Triangle Midsegment

Conjecture

C-44Trapezoid

Midesgment Conjecture


The three midsegments of a triangle divide it into 4 congruent triangles.

A midsegment of a triangle is parallel to the third side and half the length of the third side.

The converse is true – a line with half the length of the third side of a triangle that is parallel to the third side is its midsegment.

The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the bases.

The converse is true – the average of the length of the bases of a trapezoid that is parallel to the bases is its midsegment.

The opposite angles of a parallelogram are congruent.


C-45Parallelogram

Opposite Angles Conjecture

C-46Parallelogram

Consecutive Angles Conjecture

C-47Parallelogram Opposite Sides

Conjecture


The consecutive angles of a parallelogram are supplementary.

The opposite sides of a parallelogram are congruent.

The diagonals of a parallelogram bisect each other.


C-48Parallelogram

Diagonals Conjecture

C-49Double-Edged Straightedge Conjecture

C-50Rhombus Diagonals

Conjecture


If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the

parallelogram formed is a rhombus.

The diagonals of a rhombus are perpendicular and they bisect each other.

The diagonals of rhombus bisects the angles of the rhombus.


C-51Rhombus Angles

Conjecture

C-52Rectangle Diagonals

Conjecture

C-53Square Diagonals

Conjecture


The diagonals of a rectangle are congruent and they bisect each other.

The diagonals of a square are congruent, they bisect each other, and they are perpendicular.



If two chords in a circle are congruent, then their intercepted arcs are congruent.

The converse of the statement is not true. Intercepted arcs do not necessarily result in the congruence of two chords in a circle.

The perpendicular from the center of a circle to a chord is the perpendicular bisector of the chord.

The converse is true if the bisector in question emanates from the circle.

Two congruent chords in a circle are equidistant from the center of the circle.

The converse is true. Two congruent lines in a circle (which eliminate the diameter) will always be equidistant from the circle.


C-55Chord Arcs Conjecture

C-56Perpendicular to a Chord Conjecture

C-57Chord Distance to Center Conjecture


The perpendicular bisector of a chord passes through the center of the circle.

A tangent to a circle is perpedicular to the radius drawn to the point of tangency.

Tangent segments to a circle from a point outside the circle are conguent and equidistant.


C-58Perpendicular

Bisector of a Chord Conjecture

C-59Tangent Conjecture

C-60Tangent Segments

Conjecture


The measure of an angle inscribed in a circle is half the measure of its intercepted arc.

The converse is true; half the measure of an intercepted arc equals the measure of the inscribed angle.

Inscribed angles that intercept the same arc are congruent.

Angles inscribed in a semicircle are right angles.


C-61Inscribed Angle

Conjecture

C-62Inscribed Angles Intercepting Arcs

Conjecture

C-63Angles Inscribed in a Semicircle Conjecture


The opposite angles of a cyclic quadrilateral are congruent.

Parallel lines intercept congruent arcs on a circle.

If C is the circumference and d is the diameter of a circle, then there is a number pi such that C = 2pir. If d = 2r where r is the radius, then C = pid .


C-64Cyclic Quadrilateral

Conjecture

C-65Parallel Lines

Intercepted Arcs Conjecture

C-66Circumference

Conjecture


The length of an arc equals the fraction of the circumference it occupies.


C-67Arc Length Conjecture


If two chords in a circle are congruent, then they determine two central angles that are congruent.


C-54Chord Central Angles

Conjecture

geometry honors gifted 2008-2009 c-1 linear pair conjecture

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