6.6Trapezoids and Kites Last set of quadrilateral properties

Terminology:

Trapezoi d Kite

Terminology: Trapezoi d Quadrilateral with exactly one pair of parallel sides. Kite

Terminology: Trapezoi d Quadrilateral with exactly one pair of parallel sides. KiteQuadrilateral with two pairs of consecutive congruent sides, none of which are parallel.

Start with the trapezoid

O Parallel sides are called bases

Start with the trapezoid O Non parallel sides are called legs.

Start with the trapezoid O Since one pair is parallel

Start with the trapezoid O Since one pair is parallel Angles on the same leg are supplementary.

Now for the special

O Isosceles trapezoid is a trapezoid whose legs are congruent.

And now for the proof, drawing in perpendiculars A B C D E F

A B C D E F

A B C D E F

A B C D E F

As a result,  ACE   BDF by? A B C D E F

 C   D by… A B C D E F

As a result,  A   B by… A B C D E F

Theorem 6-19: If a quadrilateral is an isosceles trapezoid, then each pair of base  ’s is . A B C D E F

Make sure you can…

O Given one angle of an isosceles trapezoid, find the remaining 3 angles.

Application: page 390 Problem 2

Focusing on 1 section

AC  BD because? A B E C D

 C   D by? A B E C D

If we want to prove  ’s ACD and BCD are congruent, what do they share? A B E C D

 ACD  BCD by A B E C D

AD  BC by A B E C D

Theorem 6-20: If a quadrilateral is an isosceles trapezoid, then its diagonals are  A B E C D

The return of midsegments

The return of midsegments A midsegment of a trapezoid connects the midpoints of the legs (non parallel sides) and is the mean value of the 2 bases (parallel sides)

The return of midsegments A midsegment of a trapezoid connects the midpoints of the legs (non parallel sides) and is the mean value of the 2 bases (parallel sides)

In addition… A midsegment of a trapezoid connects the midpoints of the legs (non parallel sides) and is the mean value of the 2 bases (parallel sides)

In addition… Much like triangles, the midsegment is parallel to the sides it does not touch.

So find its length?

O Add the bases and divide by 2.

Working backwards

O Formula:

Working backwards

Plug in the length of the midsegment.

Plug in the length of a base.

Solve for the remaining base

O Or

Solve for the remaining base O Or O Arithmetically, multiply the length of the midsegment by 2 and subtract the length of the given base.

Here’s a problem I enjoy. O Given an isosceles trapezoid whose midsegment measures 50 cm and whose legs measures 24 mm. Find its perimeter.

Now to kites:

If we drew in a line of symmetry, where would it be?

And now are there   ’s?

 KEY   TEY

What new is congruent by CPCTC?

These are called the non-vertex angles, because they connect the non congruent sides

What else is congruent by CPCTC

What else is congruent by CPCTC?

The original angles, E and Y, are the vertex angles, and we can conclude they are bisected by the diagonal.

The vertex angles of a kite are the common endpoints of the congruent sides.

Summarizing

O Vertex angles connect the congruent sides and are bisected by the diagonals.

Summarizing O Vertex angles connect the congruent sides and are bisected by the diagonals. O Non vertex angles connect the non-congruent sides and are congruent.

One last property that becomes Theorem 6-22

If we draw in both diagonals…

If a quadrilateral is a kite, then its diagonals are perpendicular.

Problem solving examples

The family tree of quadrilaterals

Which group breaks down more?

And if we combine the last 2?

Those are all the definitions

O You need to remember all the properties, especially the ones that work for parallelograms, since they also work for a rhombus, rectangle, and square.

In addition… O You need to remember all the properties, especially the ones that work for parallelograms, since they also work for a rhombus, rectangle, and square.

In addition… O You need to determine the truth value (true/false) of a universal statement

In addition… O You need to determine the truth value (true/false) of a universal statement O All rectangles are parallelograms.

In addition… O You need to determine the truth value (true/false) of a universal statement O All rhombi are squares.