Question

do you understand?

1. essential question how are diagonals and angle measures related in kites and trapezoids?
2. error analysis what is nevaehs error? mp.3

by theorem 6 - 5, $overline{pr} cong overline{qs}$

1. vocabulary if $overline{xy}$ is the midsegment of a trapezoid, what must be true about point x and point y?
2. construct arguments emaan says every kite is composed of 4 right triangles. explain why emaan is correct. mp.3

do you know how?
for exercises 5 - 7, use kite wxyz to find the measures.

1. $mangle xqy$
2. $mangle yzo$
3. $wy$

for exercises 8 - 10, use trapezoid defg with $eg = 21$ ft and $mangle dgf = 77^{circ}$ to find each measure.

1. $ed$
2. $df$
3. $mangle def$
4. what is the length of $overline{pq}$?

Explanation:

1. (Essential Question)

Diagonals of kites are perpendicular; one diagonal bisects the other and the non-vertex angles. In isosceles trapezoids, diagonals are congruent, base angles are congruent, and consecutive angles between the two bases are supplementary. For non-isosceles trapezoids, only consecutive angles between bases are supplementary.

2. (Error Analysis)

Step1: Identify figure type

PQRS is a trapezoid with $\overline{PQ} \parallel \overline{SR}$, $\overline{PS} \cong \overline{QR}$ (isosceles trapezoid).

Step2: Recall trapezoid diagonal theorem

In isosceles trapezoids, the congruent segments are the legs, and the diagonals are congruent to each other ($\overline{PR} \cong \overline{QS}$ is true for isosceles trapezoids, but if PQRS is not isosceles, this does not hold. The error is misapplying the theorem to a non-isosceles trapezoid, as the markings only show $\overline{PQ} \parallel \overline{SR}$ and one pair of non-parallel sides is not marked congruent, so Theorem 6-5 (for isosceles trapezoids) does not apply.

3. (Vocabulary)

Step1: Recall midsegment definition

A trapezoid midsegment connects the midpoints of the two non-parallel legs.

Step2: State requirement for X and Y

$X$ must be the midpoint of one leg, $Y$ must be the midpoint of the other leg.

4. (Construct Arguments)

Step1: Recall kite diagonal property

The diagonals of a kite are perpendicular to each other.

Step2: Explain triangle division

The perpendicular intersection of the diagonals creates 4 triangles, each with a right angle at the intersection point, so all 4 are right triangles.

5. (Find $m\angle XQY$)

Step1: Recall kite diagonal property

Diagonals of a kite are perpendicular.

Step2: State angle measure

$\angle XQY = 90^\circ$

6. (Find $m\angle YZQ$)

Step1: Use kite angle congruence

$\angle WXQ = \angle W = 56^\circ$ (adjacent angles to congruent sides in kite are equal).

Step2: Calculate $\angle XWQ$

In $\triangle WXO$, $\angle XWQ = \frac{180^\circ - 90^\circ - 56^\circ}{1} = 34^\circ$

Step3: Use kite triangle congruence

$\triangle WXO \cong \triangle YZO$, so $\angle YZQ = \angle XWQ = 34^\circ$

7. (Find $WY$)

Step1: Identify segment length

$OY = 12$ mm, and the diagonal $WY$ is bisected by $XZ$.

Step2: Calculate $WY$

$WY = 2 \times OY = 2 \times 12 = 24$ mm

8. (Find $ED$)

Step1: Identify trapezoid side property

In trapezoid $DEFG$, $\overline{EF} \parallel \overline{DG}$, and $\overline{ED} \cong \overline{FG}$ (marked congruent).

Step2: State length of $ED$

$ED = FG = 8$ ft

9. (Find $DF$)

Step1: Recall trapezoid diagonal property

In isosceles trapezoids, diagonals are congruent.

Step2: State length of $DF$

$DF = EG = 21$ ft

10. (Find $m\angle DEF$)

Step1: Use consecutive angle property

$\angle DEF$ and $\angle DGF$ are supplementary (consecutive angles between parallel bases of trapezoid).

Step2: Calculate angle measure

$m\angle DEF = 180^\circ - 77^\circ = 103^\circ$

11. (Find length of $\overline{PQ}$)

Step1: Recall trapezoid midsegment formula

The midsegment length is $\frac{1}{2} \times (\text{sum of the two bases})$.

Step2: Substitute values

$PQ = \frac{1}{2} \times (AB + CD) = \frac{1}{2} \times (33 + 54) = \frac{1}{2} \times 87 = 43.5$ cm

Answer:

1. Diagonals of kites are perpendicular; one bisects the other and the angles. In isosceles trapezoids, diagonals are congruent, base angles are congruent, and consecutive angles between bases are supplementary.
2. Nevaeh misapplied Theorem 6-5; $\overline{PR}

ot\cong \overline{QS}$

1. $X$ is the midpoint of one leg, $Y$ is the midpoint of the other leg.
2. The diagonals of a kite are perpendicular, dividing it into 4 right triangles.
3. $90^\circ$
4. $34^\circ$
5. $24$ mm
6. $8$ ft
7. $21$ ft
8. $103^\circ$
9. $43.5$ cm