8) Given the figure below. Use the green right triangles and Pythagorean Theorem to help guide your answers.
a) Find the lengths of all four sides of the gray diagram. b) What is the special name of the gray diagram? c) Find the slopes of the red diagonals. d) What is the relationship of the diagonals? |
9) Figure \(ABCD\) is a kite with \(\overline{AB}\cong\overline{AD}\) and \(\overline{CB}\cong\overline{CD}\).
a) Find \(t\) (the angle) b) Find \(v\) c) Find \(w\) d) Find \(x\) (the parts of the diagonal) e) Find \(y\) f) Find \(z\) g) If the \(m\angle {ABC} = 69.4^{\circ}\), what is the \(m\angle{ADC}\)? |
10) Given the following diagram:
a) Prove, using mathematical reasoning, the most descriptive name for this figure. b) Find the lengths for \(\overline{AC}\) and \(\overline{BD}\). c) What is the relationship between \(\overline{AC}\) and \(\overline{BD}\)? d) What is the relationship between \(\triangle{ADC}\) and \(\triangle{BCD}\)? Justify your answer. e) Based on the answer from c), what is the relationship between \(\angle{ADC}\) and \(\angle{BCD}\)? What is the reason for this? |
11) Complete the sentences:
a) In a kite, the diagonals are __ ?__.
b) In a trapezoid, the non-base angles are __?__.
c) A kite has exactly __?__ pair(s) of opposite angles congruent.
d) In an isosceles trapezoid, the diagonals are __?__.
e) A kite has diagonals that are __?__ and __?__.
f) In an isosceles trapezoid, the base angles are __?__ to each other.
g) A kite has interior angles of \(90^{\circ}\), \(90^{\circ}\), \(80^{\circ}\), and __?__.
h) An isosceles trapezoid has interior angles of \(55^{\circ}\), __ ?__, __?__ and __?__.
i) It’s fun to fly a __?__ in the summer.
a) In a kite, the diagonals are __ ?__.
b) In a trapezoid, the non-base angles are __?__.
c) A kite has exactly __?__ pair(s) of opposite angles congruent.
d) In an isosceles trapezoid, the diagonals are __?__.
e) A kite has diagonals that are __?__ and __?__.
f) In an isosceles trapezoid, the base angles are __?__ to each other.
g) A kite has interior angles of \(90^{\circ}\), \(90^{\circ}\), \(80^{\circ}\), and __?__.
h) An isosceles trapezoid has interior angles of \(55^{\circ}\), __ ?__, __?__ and __?__.
i) It’s fun to fly a __?__ in the summer.
12) Answer each question with “Always”, “Sometimes” or “Never” true.
a) A trapezoid is always isosceles.
b) A trapezoid has congruent diagonals.
c) An isosceles trapezoid has perpendicular diagonals
d) A kite has one pair of opposite angles congruent and one pair of opposite angles not congruent.
e) A kite can be a parallelogram.
f) A kite can be a trapezoid.
g) A kite can have three of its interior angles all \(90^{\circ}\).
h) If one angle of an isosceles trapezoid is \(60^{\circ}\), then the other three are \(60^{\circ}\), \(120^{\circ}\), and \(120^{\circ}\).
i) The diagonals of a kite are perpendicular.
j) The diagonals of an isosceles trapezoid are congruent.
k) One diagonal of a kite divides the figure into two congruent triangles.
l) The sum of the measures of the interior angles of a trapezoid is \(180^{\circ}\).
m) All isosceles trapezoids have the properties of a trapezoid.
Solution Bank
a) A trapezoid is always isosceles.
b) A trapezoid has congruent diagonals.
c) An isosceles trapezoid has perpendicular diagonals
d) A kite has one pair of opposite angles congruent and one pair of opposite angles not congruent.
e) A kite can be a parallelogram.
f) A kite can be a trapezoid.
g) A kite can have three of its interior angles all \(90^{\circ}\).
h) If one angle of an isosceles trapezoid is \(60^{\circ}\), then the other three are \(60^{\circ}\), \(120^{\circ}\), and \(120^{\circ}\).
i) The diagonals of a kite are perpendicular.
j) The diagonals of an isosceles trapezoid are congruent.
k) One diagonal of a kite divides the figure into two congruent triangles.
l) The sum of the measures of the interior angles of a trapezoid is \(180^{\circ}\).
m) All isosceles trapezoids have the properties of a trapezoid.
Solution Bank