Question

1. a kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent. given kite wxyz, show that at least one of the diagonals of a kite decomposes the kite into 2 congruent triangles.
2. triangles bfm and wrh are shown.

determine whether the given information is sufficient to prove that the triangles are congruent. if so, state the appropriate congruence theorem or postulate to support that.

given information

congruent?

theorem or postulate used

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$overline{fb} cong overline{hw}$, $overline{fm} cong overline{rw}$, $angle bfm cong angle hwr$

$overline{fb} cong overline{rh}$, $angle fbm cong angle rhw$, $overline{mb} cong overline{wh}$

$overline{fb} cong overline{hw}$, $overline{bm} cong overline{hr}$, $overline{fm} cong overline{rw}$

6. wxyz is a kite. angle wxy has a measure of $133^circ$ and $angle zwx$ has a measure of $60^circ$. find the measure of $angle zyw$.

review problems

1. triangle hef is the image of $delta hgf$ after a reflection across $overline{fh}$. write a congruence statement for the 2 congruent triangles.

Explanation:

Problem 4

Step1: Recall kite side properties

By kite definition: $\overline{WX} \cong \overline{WZ}$, $\overline{XY} \cong \overline{ZY}$, and diagonal $\overline{WY}$ is common to $\triangle WXY$ and $\triangle WZY$.

Step2: Apply SSS congruence

For $\triangle WXY$ and $\triangle WZY$:
$\overline{WX} \cong \overline{WZ}$, $\overline{XY} \cong \overline{ZY}$, $\overline{WY} \cong \overline{WY}$.
By Side-Side-Side (SSS) congruence postulate, $\triangle WXY \cong \triangle WZY$. Diagonal $\overline{WY}$ decomposes the kite into two congruent triangles.

Step1: Analyze first row

Given $\overline{FB} \cong \overline{HW}$, $\overline{FM} \cong \overline{RW}$, $\angle BFM \cong \angle HWR$. This matches the Side-Angle-Side (SAS) congruence postulate (two sides and included angle congruent).

Step2: Analyze second row

Given $\overline{FB} \cong \overline{RH}$, $\angle FBM \cong \angle RHW$, $\overline{MB} \cong \overline{WH}$. This matches the Side-Angle-Side (SAS) congruence postulate (two sides and included angle congruent).

Step3: Analyze third row

Given $\overline{FB} \cong \overline{HW}$, $\overline{BM} \cong \overline{HR}$, $\overline{FM} \cong \overline{RW}$. This matches the Side-Side-Side (SSS) congruence postulate (all three corresponding sides congruent).

Step1: Use kite side congruence

In kite $WXYZ$, $\overline{WX} \cong \overline{WZ}$, so $\triangle WXZ$ is isosceles. $\angle WXZ = \angle WZX$.

Step2: Calculate $\angle WXZ$

In $\triangle WXZ$, sum of angles is $180^\circ$:
$\angle WXZ = \frac{180^\circ - \angle ZWX}{2} = \frac{180^\circ - 60^\circ}{2} = 60^\circ$.

Step3: Find $\angle XZY$

$\angle XZY = \angle WXZ + \angle WZY$. Since $\overline{XY} \cong \overline{ZY}$, $\triangle XYZ$ is isosceles, $\angle XYZ = \angle XZY$. First, find $\angle XYZ$ using quadrilateral angle sum:
Total quadrilateral angles: $360^\circ$.
$\angle XYZ + \angle WZY = 360^\circ - 133^\circ - 60^\circ = 167^\circ$.
But $\angle WZY = \angle WXZ = 60^\circ$, so $\angle XYZ = 167^\circ - 60^\circ = 107^\circ$, so $\angle XZY = 107^\circ$.

Step4: Calculate $\angle ZYW$

In $\triangle ZYW$, $\overline{ZY} \cong \overline{XY}$, $\overline{WY}$ is common. $\angle ZYW = \frac{180^\circ - \angle XYZ}{2} = \frac{180^\circ - 107^\circ}{2} = 36.5^\circ$.

Answer:

Diagonal $\overline{WY}$ decomposes kite $WXYZ$ into $\triangle WXY \cong \triangle WZY$ (by SSS congruence), proving at least one diagonal creates two congruent triangles.

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Problem 5