You are given a parallelogram ABCD with sides AB and BC measuring 6 and 10 units, respectively. The bisectors of angles A and B intersect at point E, and the bisectors of angles C and D intersect at point F. Find the length of segment EF.
|Concept||Definition||Theorem / Comment|
|Triangle||A polygon with three sides and three angles.||The sum of the angles in a triangle is 180 degrees.|
|Isosceles triangle||An isosceles triangle is a triangle that has two sides of equal length.||In an isosceles triangle, the altitudes relative to congruent sides are congruent.|
|Parallelogram||A parallelogram is a quadrilateral with both pairs of opposite sides parallel.||Opposite sides of a parallelogram are congruent; opposite angles are congruent; consecutive angles are supplementary; diagonals bisect each other.|
|Angle bisector||A line or ray that divides an angle into two congruent angles..|
|Parallel lines||Two lines in a plane that do not intersect.||If two parallel lines are cut by a transversal, then the alternate interior angles are congruent, the corresponding angles are congruent, and the consecutive interior angles are supplementary.|
|Congruence||Two triangles are said to be congruent if all corresponding sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle..||There are several ways to prove that two triangles are congruent, including the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Hypotenuse-Leg (HL) criteria.|
|Auxiliary line||Auxiliary line is a line that is added to a diagram in order to help prove a theorem or solve a problem.||Often, an auxiliary line is drawn to create additional congruent or similar triangles, to create parallel lines, or to create right angles. The use of auxiliary lines can simplify a problem or make a proof more straightforward. However, it is important to ensure that the auxiliary line does not create any new intersections or angles that were not present in the original diagram.|
In geometry, we learn of shapes,
And theorems to guide our way,
Parallelogram, a figure so sleek,
Has parallel sides that always stay.
Opposite sides and angles, congruent too,
Consecutive angles, supplementary for sure,
Diagonals bisect each other, it's true,
In a parallelogram, we can ensure.
Angle bisectors divide an angle in two,
Creating congruent angles on either side,
A helpful tool to solve problems anew,
In geometry, it's a joyride.
Parallel lines never meet,
In a plane, they stretch far and wide,
Alternate interior angles congruent,
A theorem that we can't hide.
Congruence is a notion so grand,
When triangles are the same,
All sides and angles correspond,
Proving it can be our aim.
Auxiliary lines are a guiding light,
Helping us in our geometry quest,
Creating triangles, parallel lines, just right,
Making proofs easier and theorems the best.
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