1) Given figure \(ABCD\). Use the green triangles to help find slope.
a) What is the slope of \(\overline{AB}\) and \(\overline{CD}\)? b) What is the relationship between \(\overline{AB}\) and \(\overline{CD}\)? c) What is the slope of \(\overline{AD}\) and \(\overline{BC}\)? d) What is the relationship between \(\overline{AD}\) and \(\overline{BC}\)? e) What is the relationship between \(\angle 1\) and \(\angle 2\). Justify your answer. \(\;\;\;\;\;\;\;\;\)Hint, use your answer from b) to help). f) What is the relationship between \(\angle 3\) and \(\angle 4\). Justify your answer. g) Knowing that \(\overline{AC}\) is congruent to itself, what is the relationship of \(\;\;\;\;\;\;\;\;\)\(\triangle ABC\) and \(\triangle CDA\)? Justify your answer based on your previous \(\;\;\;\;\;\;\;\;\)responses in this problem. |
2) Given figure \(ABCD\) with \(\angle{A}\cong \angle{C}\) and \(\angle{B}\cong\angle{D}\).
a) \(m\angle A + m\angle B + m\angle C + m\angle D =\) b) \(m\angle A + m\angle B =\) c) From b), what is the relationship between \(\overline{AD}\) and \(\overline{BC}\)? d) \(m\angle A + m\angle D =\) e) From d), what is the relationship between \(\overline{AB}\) and \(\overline{DC}\)? |
3) Given figure \(ABCD\) with \(\angle 1\cong\angle 2\) and \(\angle\ 3\cong\angle 4\).
a) Since \(\angle 1\cong\angle 2\), what is the relationship between \(\overline{AB}\) and \(\overline{CD}\)? b) Since \(\angle 3\cong\angle 4\), what is the relationship between \(\overline{AD}\) and \(\overline{BC}\)? c) From a) and b), justify the figure is a parallelogram. |
4) Given figure \(ABCD\) with \(\overline{AB}\cong\overline{CD}\) and \(\overline{AD}\cong\overline{BC}\).
a) What is the relationship between \(\triangle{ABD}\) and \(\triangle{CDB}\)? Why? b) From a), what is the relationship of \(\angle 1\) and \(\angle 2\) and \(\angle 3\) and \(\angle 4\). Justify your answer. c) From b) justify the figure has two pairs of opposite sides parallel, thus the figure is a parallelogram. |
5) After school, your mom asks you what you learned in geometry. You say, “well we learned that if a quadrilateral has one pair of opposite sides both congruent and parallel, it must be a parallelogram.” But, she doesn’t believe that the statement is true. Can you prove it? Use mathematical reasoning.
|
6) Given figure \(ABCD\) with a pair of bisecting diagonals \(\overline{AK}\cong\overline{KC}\) and \(\overline{BK}\cong\overline{KD}\).
a) What is the relationship between \(\angle{AKD}\) and \(\angle{CKB}\)? b) So, what can we conclude about the relationship between \(\triangle{AKD}\) and \(\triangle{CKB}\)? c) To prove \(ABCD\) is a parallelogram, we can show one pair of sides both congruent \(\;\;\;\;\)and parallel. Justify \(\overline{AD}\cong\overline{BC}\) and \(\overline{AD}\parallel\overline{BC}\). |
12) Figure \(ABCD\) (NOT drawn to scale). Determine if the markings are enough to justify that the figure is a parallelogram. If yes, name the appropriate property.
13) Given figure \(ABCD\) with \(A(4, 1)\), \(B(3, -3)\), \(C(-3, -2)\), and \(D(-2, 2)\), justify that the figure is a parallelogram. Show your work and justify your reasoning.
14) For each statement, decide (true or false) if the statement is enough to justify the quadrilateral is a parallelogram.
a) A four sided figure has all four sides congruent.
b) A four sided figure has a diagonal that divides the figure into two congruent triangles.
c) A four sided figure is equiangular.
d) A four sided figure has congruent diagonals.
e) The figure is a square.
f) A four sided figure has all pairs of adjacent angles supplementary.
g) A four sided figure has two pairs of congruent angles.
h) A four sided figure has diagonals that are perpendicular bisectors of each other.
i) A four sided figure has one pair of sides congruent and one pair of sides parallel.
j) A four sided figure has two consecutive pairs of sides congruent.
k) The figure is a kite.
l) A figure has four angles with measures of \(75^{\circ}\), \(105^{\circ}\), \(75^{\circ}\), and \(105^{\circ}\) respectively.
m) A figure has one pair of opposite sides both \(4\) inches and the other pair of opposite sides \(5\) inches.
Solution Bank
14) For each statement, decide (true or false) if the statement is enough to justify the quadrilateral is a parallelogram.
a) A four sided figure has all four sides congruent.
b) A four sided figure has a diagonal that divides the figure into two congruent triangles.
c) A four sided figure is equiangular.
d) A four sided figure has congruent diagonals.
e) The figure is a square.
f) A four sided figure has all pairs of adjacent angles supplementary.
g) A four sided figure has two pairs of congruent angles.
h) A four sided figure has diagonals that are perpendicular bisectors of each other.
i) A four sided figure has one pair of sides congruent and one pair of sides parallel.
j) A four sided figure has two consecutive pairs of sides congruent.
k) The figure is a kite.
l) A figure has four angles with measures of \(75^{\circ}\), \(105^{\circ}\), \(75^{\circ}\), and \(105^{\circ}\) respectively.
m) A figure has one pair of opposite sides both \(4\) inches and the other pair of opposite sides \(5\) inches.
Solution Bank