Question

quiz 7-2: parallelograms, rectangles, rhombi & squares

1. which quadrilaterals always have opposite angles that are congruent?

parallelograms
rectangles
rhombi
squares

1. which quadrilaterals always have diagonals that bisect opposite angles?

parallelograms
rectangles
rhombi
squares

1. if (abcd) is a parallelogram, (ad = 14), (ec = 11), (mangle abc = 64^circ), (mangle dac = 71^circ), and (mangle bdc = 25^circ), find each measure.

a) (bc = _____)
d) (mangle abd = _____)
b) (ac = _____)
e) (mangle acd = _____)
c) (mangle dab = _____)
f) (mangle adb = _____)
(image of parallelogram (abcd) with diagonals intersecting at (e))

Explanation:

Question 1

Opposite angles are congruent in all parallelograms, and rectangles, rhombi, squares are all types of parallelograms, so they share this property.

Diagonals that bisect opposite angles is a property unique to rhombi and squares (since squares are a special type of rhombus). Parallelograms and rectangles do not have this property.

Step1: Find BC (opposite sides of parallelogram are equal)

In parallelogram $ABCD$, $AD = BC$. Given $AD=14$, so $BC=14$.

Step2: Find AC (diagonals bisect each other)

Diagonals bisect each other, so $AC = 2\times EC$. Given $EC=11$, so $AC=2\times11=22$.

Step3: Find $m\angle DAB$ (consecutive angles are supplementary)

Consecutive angles in parallelogram are supplementary: $m\angle DAB + m\angle ABC = 180^\circ$. Given $m\angle ABC=64^\circ$, so $m\angle DAB=180^\circ-64^\circ=116^\circ$.

Step4: Find $m\angle ABD$ (alternate interior angles)

$AB\parallel DC$, so $\angle ABD = \angle BDC$. Given $m\angle BDC=25^\circ$, so $m\angle ABD=25^\circ$.

Step5: Find $m\angle ACD$ (alternate interior angles)

$AB\parallel DC$, so $\angle ACD = \angle BAC$. First find $\angle BAC = m\angle DAB - m\angle DAC = 116^\circ-71^\circ=45^\circ$, so $m\angle ACD=45^\circ$.

Step6: Find $m\angle ADB$ (triangle angle sum)

In $\triangle ADB$, $m\angle ADB + m\angle DAB + m\angle ABD=180^\circ$. Substitute values: $m\angle ADB=180^\circ-116^\circ-25^\circ=39^\circ$.

Answer:

• Parallelograms
• Rectangles
• Rhombi
• Squares

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Question 2