Question

mixed quadrilateral review
date:
name: joyce kachbandika
geometry
name each type of quadrilateral that can meet the given condition.
exactly one pair of congruent sides.
four right angles.
congruent diagonals.
adjacent sides that are congruent.
perpendicular diagonals.
two pairs of parallel sides.
find the values of the variables or missing values in each kite.

Explanation:

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Quadrilateral Classification Section

Step1: Classify "Exactly one pair of congruent sides"

A kite has two distinct pairs of adjacent congruent sides, while an isosceles trapezoid has exactly one pair of congruent non-parallel sides. A general trapezoid may not have congruent sides, so the correct quadrilateral is: Isosceles Trapezoid, Kite (some definitions allow kites to be interpreted this way, but isosceles trapezoid is the primary answer).

Step2: Classify "Four right angles"

Quadrilaterals with 4 right angles have opposite sides parallel and congruent. These are: Rectangle, Square (a square is a special rectangle).

Step3: Classify "Congruent diagonals"

Quadrilaterals with congruent diagonals are: Rectangle, Square, Isosceles Trapezoid.

Step4: Classify "Adjacent sides that are congruent"

Quadrilaterals with adjacent congruent sides are: Kite, Rhombus, Square (a rhombus has all sides congruent, so adjacent sides are congruent; a square is a special rhombus).

Step5: Classify "Perpendicular diagonals"

Quadrilaterals with perpendicular diagonals are: Kite, Rhombus, Square.

Step6: Classify "Two pairs of parallel sides"

Quadrilaterals with two pairs of parallel sides are parallelograms, which include: Parallelogram, Rectangle, Rhombus, Square.
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Kite Variable Calculation Section

Step1: Solve first kite (left top)

In a kite, one diagonal bisects angles, and angles in a triangle sum to $180^\circ$. The right triangle has angles $(x+6)^\circ$, $2x^\circ$, $90^\circ$.
$$(x+6) + 2x + 90 = 180$$
$$3x + 96 = 180$$
$$3x = 84$$
$$x = 28$$
Substitute $x=28$:
$(x+6) = 34^\circ$, $2x=56^\circ$
The other angles: the angle opposite $34^\circ$ is $34^\circ$, the angle opposite $56^\circ$ is $56^\circ$.

Step2: Solve second kite (right top)

In a kite, congruent adjacent sides mean opposite angles between congruent sides are equal, so $(3x+5)^\circ=(4x-30)^\circ$.
$$3x + 5 = 4x - 30$$
$$x = 35$$
Substitute $x=35$:
$(3x+5)=110^\circ$. In the triangle with angles $y^\circ$, $(2y-20)^\circ$, $110^\circ$, sum to $180^\circ$:
$$y + (2y-20) + 110 = 180$$
$$3y + 90 = 180$$
$$3y = 90$$
$$y = 30$$

Step3: Solve third kite (left bottom)

In a kite, the diagonal bisects the $45^\circ$ angle, and the right triangle has angles $45^\circ$, $90^\circ$, so the third angle is:
$$180 - 90 - 45 = 45^\circ$$
Angle 1 = $90^\circ$ (diagonals of a kite are perpendicular), Angle 2 = $45^\circ$, Angle 3 = $45^\circ$.

Step4: Solve fourth kite (right bottom)

In a kite, diagonals are perpendicular, so Angle 1 = $90^\circ$.
In the left triangle: Angle 3 = $180 - 90 - 38 = 52^\circ$
In the right triangle: Angle 5 = $180 - 90 - 53 = 37^\circ$
Angle 2 = $38^\circ$ (angle bisected by the diagonal), Angle 4 = $53^\circ$ (angle bisected by the diagonal).
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Answer:

Quadrilateral Classification:

1. Exactly one pair of congruent sides: Isosceles Trapezoid, Kite
2. Four right angles: Rectangle, Square
3. Congruent diagonals: Rectangle, Square, Isosceles Trapezoid
4. Adjacent sides that are congruent: Kite, Rhombus, Square
5. Perpendicular diagonals: Kite, Rhombus, Square
6. Two pairs of parallel sides: Parallelogram, Rectangle, Rhombus, Square

Kite Variable Solutions:

1. Top left kite: $x=28$; angles are $34^\circ, 56^\circ, 34^\circ, 56^\circ$
2. Top right kite: $x=35$, $y=30$
3. Bottom left kite: Angle 1 = $90^\circ$, Angle 2 = $45^\circ$, Angle 3 = $45^\circ$
4. Bottom right kite: Angle 1 = $90^\circ$, Angle 2 = $38^\circ$, Angle 3 = $52^\circ$, Angle 4 = $53^\circ$, Angle 5 = $37^\circ$