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. |

The Geometric World: Shapes and Theorems

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.

Geometry Problems

Open Problems

Visual Index

All Problems

Triangle

Isosceles Triangle

Parallelogram

Angle Bisector

Midpoint

Perpendicular lines

Parallel lines

Congruence

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