Trapezoids
A trapezoid is a quadrilateral with exactly ONE pair of opposite sides parallel. The two parallel sides are called the bases. Because the trapezoid has one pair of parallel sides, it only has two pairs of consecutive angles supplementary, otherwise it would become a parallelogram! (recall from the unit on Parallel and Perpendicular Lines). Manipulate the trapezoid below to see the properties in action!
A trapezoid is a quadrilateral with exactly ONE pair of opposite sides parallel. The two parallel sides are called the bases. Because the trapezoid has one pair of parallel sides, it only has two pairs of consecutive angles supplementary, otherwise it would become a parallelogram! (recall from the unit on Parallel and Perpendicular Lines). Manipulate the trapezoid below to see the properties in action!
Properties of a Trapezoid
- One pair of parallel sides.
- Consecutive angles between bases are supplementary.
If the non-bases of a trapezoid (called the legs) are congruent, then the trapezoid is called an isosceles trapezoid. Manipulate the figure below to see the properties in action.
Properties of an Isosceles Trapezoid:
- Trapezoid with legs (the non-parallel sides) that are congruent.
- Both pairs of base angles are congruent.
- Diagonals are congruent.
Example 1:
\(KATH\) is an isosceles trapezoid with bases \(\overline{KA}\) and \(\overline{TH}\). Given \(HT = 18\) cm, \(HL = 8\) cm, and the \(m\angle K=67^{\circ}\). Find: a) \(AT\) b) \(m\angle A\) c) \(m\angle {T}\) Solution: Let’s label the information given on the diagram. a) We will focus on \(\triangle KHL\) and solve for \(KH\). Recall, since the legs of an isosceles trapezoid are congruent, \(KH = AT\). \(\begin{align*} \text{sin}(67^{\circ}) &= \dfrac{8}{KH}\\ KH\cdot \text{sin}(67^{\circ}) &= \frac{8}{KH}\cdot KH\\ \frac{KH\cdot \text{sin}(67^{\circ})}{\text{sin}(67^{\circ})} &= \frac{8}{\text{sin}(67^{\circ})}\\ KH &= 8.69088...\\ KH &= AT = 8.69088... \end{align*}\) \(AT \approx 8.7\) b) Since \(\angle K\) and \(\angle A\) are base angles of an isosceles trapezoid, they must be congruent. Thus \(m\angle A = 67^{\circ}\). c) Consecutive interior angles are supplementary when lines are parallel. So: \(m\angle A + m\angle T = 180^{\circ}\) \(67^{\circ} + m\angle T = 180^{\circ}\) \(m\angle T = 113^{\circ}\) |
Kites
Our next quadrilateral is a kite, which is a quadrilateral with two distinct pairs of consecutive sides congruent. Manipulate the figure below to see the properties in action.
Our next quadrilateral is a kite, which is a quadrilateral with two distinct pairs of consecutive sides congruent. Manipulate the figure below to see the properties in action.
Properties of a Kite
- Two distinct pairs of adjacent sides congruent
- Diagonals are perpendicular.
- One diagonal is bisected.
- One pair of opposite angles is congruent.
Example 2:
Given kite \(BROW\) where \(BR = BW\), \(RO = 25\) units and \(NO = 7\) units, find \(RW\). Solution: First, let’s label the diagram given. Notice that \(RO = OW\) since both pairs of consecutive sides are congruent in a kite. The other properties we know from a kite are that the diagonals are perpendicular, and that the diagonal, \(RW\), is bisected. Now we will use the Pythagorean Theorem with \(\triangle RNO\) to find \(RN\). \(\begin{align*}\\ (RN)^2 + (NO)^2 &= (RO)^2\\ (RN)^2 + 7^2 &= 25^2\\ (RN)^2 + 49 &= 625\\ (RN)^2 &= 576\\ RN &= \pm\sqrt{576}\\ RN &= 24\;\;\;\;\;\text{length must be postitive} \end{align*}\) Since \(\overline{RW}\) is split in half, we know \(RN = NW\), so \(RW = 24 + 24 = 48\) units. |
Quick Check:
1) Given kite \(WHAT\), with \(W\:(0, -7)\) and \(A\;(-2, -3)\), what is the slope of \(\overline{HT}\)?
2) \(BCDF\) is a kite where \(BC = CD\), \(BC = 3x + 4y\), \(CD = 2x + 20\), \(BF = 12\), and \(FD = x + 2y\). Find the perimeter of the kite.
1) Given kite \(WHAT\), with \(W\:(0, -7)\) and \(A\;(-2, -3)\), what is the slope of \(\overline{HT}\)?
2) \(BCDF\) is a kite where \(BC = CD\), \(BC = 3x + 4y\), \(CD = 2x + 20\), \(BF = 12\), and \(FD = x + 2y\). Find the perimeter of the kite.
4) Fill in the following blanks with always, sometimes, or never.
a) Diagonals of a trapezoid are _?_ congruent.
b) Diagonals of a kite are _?_ perpendicular.
c) A kite is _?_ a trapezoid.
Quick Check Solutions
a) Diagonals of a trapezoid are _?_ congruent.
b) Diagonals of a kite are _?_ perpendicular.
c) A kite is _?_ a trapezoid.
Quick Check Solutions