6.6: Chapter 6 Review
Keywords and Theorems
- Polygon Sum Formula
- For any \begin{align*}n-\end{align*}gon, the sum of the interior angles is \begin{align*}(n - 2) \times 180^\circ\end{align*}.
- Equiangular Polygon Formula
- For any equiangular \begin{align*}n-\end{align*}gon, the measure of each angle is \begin{align*}\frac{(n-2)\times 180^\circ}{n}\end{align*}.
- Regular Polygon
- When a polygon is equilateral and equiangular.
- Exterior Angle Sum Theorem
- The sum of the exterior angles of any polygon is \begin{align*}360^\circ\end{align*}.
- Parallelogram
- A quadrilateral with two pairs of parallel sides.
- Opposite Sides Theorem
- If a quadrilateral is a parallelogram, then the opposite sides are congruent.
- Opposite Angles Theorem
- If a quadrilateral is a parallelogram, then the opposite angles are congruent.
- Consecutive Angles Theorem
- If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
- Parallelogram Diagonals Theorem
- If a quadrilateral is a parallelogram, then the diagonals bisect each other.
- Opposite Sides Theorem Converse
- If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.
- Opposite Angles Theorem Converse
- If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.
- Consecutive Angles Theorem Converse
- If the consecutive angles of a quadrilateral are supplementary, then the figure is a parallelogram.
- Parallelogram Diagonals Theorem Converse
- If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.
- Theorem 6-10
- Rectangle Theorem
- A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
- Rhombus Theorem
- A quadrilateral is a rhombus if and only if it has four congruent sides.
- Square Theorem
- A quadrilateral is a square if and only if it has four right angles and four congruent sides.
- Theorem 6-14
- A parallelogram is a rectangle if and only if the diagonals are congruent.
- Theorem 6-15
- A parallelogram is a rhombus if and only if the diagonals are perpendicular.
- Theorem 6-16
- A parallelogram is a rhombus if and only if the diagonals bisect each angle.
- Trapezoid
- A quadrilateral with exactly one pair of parallel sides.
- Isosceles Trapezoid
- A trapezoid where the non-parallel sides are congruent.
- Theorem 6-17
- The base angles of an isosceles trapezoid are congruent.
- Theorem 6-17 Converse
- If a trapezoid has congruent base angles, then it is an isosceles trapezoid.
- Isosceles Trapezoid Diagonals Theorem
- The diagonals of an isosceles trapezoid are congruent.
- Midsegment (of a trapezoid)
- A line segment that connects the midpoints of the non-parallel sides.
- Midsegment Theorem
- The length of the midsegment of a trapezoid is the average of the lengths of the bases
- Kite
- A quadrilateral with two sets of adjacent congruent sides.
- Theorem 6-21
- The non-vertex angles of a kite are congruent.
- Theorem 6-22
- The diagonal through the vertex angles is the angle bisector for both angles.
- Kite Diagonals Theorem
- The diagonals of a kite are perpendicular.
Quadrilateral Flow Chart
Fill in the flow chart according to what you know about the quadrilaterals we have learned in this chapter.
Sometimes, Always, Never
Determine if the following statements are sometimes, always or never true.
- A trapezoid is a kite.
- A square is a parallelogram.
- An isosceles trapezoid is a quadrilateral.
- A rhombus is a square.
- A parallelogram is a square.
- A square is a kite.
- A square is a rectangle.
- A quadrilateral is a rhombus.
Table Summary
Determine if each quadrilateral has the given properties. If so, write yes or state how many sides (or angles) are congruent, parallel, or perpendicular.
Opposite sides \begin{align*} \ || \ \end{align*} | Diagonals bisect each other | Diagonals \begin{align*}\perp\end{align*} | Opposite sides \begin{align*}\cong\end{align*} | Opposite angles \begin{align*}\cong\end{align*} | Consecutive Angles add up to \begin{align*}180^\circ\end{align*} | |
---|---|---|---|---|---|---|
Trapezoid | ||||||
Isosceles Trapezoid | ||||||
Kite | ||||||
Parallelogram | ||||||
Rectangle | ||||||
Rhombus | ||||||
Square |
Find the measure of all the lettered angles below. The bottom angle in the pentagon (at the bottom of the drawing) is \begin{align*}138^\circ\end{align*}.
Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9691.