Lesson 32-2
Proving a Quadrilateral Is a Rectangle
ACTIVITY 32
continued
• Develop criteria for showing that a quadrilateral is a rectangle.
• Prove that a quadrilateral is a rectangle.
Pacing: 1 class period
Chunking the Lesson
#1–5
#6
#7
#8
#9
Check Your Understanding
Lesson Practice
SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Create
Representations, Group Presentation, Discussion Groups
1. Complete the following definition.
A rectangle is a parallelogram with ________.
TEACH
four right angles
2. a. Complete the theorem.
Bell-Ringer Activity
Tell whether each property is a property
of every rectangle.
1. opposite sides are congruent
[yes]
2. all angles are right angles
[yes]
3. diagonals are perpendicular
[no]
Theorem If a parallelogram has one right angle, then it has
rectangle
four
________
right angles, and it is a ________.
b. Use one or more properties of a parallelogram and the definition of a
rectangle to explain why the theorem in Item 1 is true.
Since consecutive angles are supplementary in a parallelogram,
the two angles that are consecutive to the given right angle will
also be right angles. Since opposite angles of a parallelogram are
congruent, the angle opposite the given right angle will also be a
right angle. By definition, a quadrilateral with four right angles is a
rectangle.
1–5 Activating Prior Knowledge,
Think-Pair-Share, Discussion
Groups, Group Presentation,
Debriefing In Items 1 through 5,
students work through the process of
proving the theorem that if a
quadrilateral is equiangular, then it is a
rectangle. Before starting, students
should be able to recall the definition of
a rectangle. In Items 2 and 3, they are
asked to make conjectures and write an
informal (paragraph) proof for the
theorem. Students should be familiar
with identifying the hypothesis and
conclusion of a conditional statement as
they complete Item 4. A correct
response to this item is essential because
students will be using this response for a
proof in Item 5. The teacher may specify
the type of proof for Item 5 (two-column,
paragraph, flowchart) or leave the
decision to the students. Group
presentations and the resulting class
discussion will facilitate students’ attempts
at proof writing throughout this activity.
3. Given WXYZ.
a. If WXYZ is equiangular, then find the measure of each angle.
90°
b. Complete the theorem.
© 2017 College Board. All rights reserved.
rectangle
Theorem If a parallelogram is equiangular, then it is a _________.
© 2017 College Board. All rights reserved.
Lesson 32-2
PLAN
My Notes
Learning Targets:
ACTIVITY 32 Continued
X
Y
W
Z
4. Make sense of problems. Identify the hypothesis and the
conclusion of the theorem in Item 3. Use the figure in Item 3.
Hypothesis: WXYZ is equiangular.
Conclusion: WXYZ is a rectangle.
5. Write a proof for the theorem in Item 3.
Statements
Reasons
1. WXYZ is equiangular.
1. Given
2. m∠W = m∠X = m∠Y = m∠Z
2. Def. of equiangular
3. m∠W + m∠X + m∠Y +
m∠Z = 360°
3. The four angles of a
quadrilateral add to 360°.
4. 4m∠W = 360°
4. Substitution Property
5. m∠W = 90°
5. Division Property
6. WXYZ is a rectangle.
6. If a parallelogram has one
right ∠, it is a rectangle.
Activity 32 • More About Quadrilaterals
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Activity 32 • More About Quadrilaterals
553
Lesson 32-2
Proving a Quadrilateral Is a Rectangle
ACTIVITY 32
continued
My Notes
7 Think-Pair-Share, Create
Representations, Group
Presentation, Discussion Groups
Students apply the theorem and a
coordinate argument to show that a
quadrilateral is a parallelogram. If time
permits, students would benefit from
seeing multiple solutions.
6. Given OKAY with congruent diagonals, OA and KY.
O
K
Y
A
a. List the three triangles that are congruent to OYA, and the reason for
the congruence.
OYA ≅ KAY ≅ YOK ≅ AKO by SSS
b. List the three angles that are corresponding parts of congruent triangles
and congruent to ∠OYA.
∠OYA ≅ ∠KAY ≅ ∠YOK ≅ ∠AKO
c. Find the measure of each of the angles in part b.
m∠OYA = m∠KAY = m∠YOK = m∠AKO = 90°
8 Think-Pair-Share, Group
Presentation, Discussion Groups,
Debriefing Students use congruent
triangles and corresponding parts to
apply the theorem from Item 6 in a
proof. Since it is used often, check to
make sure that all students understand
CPCTC and how it can be applied.
Instead of the acronym, say the entire
sentence: Corresponding parts of
congruent triangles are congruent.
d. Complete the theorem.
Theorem If the diagonals of a parallelogram are _______, then the
parallelogram is a _______.
congruent
rectangle
7. Given quadrilateral ABCD with coordinates A(1, 0), B(0, 3), C(6, 5), and
D(7, 2).
a. Show that quadrilateral ABCD is a parallelogram.
b. Use the theorem in Item 6d to show that quadrilateral ABCD is
a rectangle.
AC = BD = 50, so the diagonals are congruent.
8. Write a two-column proof using the theorem in Item 6 as the last
reason.
9 Think-Pair-Share, Group
Presentation, Discussion Groups,
Debriefing This is an opportunity for
students to review the second part of
this activity. Lead a class discussion on
ways of proving that a quadrilateral (or a
parallelogram) is a rectangle. Either the
teacher or one of the students can
continue the “master list” from the
previous class discussion at the end of
the first part of the activity.
Given: GRAM
G
R
M
A
GRM ≅ RGA
Prove: GRAM is a rectangle.
Statements
Reasons
1. GRAM
GRM ≅ RGA
1. Given
2. RM ≅ GA
2. CPCTC
3. GRAM is a rectangle.
3. If the diagonals of a para
are ≅, the para is a rect.
9. Summarize this part of the activity by making a list of the ways to prove
that a quadrilateral (or parallelogram) is a rectangle.
Show that a quadrilateral has four right angles. Show that a
parallelogram has one right angle. Show that a quadrilateral is
equiangular. Show that the diagonals of a parallelogram are congruent.
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SpringBoard® Integrated Mathematics I, Unit 6 • Triangles and Quadrilaterals
SpringBoard® Integrated Mathematics I, Unit 6 • Triangles and Quadrilaterals
© 2017 College Board. All rights reserved.
6 Activating Prior Knowledge,
Think-Pair-Share If students are
struggling as they are led through the
key steps in this proof, suggest that they
review previous proofs in which they
used congruent triangles, corresponding
parts, and the angle sum in a
quadrilateral.
© 2017 College Board. All rights reserved.
ACTIVITY 32 Continued
Lesson 32-2
Proving a Quadrilateral Is a Rectangle
ACTIVITY 32
continued
My Notes
Check Your Understanding
10. Jamie says a quadrilateral with one right angle is a rectangle. Find a
counterexample to show that Jamie is incorrect.
11. Do the diagonals of a rectangle bisect each other? Justify your answer.
ACTIVITY 32 Continued
Check Your Understanding
Debrief students’ answers to these items
to ensure that they understand concepts
related to proving that quadrilaterals are
rectangles. When students answer
Item 11, you may also want to have
them draw a diagram to further explain
how they arrived at their answer.
Answers
10. Sample answer:
LESSON 32-2 PRACTICE
Three vertices of a rectangle are given. Find the coordinates of
the fourth vertex.
12. (−3, 2), (−3, −1), (3, −1)
13. (−12, 2), (−6, −6), (4, 2)
11. Yes. The diagonals are congruent
and the diagonals bisect each
another.
14. (4, 5), (−3, −4), (6, −1)
For Items 15–16, find the value of x that makes the parallelogram a rectangle.
15.
16.
ASSESS
(2x + 4)
Students’ answers to the Lesson Practice
items will provide a formative
assessment of their understanding of
how to prove that a quadrilateral is a
rectangle, and of students’ ability to
apply their learning.
(5x + 15)°
4x
17. Model with mathematics. Jill is building a new gate for her yard as
shown. How can she use the diagonals of the gate to determine if the
gate is a rectangle?
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
Short-cycle formative assessment items
for Lesson 32-2 are also available in the
Assessment section on SpringBoard
Digital.
Refer back to the graphic organizer the
class created when unpacking
Embedded Assessment 2. Ask students
to use the graphic organizer to identify
the concepts or skills they learned in this
lesson.
ADAPT
Activity 32 • More About Quadrilaterals
555
LESSON 32-2 PRACTICE
12.
13.
14.
15.
16.
17.
(3, 2)
(−2, 10)
(−5, 2)
x = 15
x=2
Jill can measure the lengths of the
diagonals. If their measures are
equal then the gate is a rectangle.
Check students’ answers to the Lesson
Practice to ensure that they understand
basic concepts related to proving that
quadrilaterals are rectangles and are
ready to transition to proving that
quadrilaterals are rhombuses. If students
need more practice with the concepts in
this lesson, they may find it helpful to
refer to a list of ways to prove that a
quadrilateral is a rectangle. Have them
compare and contrast that list with the
list of properties of rectangles.
See the Activity Practice on page 561
and the Additional Unit Practice in the
Teacher Resources on SpringBoard
Digital for additional problems for this
lesson.
You may wish to use the Teacher
Assessment Builder on SpringBoard
Digital to create custom assessments or
additional practice.
Activity 32 • More About Quadrilaterals
555