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 Wonders of Parallelograms and Geometry 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.

If you're interested in finding more poems with a focus on geometry, you may enjoy this collection: More geometry thematic poems.

Geometry Problems

Open Problems

Visual Index

All Problems

Triangle

Isosceles Triangle

Parallelogram

Angle Bisector

Midpoint

Perpendicular lines

Parallel lines

Congruence

View or Post a solution