5-1. Carla is thinking about parallelograms and wondering if there are as many special
properties for parallelograms as there are for triangles. She remembers that it is
possible to create a shape that looks like a parallelogram by rotating a triangle about
the midpoint of one of its sides. Carefully trace the triangle below onto paper. Be sure
to copy the angle markings as well. Then rotate the triangle to make a shape that looks
like a parallelogram.
a. Is Carla’s shape truly a parallelogram? Why? Write a convincing argument.
b. What else can the congruent triangles tell you about a parallelogram? List all
relationships you can find between the angles and sides of a parallelogram.
c. Does the diagonal of a parallelogram always split the shape into two congruent
triangles? Draw the parallelogram below on your paper. Knowing only that the
opposite sides of a parallelogram are parallel, create a flowchart to show that the
triangles are congruent. (Check your notes from chapter 5-1)
5-2. Solve for x. Show all work.
a. Point M is a midpoint of EF. If EM = 4x − 2 and MF = 3x + 9.
b. WXYZ below is a parallelogram. If m∠W = 9x − 3° and m∠Y = 3x + 15°.
5-3. A rectangle has one side of length 12 mm and a diagonal of 13 mm. Draw a
diagram of this rectangle and find its width and area.
5-4. There are often many ways to prove a statement. You have rotated triangles to
create parallelograms and used congruent parts of congruent triangles to justify that
opposite sides are parallel. But is there another way?
Angel wants to prove the statement “If a quadrilateral is a parallelogram, then
opposite angles are congruent.” He started by drawing parallelogram TUVW below.
Copy and complete his flowchart. Make sure that each statement has a reason.
5-5. Use the relationships in the diagram below to find the values of each variable.
Name which geometric relationships you used.
5-6. Use the relationships in the diagram below to find the values of each variable.
Name which geometric relationships you used.
5-7. Point M is the midpoint of
and B is the midpoint of
values of x and y? Show all work and reasoning.
. What are the
5-8. As Sergio was drawing shapes, he drew a line segment that connected the
midpoints of two sides of a triangle. This is called the midsegment of a triangle.
Describe how DE is related to AB. If DE = 12, what is AB? If AB = 5, what is DE?
5-9. Examine the diagram below.
a. Use a flowchart with justifications to prove ∆FGH
∆FIJ.
b. Name all the pairs of congruent angles in this diagram you can.
c. Are GH and IJ parallel? Explain how you know.
d. If GH = 4x − 3 and IJ = 3x + 14 , find x. Then find the length of GH.
5-10. Examine the geometric relationships in each of the diagrams below. For each
one, write and solve an equation to find the value of the variable. Name any
geometric property or conjecture that you used.
5-11. Jose started to prove that the triangles below are congruent. He was given
point E is the midpoint of segments
and
. Copy and complete his flowchart.
5-12. For each diagram below, solve for x. Show all work.
5-13. Given the information in the diagram below, prove that ΔWXY ≅ ΔYZW using a
flowchart.
5-14. Mr. Quincey likes to play a game with his class. He says, “My quadrilateral has
four right angles.” His students say, “Then it MUST BE a rectangle” and “It COULD
BE a square.” For each description of a quadrilateral below, say what special type the
quadrilateral must be and/or what special type the quadrilateral could be. Look out,
some descriptions may have no must be statements and some descriptions may have
many could be statements!
a. “My quadrilateral has four equal sides.”
b. “My quadrilateral has two pairs of opposite parallel sides.”
c. “My quadrilateral has two consecutive right angles.”
d. “My quadrilateral has two pairs of equal sides.”
5-15. What can congruent triangles tell us about the diagonals and angles of a
rhombus? Examine rhombus ABCD below. Decide how to prove that the diagonals
of a rhombus bisect the angles. Use a flowchart to prove that ∠ABD ≅ ∠CBD.
5-16. What can congruent triangles tell us about the diagonals of a rectangle?
Examine the rectangle below. Using the fact that the opposite sides of a rectangle are
congruent, prove that the diagonals of the rectangle are congruent. Use a flowchart to
prove that AC ≅ BD.
5-17. Here are some more challenges from Mr. Quincey. For each description of a
quadrilateral below, say what special type the quadrilateral must be and/or what
special type the quadrilateral could be. Look out: Some descriptions may have
no must be statements, and some descriptions may have many could be statements!
a. “My quadrilateral has a pair of equal sides and a pair of parallel sides.”
b. “My quadrilateral has congruent diagonals.”
c. “My quadrilateral has four congruent angles.”
5-18. Quadrilateral ABCD below is a rhombus. If BD = 10 units and AC = 18 units,
then what is the perimeter of ABCD? Show all work.
5-19. If ABCD is a rhombus with side length 15 mm and if diagonal BD = 24 mm,
then find the length of the other diagonal, AC. Draw a diagram and show all work.
5-20. An isosceles trapezoid is a trapezoid with a pair congruent legs. Examine
trapezoid EFGH below. How do the angles appear to be related?
a. Complete the statements: ∠
b. If
c. If
∠
∠
≅ ?, ∠ ≅ ?,
∠
∠
,
∠
∠
, find the measures of all the other angles of trapezoid EFGH.
and ∠
, then
∠
5-21. Randy has decided to study the triangle graphed below.
Use the distance formula to find RD, DN, NR. What is the best name for this triangle.
5-22. Graph quadrilateral ABCD. A(1, 4), B(5, 1), C(1, –2), and D(–3, 1).
a. What is the best name for this shape? Justify using distance formula.
b. What is the perimeter of ABCD?
5-23. For each pair of triangles below, determine if the triangles are congruent.
Complete the correspondence statement and state the congruence property,
a. ΔABC ≅ Δ____
c. ΔEDG ≅ Δ____
b. ΔSQP ≅ Δ____
d. ΔWXY ≅ Δ____
5-24. Remember that if a triangle has two equal sides, it is called isosceles. Decide
whether each triangle formed by the points below is isosceles. Show your distance
formula calculations to justify your conclusions.
a. (−3, 7), (−5, 2), (−1, 2)
b. (4, 1), (2, 3), (9, 2)
c. (1, 1), (5, −3), (1, −7)
5-25. Graph quadrilateral ABCD. A(–5, 3), B(–2 , 7), C(6, 1), and D(3, –3).
a. What is the best name for this shape? Justify using distance formula.
b. What is the perimeter of ABCD?
5-26. For each part, write and solve an equation to find the value of the variable.
b. ΔABC below is equilateral
a.
5-27. If ΔABC ≅ ΔDEC , which of the statements below must be true? Justify your
conclusion. Note: More than one statement may be true.
a.
b. m∠B = m∠D
c.
d. AB ≅ DE
5-28. Consider two line segments
and D(9, 15).
and
, given A(0, 8), B(9, 2), C(1, 3),
a. Draw these two segments on a coordinate grid. Find the slope of each segment.
b. Is
? Is
? Justify your answer.
c. Find the length of
and
.
5-29. Plot the following points on graph paper and connect them in the order given.
A(−3, 4), B(1, 6), C(5, −2), and D(1, −4)
A rectangle is a four-sided polygon with four right angles. Does the shape you
graphed appear to be a rectangle? Use slope to justify your answer.
5-30. The guidelines set forth by the National Kitchen & Bath Association
recommends that the perimeter of the triangle connecting the refrigerator (F), stove,
and sink of a kitchen be 26 feet or less. Lashayia is planning to renovate her kitchen
and has chosen the design below. Does her design conform to the National Kitchen
and Bath Association’s guidelines? Show how you got your answer.
5-31. PQRS is a rhombus with perimeter = 28 units. PR = 8 units, find b.
5-32. Shayla just drew quadrilateral SHAY, shown below. The coordinates of its
vertices are:
S(0, 0) H(0, 5) A(4, 8) Y(7, 4)
Shayla thinks her quadrilateral is a trapezoid. Is she correct? Use slope to justify
your answer.
5-33. Tomika remembers that the diagonals of a rhombus are perpendicular to each
other.
a. Graph on ABCD if A(1, 4), B(6, 6), C(4, 1), and D(−1, −1). Is ABCD a rhombus?
Show how you know. (use distance formula.)
b. Compare the slopes of AC and BD. What do you notice?
5-34. Examine the relationships in the diagrams below. For each one, write an
equation and solve for the given variable(s). Show all work.
5-35. ABCD is a parallelogram. If A(3, −4), B(6, 2), C(4, 6), then what are the
possible locations of point D? Draw a graph and justify your answer.
5-36. Each problem below gives the endpoints of a segment. Find the coordinates of
the midpoint of the segment.
a. (−3, 11) and (5, 6)
b. (1, 5) and (7, 11)
c. (−4, −1) and (8, 9)
d. (a, b) and (c, d)
5-37. On graph paper, draw quadrilateral MNPQ if M(1, 7), N(−2, 2), P(3, −1),
and Q(6, 4).
a. Find the slopes of
and
. What can you conclude about ∠MNP ?
b. What is the best name for MNPQ? Justify your answer.
c. Which diagonal is longer? Explain how you know your answer is correct.
d. Find the midpoint of
.
5-38. Which pairs of triangles below are congruent and/or similar? For each part,
explain how you know using an appropriate triangle congruence or similarity
condition. Note: The diagrams are not necessarily drawn to scale.
5-39. Cassie wants to confirm her theorem on midsegments using a coordinate grid.
She started with ΔABC, with A(0, 0), B(2, 6), and C(7, 0).
a. Graph ΔABC on graph paper.
b. Find the coordinates of P, the midpoint of AB. Likewise, find the coordinates of Q,
the midpoint of BC.
c. Show that the length of the midsegment, PQ, is half the length of AC. Use the
distance formula.
5-40. Consider ΔABC with vertices A(2, 3), B(6, 6), and C(8, –5).
a. Draw ΔABC on graph paper. What kind of triangle is ΔABC? Prove your result.
b. Reflect ΔABC across AC. Find the location of B′. What name best describes the
resulting figure? Prove your claim using the distance formula.
5-41. Mr. Quincey has some new challenges for you! For each description below,
decide what special type the quadrilateral must be and/or what special type the
quadrilateral could be. Look out: Some descriptions may have no must be statements,
and some descriptions may have many could be statements!
a. “My quadrilateral has three right angles.”
b. “My quadrilateral has a pair of parallel sides.”
c. “My quadrilateral has two consecutive equal angles.”
5-42. Solve for x in each diagram below.
a.
b.
c.
5-43. On graph paper, graph each of the lines below on the same set of axes. What is
the relationship between lines (a) and (b)? What about between (b) and (c)?
a.
c. y = − 3x – 2
b.
5-44. Graph the segment that connects the points A(−4, 8) and B(6, 3).
a. What is the slope of
?
b. Write an equation for the line that connects points A and B.
c. Write an equation for a line that is parallel to
through (0, 0).
d. Write an equation for a line that is perpendicular to
through B(6, 3).
5-45. Use what you learned about the slopes of parallel and perpendicular lines to find
the equation of a line that would meet the criteria given below.
a. Find the equation of the line that goes through the point (0, −3) and is perpendicular
to the line
.
b. Find the equation of the line that is parallel to the line −3x + 2y = 10 and goes
through the point (0, 7).
5-46. Find the equation for each line on the graph below. Remember, the general
form of any line in the slope-intercept form is y = mx + b.
What is the relationship between the two lines above? How do you know?
5-47. For the points R (−2, 7) and P (2, 1) determine each of the following:
a. The slope of the line through the points.
b. The distance between the points.
c. An equation of the line
.
d. An equation of the line perpendicular to line
and passing through point P.
5-48. Write a linear equation that represents each situation.
a. An equation for the line below.
b. An equation for a line perpendicular to the line in part (a) and passing through the
point (−1, −3).
c. An equation of the line passing through the points (4, 3) and (−1, 1).
5-49. On graph paper, graph line
a. Find the slope of
if M(−1, 1) and U(4, 5).
and write an equation for the line.
b. Find MU (the distance from M to U).
c. Are there any similarities to the calculations used in parts (a) and (b)? Any
differences?
Extra Classwork Unit 5
5-1. Examine ΔABC and ΔDEF below.
Assume the triangles above are not drawn to scale. Complete a flowchart to justify
the relationship between the two triangles. Find AC and DF.
5-2. What is the relationship of ΔABC and ΔGHJ below? Create a flowchart to justify
your conclusion.
5-3. For each pair of triangles below, decide whether the triangles are similar and/or
congruent. Justify each conclusion.
5-4. For each pair of triangles below, determine whether or not the triangles are
similar. If they are similar, show your reasoning in a flowchart. If they are not
similar, explain how you know.
5-5. For each diagram below, find the value of x, if possible. If the triangles are
congruent, state which triangle congruence condition was used. If the triangles are
not congruent or if there is not enough information, state, “Cannot be determined.”
5-6. With the class or your team, create a flowchart to prove your answer to part (b) of
problem 7-73. That is, prove that
. Be sure to include a diagram for your
proof and reasons for every statement. Make sure your argument is convincing and
has no “holes.”
5-7. Identify if each pair of triangles below is congruent or not. Remember that the
diagram may not be drawn to scale. Justify your conclusion
5-8. For either part (a), (c) or (d) of problem 7-75, create a flowchart to prove your
conclusion. Remember to start with the given information and include a reason or
justification for each “bubble” in your flowchart.
5-9. On graph paper, graph quadrilateral ABCD if A(0, 0), B(6, 0), C(8, 6), and D(2,
6).
What is the best name for ABCD? Justify your answer.
5-10. Graph quadrilateral ABCD. A(1, 4), B(5, 1), C(1, –2), and D(–3, 1).
a. What is the best name for this shape? Justify using distance formula.
b. What is the perimeter of ABCD?
5-11. On graph paper, plot ABCD if A(–1, 2), B(0, 5), C(2, 5), and D(6, 2).
What type of shape is ABCD? Justify your answer.
5-12. Here are some more challenges from Mr. Quincey. For each description of a
quadrilateral below, say what special type the quadrilateral must be and/or what
special type the quadrilateral could be. Look out: Some descriptions may have
no must be statements, and some descriptions may have many could be statements!
My quadrilateral has a pair of equal sides and a pair of parallel sides.
The diagonals of my quadrilateral bisect each other
5-13. Here are some more challenges from Mr. Quincey. For each description of a
quadrilateral below, say what special type the quadrilateral must be and/or what
special type the quadrilateral could be. Remember: Some descriptions may have
no must be statements, and some descriptions may have many “could be statements!
The diagonals of my quadrilateral are equal.
My quadrilateral has one right angle.
My quadrilateral has one pair of equal adjacent sides.
5-14. Plot the following points on another sheet of graph paper and connect them in
the order given. Then connect points A and D.
A(−3, 4), B(1, 6), C(5, −2), and D(1, −4)
A rectangle is a four-sided polygon with four right angles. Does the shape you
graphed appear to be a rectangle? Use slope to justify your answer.
If ABCD is rotated 90° clockwise ( ) about the origin to form A′B′C′D′, what are the
coordinates of the vertices of A′B′C′D′?