2) \(ABCD\) is a parallelogram. The green triangles that surround \(ABCD\) are right triangles and have their measurements given.
a) Use the green triangles to determine the slopes of: \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{AD}\) b) Given your response from a), what can we say about the opposite sides? c) Use the green triangles and Pythagorean Theorem to determine the lengths of: \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{AD}\) d)Given your response from c), what can we say about the opposite sides? |
5) \(ABCD\) is a parallelogram with diagonals \(\overline{AC}\) and \(\overline{BD}\).
a) Use the given green right triangles and Pythagorean Theorem to determine the lengths of \(\overline{AK}\) and \(\overline{CK}\) b) Is \(\overline{AC}\) bisected? Explain. c) Use a similar procedure to determine the lengths of \(\overline{BK}\) and \(\overline{DK}\) d) Is \(\overline{BD}\) bisected? Explain. e) Can you conclude that the diagonals of parallelogram \(ABCD\) bisect each other? |
6) \(ABCD\) is a parallelogram with
a) \(\overline{AB}\parallel\overline{DC}\). What is the relationship between \(\angle A\) and \(\angle D\)? b) \(\overline{AD}\parallel\overline{BC}\). What is the relationship between \(\angle A\) and \(\angle B\)? c) What is the relationship between \(\angle D\) and \(\angle B\)? d) What can we also say about the relationship between \(\angle A\) and \(\angle C\)? |
11) \(ABCD\) is a parallelogram.
a) What are the lengths of all the sides in FIGURE \(A\)?
b) What are the lengths of all the sides in FIGURE \(B\)?
c) Two students argue. They both agree that both figures show a parallelogram, but they disagree as to which one is
also a rhombus. Is FIGURE \(A\) or FIGURE \(B\) a rhombus? Explain your reasoning.
a) What are the lengths of all the sides in FIGURE \(A\)?
b) What are the lengths of all the sides in FIGURE \(B\)?
c) Two students argue. They both agree that both figures show a parallelogram, but they disagree as to which one is
also a rhombus. Is FIGURE \(A\) or FIGURE \(B\) a rhombus? Explain your reasoning.
14) Given the following diagram of a quadrilateral with three of the vertices at \((8, 3)\), \((10, -2)\), and \((5, -4)\). The fourth vertex is part of a piece of the quadrilateral that has been "broken off."
a) What is the missing point for this figure to be a rhombus. Use slope or distance of sides to guide you.
b) Find the slope of the diagonals of the rhombus. What is the relationship of these two slopes?
a) What is the missing point for this figure to be a rhombus. Use slope or distance of sides to guide you.
b) Find the slope of the diagonals of the rhombus. What is the relationship of these two slopes?
15) Three of the four vertices of a rhombus are \((2, -1)\), \((-1, -3)\), and \((-4, -1)\).
a) What is the fourth vertex?
b) What is the length of all four sides?
a) What is the fourth vertex?
b) What is the length of all four sides?
16) \(ABCD\) is a rectangle. The green triangles and Pythagorean Theorem can guide you in finding the slopes of the corresponding sides.
a) What is the slope of \(\overline{AB}\), \(\overline{BC}\)? b) What is the relationship of the slopes of \(\overline{AB}\) and \(\overline{BC}\)? c) What is the relationship between the segments \(\overline{AB}\) and \(\overline{BC}\)? d) What are the slopes of \(\overline{CD}\) and \(\overline{AD}\)? e) What is the relationship of the slopes of \(\overline{CD}\) and \(\overline{AD}\)? f) What is the relationship between the segments \(\overline{CD}\) and \(\overline{AD}\)? |
17) \(ABCD\) is a parallelogram.
a) Use the slopes of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{AD}\) to show that the figure is a rectangle. b) What is the length of \(\overline{AC}\)? c) What is the length of \(\overline{BD}\)? d) What can you say about the diagonals of this figure? |
19) Given figure \(ABCD\):
a) Find both the slope and length of \(\overline{AB}\). b) Find both the slope and length of \(\overline{BC}\). c) Find both the slope and length of \(\overline{CD}\). d) Find both the slope and length of \(\overline{AD}\). e) Name the relationships of the sides of \(ABCD\). f) Find both the slope and length of diagonal \(\overline{AC}\). g) Find both the slope and length of diagonal \(\overline{BD}\). h) Name the relationships of the diagonals \(\overline{AC}\) and \(\overline{BD}\). i) Is this figure a rhombus? j) Is this figure a rectangle? k) What is the most descriptive name for this figure? l) What are \(m\angle ABC\), \(m\angle AKB\), and \(m\angle ABK\)? |
20) Answer with “Always”, “Sometimes”, or “Never” true.
a) A parallelogram is a quadrilateral with two pairs of opposite sides parallel
b) All parallelograms have all four sides congruent
c) A rhombus has all the properties of a parallelogram
d) A rhombus has congruent diagonals
e) A square has congruent diagonals
f) The diagonals of a rectangle bisect their interior angles
g) A rhombus has diagonals that intersect at right angles
h) A rectangle has congruent diagonals
i) A rhombus is a square
j) Both diagonals of a square divide the square into four congruent \(30-60-90\) triangles.
k) A square is equilateral and equiangular
l) Parallelograms, rectangles, rhombuses and squares have two diagonals
m) Parallelograms, rectangles, rhombuses and squares all have interior angles that add to \(360^{\circ}\)
a) A parallelogram is a quadrilateral with two pairs of opposite sides parallel
b) All parallelograms have all four sides congruent
c) A rhombus has all the properties of a parallelogram
d) A rhombus has congruent diagonals
e) A square has congruent diagonals
f) The diagonals of a rectangle bisect their interior angles
g) A rhombus has diagonals that intersect at right angles
h) A rectangle has congruent diagonals
i) A rhombus is a square
j) Both diagonals of a square divide the square into four congruent \(30-60-90\) triangles.
k) A square is equilateral and equiangular
l) Parallelograms, rectangles, rhombuses and squares have two diagonals
m) Parallelograms, rectangles, rhombuses and squares all have interior angles that add to \(360^{\circ}\)