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Secondary II
Geometric Reasoning
Teacher Edition
Unit 2
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Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
NUCC| Secondary II Math i
Table of Contents
2.1 BASIC ANGLE THEOREMS .......................................................................................................................1
Teacher Notes ..................................................................................................................................................1
Mathematics Content .......................................................................................................................................3
Roads A Develop Understanding Task 1 .........................................................................................................4
Ready, Set, Go! ................................................................................................................................................5
Solutions ..........................................................................................................................................................8
2.2 PARALLEL LINES AND TRANSVERSALS .............................................................................................9
Teacher Notes ..................................................................................................................................................9
Mathematics Content .....................................................................................................................................11
Ready, Set, Go! ..............................................................................................................................................12
Solutions ........................................................................................................................................................15
2.3 RELATIONSHIPS BETWEEN PERPENDICULAR LINES .....................................................................16
Teacher Notes ................................................................................................................................................16
Mathematics Content .....................................................................................................................................19
A Hiker in Distress A Develop Understanding Task 2 ..................................................................................20
Ready, Set, Go! ..............................................................................................................................................21
Solutions ........................................................................................................................................................24
2.4 INTERIOR ANGLES OF A TRIANGLE ...................................................................................................25
Teacher Notes ................................................................................................................................................25
Mathematics Content .....................................................................................................................................26
Interior Angles of a Triangle A Develop Understanding Task 3 ...................................................................27
Ready, Set, Go! ..............................................................................................................................................30
Solutions ........................................................................................................................................................32
2.5 ISOSCELES TRIANGLE ............................................................................................................................33
Teacher Notes ................................................................................................................................................33
Mathematics Content .....................................................................................................................................34
Isosceles Triangles A Develop Understanding Task 4 ...................................................................................35
Ready, Set, Go! ..............................................................................................................................................37
Solutions ........................................................................................................................................................40
2.6 MIDSEGMENT OF A TRIANGLE ............................................................................................................41
Teacher Notes ................................................................................................................................................41
Mathematics Content .....................................................................................................................................42
Midsegment of a Triangle A Develop Understanding Task 5 ........................................................................43
NUCC| Secondary II Math ii
Ready, Set, Go! ..............................................................................................................................................45
Solutions ........................................................................................................................................................47
2.7 CENTROID OF A TRIANGLE ..................................................................................................................48
Teacher Notes ................................................................................................................................................48
Mathematics Content .....................................................................................................................................49
Medians of a Triangle A Develop Understanding Task 6 ..............................................................................50
Ready, Set, Go! ..............................................................................................................................................52
Solutions ........................................................................................................................................................54
2.8 PARALLELOGRAMS ................................................................................................................................55
Teacher Notes ................................................................................................................................................55
Mathematics Content .....................................................................................................................................57
Setting up the Basketball Hoop A Develop Understanding Task 7 ...............................................................58
Ready, Set, Go! ..............................................................................................................................................59
Solutions ........................................................................................................................................................62
2.9 MORE DISCOVERIES IN PARALLELOGRAMS ...................................................................................63
Teacher Notes ................................................................................................................................................63
Parallelograms A Develop Understanding Task 8 .........................................................................................64
Mathematics Content .....................................................................................................................................67
Ready, Set, Go! ..............................................................................................................................................68
Solutions ........................................................................................................................................................71
ON-LINE RESOURCES: ..................................................................................................................................72
Practice Exam ....................................................................................................................................................74
Solutions ........................................................................................................................................................79
Exam ..................................................................................................................................................................80
Solutions ........................................................................................................................................................86
NUCC| Secondary II Math iii
Unit 2.1
2.1 BASIC ANGLE THEOREMS
Teacher Notes
Time Frame:
Materials Needed: Classroom set of protractors
Purpose: Get students to identify the geometric vocabulary and visualize the angle relationships.
Justify the basic angle theorems associated with vertical angles, adjacent angles, and linear pairs
and use the theorems to solve mathematical problems involving algebra.
Related Standards: G.CO.9 Prove theorems about lines and angles. Identify and verify that
vertical angles are congruent, angles that form a linear pair are supplementary, and that adjacent
angles that form a right triangle are complimentary.
Vocabulary:
Complimentary angles: A pair of angles that add to 90 degrees.
Supplementary angles: A pair of angles that add to 180 degrees.
Perpendicular angles: A pair of angles that intersect at 90 degree angles or whose slopes are
negative reciprocals of each other
Linear pair: A pair of adjacent angles that together forms a straight line.
Congruent angles: Angles that have the same angle measure.
Vertical angles: A pair of angles across from each other in a pair of intersecting lines.
Adjacent angles: A pair of angles that share a vertex and one side.
Skills:
Are students able to draw to angles that are adjacent, but not congruent?
Are students able to draw two angles that are supplementary, but that do not form a linear pair?
Launch (Whole Class): Refer to the map and the task page and define and identify the
following terms in the map: right angles, complimentary angles, supplementary angles, vertical
angles, linear pair, adjacent angles, parallel lines, and perpendicular lines.
Explore (Individual, small group or pairs): Use the following picture and have students identify
C
B
A
30
D
E
H
F
60
G
a pair of vertical angles
a pair of complimentary angles
a pair of supplementary angles
a pair of obtuse vertical angles
a set of angles that form a linear pair.
NUCC| Secondary II Math 1
Unit 2.1
Discuss (Whole Class or Group): How can these definitions be used to solve problems.
B
C
(6x)
(3x)
A
D
(12y - 10) F
(5x - 2)
(2x + 34)
E
NUCC| Secondary II Math 2
Unit 2.1
Mathematics Content
Cluster Title: Prove geometric theorems.
Standard G.CO.9 Prove theorems about lines and angles. (Theorems include: vertical angels are
congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.)
Concepts and Skills to Master
Prove and use theorems about lines and angles, including but not limited to:
 Vertical angles are congruent.
 When parallel lines are cut by a transversal congruent angle pairs are created.
 When parallel lines are cut by transversal supplementary angle pairs are created.
 Points on the perpendicular bisector of a line segment are equidistant from the segment’s
endpoints.
Critical Background Knowledge

Know properties of supplementary, complementary, vertical, and adjacent angles (7.G.5).
Academic Vocabulary
proof, vertical angles, parallel lines, transversal, alternate interior angles, perpendicular bisector,
equidistant
Suggested Instructional Strategies



Use multiple formats to write proofs; narrative paragraphs, flow diagrams, two-column
format, and diagrams without words.
Focus on the validity of the underlying reasoning while writing proofs.
Use dynamic geometry software to explore angle relationships.

Connect angle relationships to the creation of tessellation patterns.
Skills Based Task:


Are students able to draw to angles that
are adjacent, but not congruent?
Are students able to draw two angles that
are supplementary, but that do not form a
linear pair?
Problem Task:
Find as many angle
relationships as possible in
this pattern.
Some Useful Websites:
http://www.khanacademy.org/math/geometry/angles/v/complementary-and-supplementaryangles
http://www.khanacademy.org/math/geometry/angles/e/vertical_angles
http://www.khanacademy.org/math/geometry/angles/e/vertical_angles_2
NUCC| Secondary II Math 3
Unit 2.1
Roads
A Develop Understanding Task 1
Examine the following map of Salt Lake City streets. In Utah, most major streets run either
north-south or east-west. Look at the following streets and measure the angles made at their
intersection points.
mtop 
700 East and 3000 South
mright 
mbottom 
mtop 
2300 East and 2700 South
mright 
mbottom 
mleft 
mleft 
Now look at Highland Drive. Does it follow a north-south or east-west alignment? No. Use a
protractor to measure the angles at the following intersections.
mtop 
Highland Dr. and 2300 South
mright 
mbottom 
mtop 
Highland Dr. and 3000 South
mleft 
What do you notice about the angles? Are they related to each other?
mright 
mbottom 
mleft 
NUCC| Secondary II Math 4
Unit 2.1
Ready, Set, Go!
Ready
1. From the following illustration:
A
C
a. Identify two acute vertical angles.
B
b. Identify two obtuse adjacent angles.
D
F
E
2. The measure of the supplement of an angle is 40 less than three times the compliment of the
angle. Find the angle.
Set
3. If two angles are supplementary and one of them is acute, the other must be obtuse. Is this
always, sometimes or never true?
4. The measures of two complimentary angles are 16y + 13 and 4y – 3. Find the measure of the
two angles.
5. The measure of an angles supplement is 32 less than the measure of the angle. Find the
measure of the angle and its supplement.
6. Which of the following two statements can be justified from the given picture?
a. SRP and PRT are complimentary.
P
b. QPT and TPR are adjacent, but neither
complimentary or supplementary.
Q
T
S
R
NUCC| Secondary II Math 5
Unit 2.1
Go!
7. Use the figure below. PTR and QTS are right angles.
Q
P
R
S
T
a. If mRTS  36 find mQTR and mPTQ .
b. If mQTR  77 find mRTS and mPTQ .
c. If mPTQ  21 find mQTR and mRTS .
d. If mPTS  124 find mPTQ and mTQR .
e. If mRTS  x find mQTR and mPTS .
8. From the picture above, justify mPTQ  mRTS .
9. In the following picture, what value of x guarantees that ABD and DBC form a linear
pair?
D
(2x - 5)
A
(8x + 15)
C
B
10. Given that ABE is a right angle and mEBD  34 , find mFEB.
C
B
A
30
D
E
H
F
60
G
NUCC| Secondary II Math 6
Unit 2.1
11. Given that m1  32 , and mR  90 find the measure of the other seven numbered angles.
R
2
3
1
4
5
6
8 7
12. If X is supplementary to Y and Y is supplementary to Z , then X is
supplementary to Z . Is this always, sometimes or never true?
13. If AB  BC , then ABC is acute. Is this always, sometimes or never true?
14. Two angles are supplementary. One angle measures 32 more than the other. What are the
two angles?
15. Two angles form a linear pair. The measure of one of the angles is twenty less than three
times the angle. What are the two angles?
NUCC| Secondary II Math 7
Unit 2.1
Solutions
1.
2.
3.
4.
5.
6.
7.
a. ABF , CBD
b. ABC , FBD
25
Always
77, 13
106, 74
a. no
b. yes
a. 54 ,36
b. 13 ,13
c. 69 ,21
d. 34 ,56
e. (90  x) ,(90  x)
8.
9.
10.
11.
12.
13.
14.
15.
mPTQ  mQTR  90
mQTR  mRTS  90
then
mPTQ  mQTR  mQTR  mRTS
then
mPTQ  mRTS
17
106
m2  148 , m3  132 , m4  148 , m5  68
m6  112 , m7  68 , m8  112
Sometimes (both must be 90). They will always be equal to each other.
Never
74, 106
50, 130
NUCC| Secondary II Math 8
Unit 2.2
2.2 PARALLEL LINES AND TRANSVERSALS
Teacher Notes
Time Frame:
Materials Needed: Classroom set of protractors.
Purpose: Get students to identify the geometric vocabulary and visualize the angle relationships
in parallel lines and their transversals. Justify the basic angle theorems associated with
corresponding, alternate interior, alternate exterior and consecutive interior angles.
Vocabulary:
Corresponding angles: A pair of angles formed by a pair of lines and a transversal that share the
same locations in both intersections. Top left and top left, bottom right and bottom right, etc.
Alternate interior angles: A pair of angles formed by a pair of lines and a transversal that are both
on the interior of the pair of lines, but on opposite sides of the transversal.
Alternate exterior angles: A pair of angles formed by a pair of lines and a transversal that are
both on the exterior of the pair of lines, but on opposite sides of the transversal.
Same side interior angles: (Also known as consecutive interior angles) A pair of angles formed
by a pair of lines and a transversal that are both on the interior of the pair of lines, but on the
same side of the transversal.
Launch (Whole Class): Refer to the diagram below and define and identify the following terms:
corresponding angles (4 pairs), alternate exterior angles (4 pairs), alternate interior angles (4
pairs), and consecutive interior angles (2 pair). (Notice in this sketch the pair of lines is not
parallel.)
1 2
3 4
5 6
8 7
If the diagram is drawn in a program such as smartboard notebook, or Geometer’s Sketchpad,
adjust the slope of the lines and the transversal and measure all of the angles. Help the students
identify the properties listed above are true only when the lines are parallel.
Explore (Individual, small group or pairs): Have students define the four pairs of related
angles; corresponding angles, alternate interior angles, alternate exterior angles, and consecutive
interior angles. Have them label from the following illustration with the following terms:
alternate interior, alternate exterior, consecutive interior, corresponding, or none of these.
NUCC| Secondary II Math 9
Unit 2.2
5 6
8 7
1 2
4 3
5 are ____________.
3 and 6 are ____________.
2 and 8 are ____________.
1 and 8 are ____________.
4 and 6 are ____________.
1 and
Discuss (Whole Class or Group): These geometric theorems are bi-conditional, that is they work
both directions. If a pair of parallel lines is cut by a transversal, then the corresponding angles are
congruent. If a pair of corresponding angles are congruent, then the two lines cut by the
transversal are parallel.
What must the value of x be so that the two lines would be parallel?
(2x + 37)
(5x + 13)
NUCC| Secondary II Math 10
Unit 2.2
Mathematics Content
Cluster Title: Prove geometric theorems.
Standard G.CO.9 Prove theorems about lines and angles. (Theorems include: vertical angels are
congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.)
Concepts and Skills to Master
Prove and use theorems about lines and angles, including but not limited to:
 Vertical angles are congruent.
 When parallel lines are cut by a transversal congruent angle pairs are created.
 When parallel lines are cut by transversal supplementary angle pairs are created.
 Points on the perpendicular bisector of a line segment are equidistant from the segment’s
endpoints.
Critical Background Knowledge

Know properties of supplementary, complementary, vertical, and adjacent angles (7.G.5).
Academic Vocabulary
proof, vertical angles, parallel lines, transversal, alternate interior angles, perpendicular bisector,
equidistant
Suggested Instructional Strategies
 Use multiple formats to write proofs; narrative paragraphs, flow diagrams, two-column
format, and diagrams without words.
 Focus on the validity of the underlying reasoning while writing proofs.
 Use dynamic geometry software to explore angle relationships.
 Connect angle relationships to the creation of tessellation patterns.
Skills Based Task:
Problem Task:
 Are students able to draw two angles that Find as many angle
relationships as possible in
are adjacent, but not congruent?
 Are students able to draw two angles that this pattern.
are supplementary, but that do not form a
linear pair?
Some Useful Websites:
http://www.khanacademy.org/math/geometry/angles/e/corresponding_angles
http://www.khanacademy.org/math/geometry/angles/e/corresponding_angles_2
http://www.khanacademy.org/math/geometry/angles/e/alternate_interior_angles
http://www.khanacademy.org/math/geometry/angles/e/alternate_exterior_angles
http://www.khanacademy.org/math/geometry/angles/e/alternate_interior_angles_2
http://www.khanacademy.org/math/geometry/angles/e/alternate_exterior_angles_2
http://www.khanacademy.org/math/geometry/angles/e/same_side_interior_angles
http://www.khanacademy.org/math/geometry/angles/e/same_side_interior_angles_2
NUCC| Secondary II Math 11
Unit 2.2
Ready, Set, Go!
Ready
1. If two lines are both perpendicular to the transversal, are they parallel to each other? Why?
2. If two lines are cut by a transversal, the alternate interior angles are supplementary? Is this
always, sometimes or never true?
3. If two lines form alternate exterior angles that are complimentary, the two lines are parallel.
Is this always, sometimes or never true?
Set
4. Identify:
a. DEG and ABE are ________________
b. FEB and EBC are ________________
c. FEB and ABE are ________________
G
D
A
E
F
C
B
5. Use the following picture to answer the following questions.
a. What value of x will guarantee that lines a and b are
parallel?
b. What value of y will guarantee that lines a and b are
parallel?
(8x + 4)
a
(14y + 12)
(5y + 16)
b
(9x - 11)
NUCC| Secondary II Math 12
Unit 2.2
Go!
6. For the following picture find all eight angles for the given information.
1 2
4 3
6
5
8
7
Given m3  113 find the measure of
all 8 angles.
a.
b. Given m8  50 find the measure of all 8
angels.
m1 
m1 
m2 
m3 
m4 
m2 
m3 
m4 
m5 
m6 
m7 
m8 
m5 
m6 
m7 
m8 
7. Given that WXYZ is a parallelogram and mYWZ  37 , find mWYX .
X
W
Y
Z
8. Which of the following statements guarantee that lines m and n are parallel?
a.
b.
c.
d.
e.
f.
m1  42 and m5  42
m4  64 and m5  64
m3  118 and m6  62
m2  57 and m8  57
m1  42 and m7  138
m2  (3x  7) and m6  (3x  7)
g. m4  3 x and m5  (180  3x)
h. m3  y and m7  (180  y)
m
1 2
4 3
6
5
8
n
7
NUCC| Secondary II Math 13
Unit 2.2
9. If lines m and n are parallel, find the value of x and y.
(3x + 12)
(6y + 3)
(7x - 48)
10. Identify which, if any, of the lines in the picture are parallel with the following information.
a
b
9 10
12 11
1 2
4 3
13 14
16 15
5 6
7 8
a.
b.
c.
d.
e.
f.
g.
m
n
1  11
9  15
6  12
m2  m9  180
5  14
7  13
m3  m6  180
11. Identify which, if any, of the lines in the picture are parallel with the following information.
E
F
G
D
A
B
C
a.
b.
c.
d.
AEF  BFG
EAB  DBC
EFB  CBF
mGFD  mCBD  180
NUCC| Secondary II Math 14
Unit 2.2
Solutions
1.
2.
3.
4.
Yes
Sometimes (at 90 degrees)
Never
a. corresponding
b. consecutive interior
c. alternate interior
5. x = 15, y = 8
6.
m1  113
a.
m1  130
m2  67
m2  50
m3  113
m3  130
m4  67
m5  113
b.
m4  50
m5  130
m6  67
m6  50
m7  113
m7  130
m8  67
m8  50
7. 37
8. a. yes
b. no
c. yes
d. yes
e. no
f. yes
g. no
h. yes
9. x = 15, y = 20
10. a.
b.
c.
d
e
f
g.
a,b
m,n
none
a,b
none
none
m,n
11. a.
b.
c.
d.
AE and BF
AE and BF
AB and EF
AB and EF
NUCC| Secondary II Math 15
Unit 2.3
2.3 RELATIONSHIPS BETWEEN
PERPENDICULAR LINES
Teacher Notes
Time Frame:
Materials Needed: Classroom set of rulers or straight edges.
Purpose: Get students to identify the geometric vocabulary and visualize the relationships
between perpendicular lines and the distances between them. Justify the basic theorems
associated with perpendicular bisectors and use the theorems to solve mathematical problems
involving algebra.
Vocabulary:
Segment: A measurable length of a piece of a line.
Perpendicular: Two lines that intersect at 90 degree angles.
Bisector: A line or point that cuts the identified item (the line segment) in half.
Skills:
Distance Formula d  ( x2  x1 )2  ( y2  y1 )2
Pythagorean Theorem a 2  b 2  c 2
Launch (Whole Class): Start with a look at line symmetry of a few letters; W, D, and X. When
comparing the points to their distance from the line of symmetry, what do you notice? How does
point A compare to point B on the first graph? How does point X compare to point Y on the first
graph? How does point C compare to point D on the second graph? How does point E compare
to point F or point G? Look at the first graph again. Are points A and X related? Why not?
A
W
X
B
Y
C
D
E
G
X
F
H
D
Once you draw the lines of symmetry, you can draw some comparisons about comparable points
on the letters. This in turn will lead to an understanding that a segment cut by a perpendicular
line is symmetric.
NUCC| Secondary II Math 16
Unit 2.3
Look at the example of a suspension bridge and discuss the bridge and its symmetry. Compare
how points A and B are not only equidistant from the vertical support, but that they are also the
same height above the bridge.
A
B
Explore (Individual, small group or pairs): Have students draw a pair of perpendicular lines on
their papers. Measure out one inch from the intersection on one of the lines and label these
points X and Y. On the other line label points A, B and C at different lengths and measure the
following lengths: AX compared to AY, BX compared to BY, and CX compared to CY.
X
Y
A
B
C
Students should discover that AX = AY, BX = BY and CX = CY. Have them justify this
property for any other point on line AB.
Discuss (Whole Class or Group): Review the student’s justifications and determine an accurate
method to prove the theorem. Review the distance formula or Pythagorean Theorem as one
method to prove the theorem. For example, they should find the distance between the points,
(-4, 8) and (2, 16). Then find the distance between the points (-a, 0) and (0, -c) compared to (a,
0) and (0, -c).
(a,0)
(-a,0)
(0,-c)
NUCC| Secondary II Math 17
Unit 2.3
Have students review solving some simple systems of linear equations. Some examples would
y  2x  4
y  4x 1
be:
and
1
1
y
x3
y
x2
2
4
Use this solving method to find the distance between the two lines: y = 3x – 6 and y = 3x +2.
Now try the distance between the lines
2x  3y  9
2 x  3 y  6
NUCC| Secondary II Math 18
Unit 2.3
Mathematics Content
Cluster Title: Prove geometric theorems.
Standard G.CO.9 Prove theorems about lines and angles. (Theorems include: vertical angels are
congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.)
Concepts and Skills to Master
Prove and use theorems about lines and angles, including but not limited to:
 Vertical angles are congruent.
 When parallel lines are cut by a transversal congruent angle pairs are created.
 When parallel lines are cut by transversal supplementary angle pairs are created.
 Points on the perpendicular bisector of a line segment are equidistant from the segment’s
endpoints.
Critical Background Knowledge

Know properties of supplementary, complementary, vertical, and adjacent angles (7.G.5).
Academic Vocabulary
proof, vertical angles, parallel lines, transversal, alternate interior angles, perpendicular bisector,
equidistant
Suggested Instructional Strategies



Use multiple formats to write proofs; narrative paragraphs, flow diagrams, two-column
format, and diagrams without words.
Focus on the validity of the underlying reasoning while writing proofs.
Use dynamic geometry software to explore angle relationships.

Connect angle relationships to the creation of tessellation patterns.
Skills Based Task:


Are students able to draw to angles that
are adjacent, but not congruent?
Are students able to draw two angles that
are supplementary, but that do not form a
linear pair?
Problem Task:
Find as many angle
relationships as possible in
this pattern.
Some Useful Websites:
http://www.khanacademy.org/test-prep/iit-jee/v/distance-between-planes
http://www.khanacademy.org/test-prep/iit-jee/v/point-distance-to-plane
http://www.khanacademy.org/math/algebra/systems-of-eq-andineq/e/distance_between_point_and_line
http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/distance-formula
NUCC| Secondary II Math 19
Unit 2.3
A Hiker in Distress
A Develop Understanding Task 2
A hiker has gotten lost and is need of rescue. He knows that he is somewhere on the map below.
He uses his cell phone to make the call and it is received by 4 cell phone towers noted as towers
A, B, C, and D. If the phone companies towers relate that the lost hiker is the same distance from
tower A as they are from tower B and that they are also the same distance from tower C as they
are from tower D, find the coordinates of the lost hikers. The idea is to draw two lines, one that is
equidistant from points A and B, and another that is equidistant from points C and D. The
coordinate of the lost hiker is approximately (-2, -2).
8
A(-2, 7)
6
C(1, 6)
4
2
-15
-10
-5
5
10
15
-2
-4
D(6, -5)
B(-10, -6)
-6
-8
NUCC| Secondary II Math 20
Unit 2.3
Ready, Set, Go!
Ready
1. Is point C on the perpendicular bisector of line AB?
5
C(1, 5)
4
A(-6, 4)
3
2
1
-10
-8
-6
-4
-2
2
4
6
8
10
-1
B(2, -2)
-2
-3
-4
-5
2. Draw the segment that would indicate the distance in the following pictures.
a. W to YZ
b. P to QR
c. A to FE
E to BC
B
A
P
D
X
W
R
Y
Z
C
F
E
Q
3. Find the distance between the points.
a. (5, 4) and (-2, 7)
b. (-3, -6) and (0, 5)
c. (-2, 5) and (a, b)
Set
4 Is point C on the perpendicular bisector of AB? Why or why not?
C(-1, 5)
B(6, 5)
A(-2, -3)
(The answer is no.) So, what value of C would be on the perpendicular bisector?
NUCC| Secondary II Math 21
Unit 2.3
Go!
5. If AE = EB = 5cm, ED = 12 cm, and BC = 8 cm, find AC and AD.
C
A
E
B
D
6. Draw the segment that would indicate the distance in the following pictures.
a. B to FE and A to BC
B
b. I to GH and J to IH
H
C
A
G
D
I
E
F
J
7. What is the distance between the pairs of parallel lines?
a. y = -2 and y = 6
b. y = 2x -2 and y = 2x +7
c. y = -2x +1 and y = -2x +7
8. Prove that point A is on the perpendicular bisector of XY.
8
A (5, 7)
6
X (-1, 4)
4
2
-15
-10
-5
Y (8,1)
5
10
15
-2
-4
-6
-8
NUCC| Secondary II Math 22
Unit 2.3
9. Describe how you would or would not find the distance between two lines that are not
parallel.
10. Describe how you would find the distance in three dimensional space between a plane and a
point not on the plane.
11. The distance between a line and a plane can be found. Is this statement always, sometimes, or
never true? Why?
12. Find the distance between the two parallel lines AB and CD, given; AB = 40 ft and AD = 41
ft and AB  BD .
A
B
C
D
13. Given the following graph, how would you justify that the two lines are not parallel?
NUCC| Secondary II Math 23
Unit 2.3
Solutions
2.
Yes
W perpendicular to YZ, etc.
3.
a.
1.
b.
58
130
c.
(a  2) 2  (b  5) 2
4.
5.
6.
7.
65  49 , (-2, 5)
No,
AC = 8, AD = 13
B perpendicular to FE, etc.
a.
8
b.
4.2
c.
2.7
8.
9.
10.
11.
12.
13.
Show that AX = 3 5 and AY = 3 5
Impossible
Find the segment through the point perpendicular to the plane.
Sometimes, if the line is perpendicular to the plane.
9
NUCC| Secondary II Math 24
Unit 2.3
2.4 INTERIOR ANGLES OF A TRIANGLE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Facilitate the discovery that the sum of the three interior angles is always 180 degrees.
Students will prove this in three different ways: 1) Graphically by lining up the three angles next
to each other to see that they form a straight angle. 2) Inductively by drawing various triangles
and measuring the sums of the angles. 3) Deductively by comparing interior/exterior angles
along with using alternate interior angles.
Vocabulary:
Vertex: The point where the two sides of an angle meet
Transversal: A line that intersects two or more different lines.
Launch (Whole Class): The instructor should have rulers and scissors (optional) ready for the
class. The class could possibly make predictions here about what they think will happen when
you add the measures of the three interior angles of a triangle together.
Explore (Individual, small group or pairs): Items 1-8 of this activity could be completed as
individuals, pairs, or small groups. The instructor should listen/watch what different
individuals/groups are doing and guide them as needed.
Discuss (Whole Class or Group): Between items 8 and 9 on the activity might be a good place
to stop and have students’ share what they have discovered so far. Hopefully they have been
able to discover that the sum of the three interior angles of a triangle is 180 degrees. Discuss
things like rounding error with protractors, why the sum couldn’t be bigger or smaller than 180
degrees etc.
Item 9 in the activity could be completed as a large class group, or back in pairs/small groups.
The students should be able to recall what they have learned about alternate interior angles and
figure out the following:
ABC  CAY
BCA  BAX
The students can then figure out that XAB  BAC  YAC form a straight angle (180
degrees), leading them to prove that the sum of the measures of the three interior angles of a
triangle must always add to 180 degrees.
NUCC| Secondary II Math 25
Unit 2.3
Mathematics Content
Cluster Title: Prove geometric theorems.
Standard G.CO.10 Prove theorems about triangles. (Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.)
Concepts and Skills to Master
Prove and use theorems about triangles, including but not limited to:
 Prove that the sum of the interior angles of a triangles = 180.
 Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles
of a triangle are congruent, the triangle is isosceles.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length.
 Prove the medians of a triangle meet at a point.
Critical Background Knowledge

Prove theorems about lines and angles (II.5.G.CO.9)
Academic Vocabulary
Interior/exterior angles of a triangle, supplementary angles, linear pairs, isosceles, base, legs,
base angles, vertex angles, midpoint, median of a triangle, auxiliary line
Suggested Instructional Strategies




Use paper folding to demonstrate relationships in triangles.
Use similar triangles or dilations to show that the mid-segment is parallel and half the
length of the third side of a triangle.
Use dynamic geometry software to explore relationships in triangles.
Write proofs in a variety of formats.
Skills Based Task:
 Prove that the base angles of an isosceles
triangle are congruent.
Problem Task:
Write a paragraph explaining why the segment
joining midpoints of two sides of a triangle is
parallel to the third side.
Some Useful Websites:
http://www.khanacademy.org/math/geometry/triangles/v/proof---sum-of-measures-ofangles-in-a-triangle-are-180
http://www.khanacademy.org/math/geometry/triangles/v/triangle-angle-example-1
http://www.khanacademy.org/math/geometry/triangles/v/triangle-angle-example-2
http://www.freemathhelp.com/feliz-angles-triangle.html
NUCC| Secondary II Math 26
Unit 2.3
Interior Angles of a Triangle
A Develop Understanding Task 3
1. Take a piece of graph paper. Draw any type of triangle on it. Make sure it looks at least a
little different from the example below:
2. Cut or tear each angle off of your triangle.
3. Take the three angles that you cut off of your triangle and line them up so that the vertices
are touching each other. Write a paragraph explaining what you notice.
NUCC| Secondary II Math 27
Unit 2.3
4. On the grids below, draw two more triangles that look different than the triangle you used
before. Label the angles A, B, and C on each one.
Triangle 1
Triangle 2
5. Using a protractor, measure the angles of each of the triangles that you drew on #4. Write
your results below:
Triangle 1: Angle A: ____________, Angle B: _______________,
Angle C: ____________
Triangle 2: Angle A: ____________, Angle B: _______________,
Angle C: ____________
6. What do you notice when you add the three angles of each of your triangles together? Write a
paragraph to describe if or how this relates to what you wrote on #3.
7. Check out the triangles that two other people/groups have drawn. Are their results similar to
yours? List any similarities or differences below:
NUCC| Secondary II Math 28
Unit 2.3
8. Take all the data that you’ve collected above and write a statement about what you think
happens when you add the three angles of a triangle together:
9. Let’s see if you can prove generally the statement you just wrote on #8. Below, are two
parallel lines (XY and BC) with a couple of transversals running through them (AB and AC).
Label some of the alternate interior angles. Individually, or as a class, try and prove what you
just wrote on #8.
Thoughts/Insights:
NUCC| Secondary II Math 29
Unit 2.5
Ready, Set, Go!
Ready
1. Find the missing angle:
2. Find the missing angle:
3. Is the following really a triangle? Explain why or why not.
4. Solve for x:
x  x  3  3x  2  180
Set
1. Find the measure of the missing angle.
2. Solve for x:
3. Find the measure of angle A
4. Find the measure of the missing angle:
NUCC| Secondary II Math 30
Unit 2.5
Go!
In exercises 1 and 2, solve for x:
1.
2.
In exercises 3 and 4, find the measure of angle A:
3.
4.
In exercises 5 and 6, find the measure of the missing angle:
5.
6.
NUCC| Secondary II Math 31
Unit 2.5
Solutions
1.
2.
3.
4.
5.
6.
5
4
45°
24°
75°
39°
NUCC| Secondary II Math 32
Unit 2.5
2.5 ISOSCELES TRIANGLE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: To understand and be able to justify that the measures of the two base angles in an
isosceles triangle are always equal. Students will justify this inductively through the use of
examples and then deductively at the end of the activity.
Vocabulary:
Isosceles Triangle: A triangle with at least two sides that are congruent.
Launch (Whole Class): Before starting the task, it could be helpful to review the concept of an
isosceles triangle. Make sure that the students understand that an isosceles triangle has two
congruent sides. The instructor could also check to see how well the students remember SSS
congruency. (It is normally used to prove that the two base angles in an isosceles triangle are
congruent.)
Explore (Individual, small group or pairs): Students should have access to a blank piece of
paper, a protractor, and a pair of scissors for this activity. Items 1-6 of the activity can be
completed as individuals, small groups, or pairs. Students will compare their findings with other
groups/individuals on items 3 and 5. During this portion of the activity, the instructor could
circulate around the room to check for understanding and guide the students as needed.
Discuss (Whole Class or Group): After the students have completed item 6, the instructor could
have the class have a group discussion for the students to share their findings. Hopefully they
have concluded that the base angles of an isosceles triangle are congruent. Proving/justifying that
the base angles are congruent on item 7 could then be completed as individuals/groups, or as a
whole class.
NUCC| Secondary II Math 33
Unit 2.5
Mathematics Content
Cluster Title: Prove geometric theorems.
Standard G.CO.10 Prove theorems about triangles. (Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.)
Concepts and Skills to Master
Prove and use theorems about triangles, including but not limited to:
 Prove that the sum of the interior angles of a triangles = 180.
 Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles
of a triangle are congruent, the triangle is isosceles.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length.
 Prove the medians of a triangle meet at a point.
Critical Background Knowledge

Prove theorems about lines and angles (II.5.G.CO.9)
Academic Vocabulary
Interior/exterior angles of a triangle, supplementary angles, linear pairs, isosceles, base, legs,
base angles, vertex angles, midpoint, median of a triangle, auxiliary line
Suggested Instructional Strategies




Use paper folding to demonstrate relationships in triangles.
Use similar triangles or dilations to show that the mid-segment is parallel and half the
length of the third side of a triangle.
Use dynamic geometry software to explore relationships in triangles.
Write proofs in a variety of formats.
Skills Based Task:
 Prove that the base angles of an isosceles
triangle are congruent.
Problem Task:
Write a paragraph explaining why the segment
joining midpoints of two sides of a triangle is
parallel to the third side.
Some Useful Websites:
http://www.khanacademy.org/math/geometry/triangles/v/congruent-legs-and-base-anglesof-isosceles-triangles
http://www.khanacademy.org/math/geometry/triangles/v/equilateral-and-isoscelesexample-problems
http://www.khanacademy.org/math/geometry/triangles/v/another-isosceles-exampleproblem
http://www.mathopenref.com/isosceles.html
NUCC| Secondary II Math 34
Unit 2.5
Isosceles Triangles
A Develop Understanding Task 4
1. On a piece of blank paper, with the help of a ruler, draw a large isosceles triangle. Make sure
it looks at least a little bit different than the one below. Cut the triangle out and fold it as
shown.
2. After you fold the triangle in half, what do you notice about the two halves?
____________________
___________________________________________________________________________
______
3. Compare your folded triangle with another individual/group’s triangle. Talk about any
similarities and differences you see.
4. Below, draw two more isosceles triangles. Measure the base angles (we called them A and C
above) with a protractor. Label those angles with their measures.
Triangle 1
Triangle 2
NUCC| Secondary II Math 35
Unit 2.5
5. Compare your results from #4 with another individual/group. Discuss any similarities and/or
differences that you notice.
6. What do you think will always be true about the base angles of any isosceles triangle?
___________________________________________________________________________
___________________________________________________________________________
7. Try to prove what you wrote in #6 generally (for any isosceles triangle). Below is a generic
isosceles triangle. Point x is the midpoint of line segment AC. In groups, or as an entire class,
try to prove that the measure of angle A will always equal the measure of angle C. (Hint:
Remember SSS?)
Thoughts/Insights:
NUCC| Secondary II Math 36
Unit 2.5
Ready, Set, Go!
Ready
Refer to the figure on the right to answer the following.
1. If AB  AD , name two congruent angles.
2.
3.
4.
5.
6.
If
If
If
If
If
GA  HA , name two congruent angles.
AF  AI , name two congruent angles.
ACD  ADC , name two congruent sides.
AHI  AIH , name two congruent sides.
DBA  BDA , name two congruent sides.
Set
Problems 1-3 solve for x:
1.
2.
3.
4. Given that triangle QRT is isosceles and
S is the midpoint of segment RT, explain
why angles RSQ and TSQ would have to
be 90 degrees.
NUCC| Secondary II Math 37
Unit 2.5
Go!
Problems 1-7 solve for x:
1.
2.
3.
4.
5.
6.
NUCC| Secondary II Math 38
Unit 2.5
7.
8. If triangle LMN is an isosceles triangle
and the measure of angle L is 60
degrees, explain why all three angles
must be congruent.
9. If the measure of angle CDE is 28 degrees, find each of the remaining angles:
10. Solve for x (Remember what the sum of the three angles of a triangle is.):
NUCC| Secondary II Math 39
Unit 2.5
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
x = 60°
x = 45°
x=6
x = -9
x = 96°
x = 84°
x = -6
Explanations will differ slightly.
mDCE  28
mDEC  124
9. mECF  68
mCEF  56
mCFE  56
10. x = 198/13
NUCC| Secondary II Math 40
Unit 2.5
2.6 MIDSEGMENT OF A TRIANGLE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Facilitate the discovery that a midsegment of a triangle is half as long as the base of a
triangle and parallel to the base. Students will try to see this by graphing triangles and measuring
the midsegment compared to the base. At the end of the activity they will try to prove it
generally.
Related Standards: G.CO.10 Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
Launch (Whole Class): Students should have a ruler to measure the lines of the triangles in item
#1. Alternatively, the students could find the midpoints using the midpoint formula. Time could
be taken here to review the midpoint formula.
Explore (Individual, small group or pairs): Students can complete items 1-4 as individuals
or small groups. The instructor could wander the room and get a feel for how the students
are progressing, and give guidance as needed.
Discuss (Whole Class or Group): After students have completed item 4, the instructor could
bring everyone together and have a discussion as a large group on what conjectures the students
have come up with. The instructor can give guidance and correction as needed here while also
having the students share what they have found.
NUCC| Secondary II Math 41
Unit 2.5
Mathematics Content
Cluster Title: Prove Geometric Theorems
Standard G.CO.10 Prove theorems about triangles. (Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.)
Concepts and Skills to Master
Prove and use theorems about triangles, including but not limited to:
 Prove that the sum of the interior angles of a triangles = 180.
 Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles
of a triangle are congruent, the triangle is isosceles.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length.
 Prove the medians of a triangle meet at a point.
Critical Background Knowledge
 Prove theorems about lines and angles (II.5.G.CO.9)
Academic Vocabulary
Triangle Midsegment: A segment with endpoints that are the midpoints of two sides of a
triangle.
Suggested Instructional Strategies
Skills:
 Use paper folding to demonstrate
relationships in triangles.
 Use similar triangles or dilations to
show that the mid-segment is parallel
and half the length of the third side of a
triangle.
 Use dynamic geometry software to
explore relationships in triangles.
 Write proofs in a variety of formats.
Some Useful Websites:
http://www.mathopenref.com/trianglemidsegment.html
http://www.geom.uiuc.edu/~dwiggins/conj20.html
NUCC| Secondary II Math 42
Unit 2.5
Midsegment of a Triangle
A Develop Understanding Task 5
1. On the first graph below, look at triangle ABC. Find and mark the midpoint of side AC. Find
and mark the midpoint of side BC. Connect the two midpoints that you just found. Call this
line segment XY. On the second graph below, draw another triangle that looks different from
the first one. Complete the same steps as you did on the first graph.
2. In each triangle, compare line segment XY with line segment AB. Write what you notice
below:
3. Compare your results from #2 with another individual, group, or with the whole class. Write
the similarities and differences below:
4. The line segments that we called XY above are called midlines or midsegments of a triangle.
Write a conjecture below about what you think will always be true when you compare the
midline of a triangle to its base.
NUCC| Secondary II Math 43
Unit 2.5
5. As individuals, groups, or as a class. Try to prove that in any triangle, your conjecture from
above will hold true.
Thoughts/Insights:
NUCC| Secondary II Math 44
Unit 2.6
Ready, Set, Go!
Ready
In exercises 1 and 2, explain why line segment DE would not be the midsegment of triangle
ABC, then redraw each picture so DE would be the midsegment of triangle ABC.
1.
2.
Set
1. Given that line segment XY is a midsegment of triangle PQR, solve for x, y and z.
2. Given that line segment AB is a midsegment of the triangle below, find all of the missing
interior angles:
NUCC| Secondary II Math 45
Unit 2.6
Go!
1. State two reasons why line segment LM is not a midsegment of the triangle below, then
redraw the triangle so that LM is a midsegment.
Exercises 2 and 3: Given that line segment XY is a midsegment of triangle PQR, solve for x, y
and z.
2.
3.
In exercises 4 and 5, given that line segment AB is a midsegment of the triangle, find the missing
interior angles.
4.
5.
NUCC| Secondary II Math 46
Unit 2.6
Solutions
1. The midsegment does not bisect the sides of the triangle and it is not parallel with the third
side.
2. x = 8, y = 2, z = 6
3. x = 3, y = 4, z = 1
4.
5.
NUCC| Secondary II Math 47
Unit 2.7
2.7 CENTROID OF A TRIANGLE
Teacher Notes
Time Frame:
Materials Needed:
Purpose: To facilitate the discovery that the centroid of a triangle is always two thirds of the
distance from a vertex to the midpoint of the opposite side of a triangle, and to prove that the
medians of a triangle are always concurrent.
Launch (Whole Class): Discuss the definition of the median of a triangle. The students will
need rulers, or straightedges and compasses. You could have students construct the medians
using some of the construction techniques that they learned in Secondary Mathematics I, or you
could have them measure where the midpoint of each side of the triangle is.
Explore (Individual, small group or pairs): Students could complete items 1-6 as individuals, or
small groups. The instructor should listen for discoveries and the chance to help students who are
having trouble.
Discuss (Whole Class or Group): After students have completed item 6, the instructor should
call the students back as a whole class and discuss everyone’s findings. The instructor could then
help the students as much or as little as needed on the general proofs in items 7 and 8. Different
instructors may have different styles of proofs that they would like to teach the students, so the
items are left very open ended.
NUCC| Secondary II Math 48
Unit 2.7
Mathematics Content
Cluster Title: Prove Geometric Theorems
Standard G.CO.10 Prove theorems about triangles. (Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment
joining midpoints of two sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.)
Concepts and Skills to Master
Prove and use theorems about triangles, including but not limited to:
 Prove that the sum of the interior angles of a triangles = 180.
 Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles
of a triangle are congruent, the triangle is isosceles.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side
and half the length.
 Prove the medians of a triangle meet at a point.
Critical Background Knowledge
 Prove theorems about lines and angles (II.5.G.CO.9)
Academic Vocabulary
Median: A line segment whose endpoints are a vertex of a triangle and the midpoint of the side
opposite the vertex.
Concurrent Lines: Three or more lines that meet at a single point.
Suggested Instructional Strategies
Skills:
 Use paper folding to demonstrate
relationships in triangles.
 Use similar triangles or dilations to
show that the mid-segment is parallel
and half the length of the third side of a
triangle.
 Use dynamic geometry software to
explore relationships in triangles.
 Write proofs in a variety of formats.
Some Useful Websites:
http://www.khanacademy.org/math/geometry/triangles/v/triangle-medians-and-centroids
(proved in 3-D)
http://www.khanacademy.org/math/geometry/triangles/v/triangle-medians-and-centroids-2d-proof
(proved with coordinate geometry)
http://demonstrations.wolfram.com/TheMediansOfATriangleAreConcurrentAVisualProof/
(need to download the player)
http://www.cut-the-knot.org/triangle/medians.shtml
NUCC| Secondary II Math 49
Unit 2.7
Medians of a Triangle
A Develop Understanding Task 6
1. Using triangle ABC below, construct the three medians of the triangle. (A median is a line
segment that goes from the midpoint of one side of the triangle to the opposite vertex of the
triangle)
2. If you constructed the medians correctly, they should all meet at a single point. The point
where the three medians meet is called the centroid of the triangle. Call the centroid of your
triangle above G.
3. Look at one of your medians again. Measure the length from the midpoint of the line to the
centroid, then measure from the centroid to the vertex. (See below) Compare your two
measurements. Write below what you notice when you compare these two measurements.
NUCC| Secondary II Math 50
Unit 2.7
4. Repeat the process from #3 with the other two medians. Do you notice the same thing with
the other two medians?
5. Write down this conjecture below:
6. Compare your conjecture with another individual/group’s conjecture. Discuss your
measurements and findings with them.
7. See if you can prove your conjecture generally.
8. Do you think that the three medians will always meet at one point? Try to prove that they
will always meet at the same point.
NUCC| Secondary II Math 51
Unit 2.7
Ready, Set, Go!
Ready
1. Explain why the “medians” of the triangle below are not concurrent (why don’t they all meet
at one point)? Draw the triangle again with the correct medians. Are the medians in your new
triangle concurrent?
Set
For 1 and 2, the centroid of the triangle is point C. Solve for each variable. (Triangles are not to
scale)
1.
2.
NUCC| Secondary II Math 52
Unit 2.7
Go!
1. Explain why the “medians” of the triangle below are not concurrent (why don’t they all meet at one
point)? Draw the triangle again with the correct medians. Are the medians in your new triangle
concurrent?
For 2 and 3, the centroid of the triangle is point C. Solve for each variable. (Triangles are not to
scale)
2.
3.
NUCC| Secondary II Math 53
Unit 2.7
Solutions
1.
2.
3.
Answers/drawings will vary
x = 4, y = 5.8, z = 0.575
a = 10, b = 5, c = 4.5
NUCC| Secondary II Math 54
Unit 2.8
2.8 PARALLELOGRAMS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Have students identify the geometric vocabulary and visualize the relationships in
parallelograms. Justify the basic theorems associated with parallelograms such as opposite
angles are congruent and use the theorems to solve mathematical problems involving algebra.
Required Materials: Students will need graph paper to draw or straight edges to model
parallelograms with to construct the concepts of parallelograms.
Launch (Whole Class): Have students work with and model different representations of heights
associated with the parallelograms involved in the basketball support. The properties that you
are looking for are: Opposite sides are always congruent, opposite angle are always congruent,
adjacent angles are always supplementary, and that the opposite sides are always parallel.
Another important property is that a rectangle is a special case of a parallelogram. If your
students see it, you could also introduce the concept of limits here. If the basketball hoop could
adjust forever, as the lower right angle approaches 0 degrees, what is happening to the upper
right angle?
Have students adjust their models and justify to themselves that the above mentioned properties
are true and go over the definitions listed in the Mathematics Content page.
b
a
d
ma  mb  180
c
etc.
Explore: Given WXYZ is a parallelogram, prove that W  Y and X  Z
X
W
Z
Y
Discuss: The various ideas for proofs that the students found in the explore section. Common
proofs include the ideas of alternate interior angles are congruent since you have parallel lines
that have been cut by a transversal.
Practice a couple of problems using either the distance formula or Pythagorean Theorem.
(3,7) (5,1)
(2,9) (7, 4)
(4,6) (a, b)
NUCC| Secondary II Math 55
Unit 2.8
Practice a couple of problems using the slope formula.
(4, 2) (5,9)
(7, 1) (4, 3)
(c, d ) ( 4,9)
Use some algebraic skills to solve for variables.
x+23
A
F
B
(7x+11)
6x
G
2y-1
y+7
D
5x-9
C
(5y-12)
E
(10z+8)
H
NUCC| Secondary II Math 56
Unit 2.8
Mathematics Content
Cluster Title: Prove Geometric Theorems
Standard G.CO.11 Prove theorems about parallelograms. (Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.)
Concepts and Skills to Master
Prove and use theorems about parallelograms including, but not limited to:
 Opposite sides of a parallelogram are congruent.
 Opposite angles of a parallelogram are congruent.
 The diagonals of a parallelogram bisect each other.
 Rectangles are parallelograms with congruent diagonals.
Critical Background Knowledge
 Know the definition and properties of parallelograms
Academic Vocabulary
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Opposite angles: The angles that are across from each other in the quadrilateral.
Opposite sides: The sides that are across from each other in the quadrilateral.
Bisect: To be cut in half.
Quadrilateral: A four sided polygon.
Suggested Instructional Strategies
 Use geometry software to reflect a
triangle across one of its sides.
 Show that any two intersecting
segments that bisect each other create
the diagonals of a parallelogram.
 Use properties of a parallelograms to
find missing measures in geometric
figures.
Some Useful Websites:
Skills:
Distance Formula d  ( x2  x1 )2  ( y2  y1 )2
Pythagorean Theorem a 2  b 2  c 2
y2  y1
Slope Formula m 
x2  x1
Use algebra to compute missing angles and
sides
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof---opposite-anglesof-parallelogram-congruent
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof---opposite-sides-ofparallelogram-congruent
NUCC| Secondary II Math 57
Unit 2.8
Setting up the Basketball Hoop
A Develop Understanding Task 7
When setting up their new basketball hoop, the Johnson’s were able to adjust the height from 6.5
feet up to 10 feet at the rim. The first example has the rim at 8 feet, the second example has the
rim at 6.5 feet, and the third example has the rim at 10 feet in height.
Using some graph paper or a set of four straight edges, 2 long 2 short, model each of the above
support brackets and identify some properties that you see. Be ready to share your observations
with the class.
NUCC| Secondary II Math 58
Unit 2.9
Ready, Set, Go!
Ready
1. Given mW  63 , find the measures of the other three angles.
X
W
Y
Z
2. Find the values for x and y that makes the following shape a parallelogram.
(2x+3)
(3y+42)
(7x-12)
3. Given that the following shape is a parallelogram, solve for x.
105
75
(2x-1)
Set
4. Is the following a parallelogram? Why?
(a+b, c)
(b, c)
(0,0)
(a,0)
5. Verify that the opposite sides are the same length.
(2, 5)
(-1, 2)
(6, 5)
(2, 2)
NUCC| Secondary II Math 59
Unit 2.9
Go!
6. Given that ABDC is a parallelogram, find the values of x, y, and z.
5x
A
B
33
95
(2y-5)
(y+9)
C
3z
D
27
7. Use the following sketch of parallelogram ABCD to find the measures of:
a. mA 
b.
mB 
c.
AB 
d.
AD 
B
A
4.2
123
D
C
6
8. Prove that a rectangle is a parallelogram.
9. Given that PXQR is a parallelogram and XY  YZ , prove that QRP  Z .
X
Q
P
Z
R
Y
10. Provide a counterexample to the following statement. Two parallelograms are always
congruent if all four sides of parallelogram #1 are congruent to the four sides of
parallelogram #2.
11. List three pieces of information that would guarantee that ABCD is a parallelogram.
NUCC| Secondary II Math 60
Unit 2.9
A
B
95
D
C
12. Solve for x.
F
(7x+11)
6x
G
H
E
13. The scissor lift uses parallelograms to guarantee a straight lift every time. Identify all of the
angles that are congruent to angle C. Then identify all of the angles congruent to angle EKF.
A
B
J
D
C
K
E
F
L
G
H
14. Solve for a and b.
3b-17
A
B
3a+11
a+15
D
4a+2
C
NUCC| Secondary II Math 61
Unit 2.9
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
9.
mX  117 .
mY  63 , mZ  117
x = 21, y = 49
x = 38
Yes
Use the slope formula to show opposite sides are parallel or the distance formula to show the
opposite sides are congruent.
x = 5.4, y = 16, and z = 11
a = 57, b = 123, c = 6, and d = 4.2
Opposite sides are parallel, opposite angles are congruent, or opposite sides are congruent.
Since XPRQ is a parallelogram, mX  mQRP . If XY = YZ, the triangle XYZ is an
isosceles triangle, so mX  mY . By substitution, mQRP  mZ and then QRP  Z
.
10.
10
10
3
3
10
3
3
10
11. mB  85 , mC  95 , mD  85 (or others)
12. x = 13
13. a. angles D, E, and F
b. angles AJB, CJD, CKD, ELF, GLH
NUCC| Secondary II Math 62
Unit 2.9
2.9 MORE DISCOVERIES IN PARALLELOGRAMS
Teacher Notes
Time Frame:
Materials Needed:
Purpose: Facilitate the discoveries that in parallelograms, opposite sides are congruent, opposite
angles are congruent and diagonals bisect each other. Also, to facilitate the discovery that in
rectangles, diagonals are congruent.
Launch (Whole Class): Discuss the definitions in item 1. The instructor could possible also
complete items 2 and 3 as a class as well.
Explore (Individual, small group or pairs): Allow the students to complete items 4, 5, and 6 in
groups or individually. The instructor should wander the room listening to the progress of
groups and looking for discoveries the students have made and opportunities to help.
Discuss (Whole Class or Group): Reconvene as a whole class and go through items 7, 8, and 9
together.
NUCC| Secondary II Math 63
Unit 2.9
Parallelograms
A Develop Understanding Task 8
1. Together as a class, talk about the following vocabulary words:
Quadrilateral
Parallel
Parallelogram
2. Look at parallelogram ABCD below. Using a ruler, draw a line from point A to point C. This line is
called a diagonal of the parallelogram. Could you prove that triangle ACD is congruent to triangle
ACB? Try it out.
3. Once you’ve proved that triangle ACD is congruent to triangle ACB, you can come up with a couple
of conjectures about parallelograms in general. Fill in the blanks below:
*Opposite sides of a parallelogram are ___________
*Opposite angles in a parallelogram are __________
NUCC| Secondary II Math 64
Unit 2.9
4. Again, look at parallelogram ABCD below. Both diagonals have been drawn in for you. The point
where the two diagonals intersect is point E. Compare lengths AE and EC. Also compare lengths DE
and EB.
How long is segment AE? ______
How long is segment EC? ______
How long is segment DE? ______
How long is segment EB? ______
5. Write down a conjecture about the diagonals of a parallelogram.
6. Compare your measurements from item 4 and your conjecture from item 5 with another
group/individual’s.
7. Can you prove your conjecture generally? As a class or in groups, try it out.
8. A rectangle is a special kind of parallelogram where each angle is a right angle. Look at rectangle
WXYZ below. Measure the whole diagonals (WY and XZ). What do you notice?
NUCC| Secondary II Math 65
Unit 2.9
9. Talk about what you noticed as a class and try to prove it generally below:
NUCC| Secondary II Math 66
Unit 2.9
Mathematics Content
Cluster Title: Prove Geometric Theorems
Standard G.CO.11 Prove theorems about parallelograms. (Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.)
Concepts and Skills to Master
Prove and use theorems about parallelograms including, but not limited to:
 Opposite sides of a parallelogram are congruent.
 Opposite angles of a parallelogram are congruent.
 The diagonals of a parallelogram bisect each other.
 Rectangles are parallelograms with congruent diagonals.
Critical Background Knowledge
 Know the definition and properties of parallelograms
Academic Vocabulary
Quadrilateral: A polygon with four sides
Parallel: Lines that never intersect
Parallelogram: A quadrilateral with parallel opposite sides
Diagonal (of a quadrilateral): A line that connects opposite vertices
Rectangle: A parallelogram with four right angles
Suggested Instructional Strategies
Skills:
 Use geometry software to reflect a
Distance Formula d  ( x2  x1 )2  ( y2  y1 )2
triangle across one of its sides.
Pythagorean Theorem a 2  b 2  c 2
 Show that any two intersecting
y2  y1
segments that bisect each other create
Slope Formula m 
x2  x1
the diagonals of a parallelogram.
 Use properties of a parallelograms to
Use algebra to compute missing angles and
find missing measures in geometric
sides
figures.
Some Useful Websites:
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-angles-of-parallelogram-congruent
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-sides-of-parallelogram-congruent
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--diagonals-of-a-parallelogram-bisect-each-other
http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/
NUCC| Secondary II Math 67
Unit 2.9
Ready, Set, Go!
Ready
Are the following shapes necessarily parallelograms? Why or why not?
1.
2.
Set
In items 1-5, all shapes are parallelograms
1. Find the missing angle.
2. Solve for x:
3.
4. Solve for x:
5.
NUCC| Secondary II Math 68
Unit 2.9
Go!
1. Is the shape below necessarily a parallelogram? Explain why or why not.
In Items 2-11, all shapes are parallelograms
2. Find the missing angle
3. Find the missing angle
4. Solve for x:
5. Solve for x:
6.
7.
NUCC| Secondary II Math 69
Unit 2.9
8. Solve for x:
9. Solve for x:
10.
11.
NUCC| Secondary II Math 70
Unit 2.9
Solutions
1. Yes, explanations will vary
2. 47°
3. 30°
4. 5
5. 6
6. 25.8
7. 23.2
8. 2
9. 12
10. 14
11. 10
NUCC| Secondary II Math 71
Unit 2
ON-LINE RESOURCES:
2.1



2.2
2.3












2.4




2.5




2.6
2.7





2.8
Basic Angle Theorems
http://www.khanacademy.org/math/geometry/angles/v/complementary-and-supplementaryangles
http://www.khanacademy.org/math/geometry/angles/e/vertical_angles
http://www.khanacademy.org/math/gelometry/angles/e/vertical_angles_2
Parallel Lines and Transversals
http://www.khanacademy.org/math/geometry/angles/e/corresponding_angles
http://www.khanacademy.org/math/gelometry/angles/e/corresponding_angles_2
http://www.khanacademy.org/math/geometry/angles/e/alternate_interior_angles
http://www.khanacademy.org/math/geometry/angles/e/alternate_exterior_angles
http://www.khanacademy.org/math/geometry/angles/e/alternate_interior_angles_2
http://www.khanacademy.org/math/geometry/angles/e/alternate_exterior_angles_2
http://www.khanacademy.org/math/geometry/angles/e/same_side_interior_angles
http://www.khanacademy.org/math/geometry/angles/e/same_side_interior_angles_2
Relationships Between Perpendicular Lines
http://www.khanacademy.org/test-prep/iit-jee/v/distance-between-planes
http://www.khanacademy.org/test-prep/iit-jee/v/point-distance-to-plane
http://www.khanacademy.org/math/algebra/systems-of-eq-andineg/e/distance_between_point_and_line
http://www.khanacademy.org/math/algebra/ck12-algebra-1/v/distance-formula
Interior Angels of a Triangle
http://www.khanacademy.org/math/geometry/triangles/v/proof---sum-of-measures-of-angles-ina-triangle-are-180
http://www.khanacademy.org/math/geometry/triangles/v/triangle-angle-example-1
http://www.khanacademy.org/math/geometry/triangles/v/triangle-angle-example-2
http://www.freemathhelp.com/feliz-angles-triangle.html
Isosceles Triangle
http://www.khanacademy.org/math/geometry/triangles/v/congruent-legs-and-base-angles-ofisosceles-triangles
http://www.khanacademy.org/math/geometry/triangles/v/equilateral-and-isosceles-exampleproblems
http://www.khanacademy.org/math/geometry/triangles/v/another-isosceles-example-problems
http://www.mathopenref.com/isosceles.html
Midsegment of a Triangle
http://www.mathopenref.com/trianglemidsegment.html
http://www.geom.uiuc.edu/~dwiggins/conj20.html
Centroid of a Triangle
http://www.khanacademy.org/math/geometry/triangles/v/triangle-medians-and-centroids (Proved
in 3-D)
http://www.khanacademy.org/math/geometry/triangles/v/triangle-medians-and-centroids-2dproof (proved with coordinate geometry)
http://www.cut-the-knot.org/triangle/medians.shtml
Parallelograms
NUCC| Secondary II Math 72
Unit 2


2.9




http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-angles-of-parallellogram-congruent
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-sides-of-parallelogram-congruent
More Discoveries in Parallelograms
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-angles-of-parallellogram-congruent
http://www.khanacademy.org/math/geometry/polygons-quads-parallelograms/v/proof--opposite-sides-of-parallelogram-congruent
http://www.khanacademy.org/math/geometry/ploygons-quads-parallelograms/v/proof--diagonals-of-a-parallelogram-bisect-each-other
http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/
NUCC| Secondary II Math 73
Unit 2
Practice Exam
Name: _____________________________________________________
Hour: _______
1. Identify the following and know the definitions of: vertical angles, a linear pair, and complimentary
angles.
E
A
B
D
F
C
2. Identify and know the meanings of: corresponding angles, alternate interior angles, alternate exterior
angles, and consecutive interior angles.
1 5
2 6
3 7
4 8
3. Find the distance between the lines y = x – 6 and y = x + 4.
4. Solve for x.
102
134
x
5. Solve for x.
(5x-4)
(3x+6)
NUCC| Secondary II Math 74
Unit 2
6. Solve for x.
3x + 7
8x + 11
7. Know the definitions and properties associated with medians and centroids, angle bisectors and
incenters, perpendicular bisectors and circumcenters, and altitudes and orthocenters.
8. Solve for x and y.
A
B
(3y+12)
7x-15
5x+11
42
D
C
9. Describe the properties associated with parallelograms involving their sides and angles and be able
to use them algebraically to solve problems.
10. List a few points that are on the perpendicular bisector of the following line.
8
6
4
2
-15
-10
-5
5
-2
10
15
y = -x + 2
-4
-6
-8
NUCC| Secondary II Math 75
Unit 2
11. The measure of an angle’s supplement is 75 less than twice the measure of the angle. Find the
measure of the angle and its supplement.
12. Know the properties associated with parallel lines and transversals. That is which angles are
congruent, which angles are supplementary, which angles are complimentary etc.
13. Find BD.
B
9 cm
D
A
C
4 cm
14. Solve for x.
27
44
127
x
NUCC| Secondary II Math 76
Unit 2
15. Prove that angle J is congruent to angle K.
L
M
J
K
16. Prove that the measure of angle 1 will always equal the sum of the measures of angle2 and angle 3.
2
1
3
17. Find the length of segment PR.
P
S
6x-10
R
Q
8x-13
5x+20
18. If ABCH, CDGH, and DEFG are all parallelograms prove that AB = FE.
B
C
A
H
D
G
E
F
NUCC| Secondary II Math 77
Unit 2
19. Solve for x.
(3x+24)
(2x+7)
(5x-11)
20. Prove XY = YZ if AW is the angle bisector of angle BAC.
B
X
W
Y
C
A
Z
NUCC| Secondary II Math 78
Unit 2
Solutions
1. Angles AFB and DFC; AFD and DFC; EFD and DFC. There are other solutions as well.
2. Angles 1 and 3, 2 and 7, 5 and 4, 6 and 7, or other pairings.
3.
4.
5.
6.
7.
8.
9.
3 2
148
14
1.5
x = 13, y = 10
Opposite sides are congruent, opposite angles are congruent, alternate interior angles are congruent.
Etc.
10. (0, -1) (1, 0) etc.
11. angle = 85, supplement = 95.
12. Corresponding = congruent; alternate interior = congruent; alternate exterior = congruent,
consecutive interior = supplementary.
13. 65
14. 36
15. Since triangle JLM and triangle KLM are congruent by HL or SSS (with some work), the angle must
be congruent.
16. The sum of angle 1 and angle x = 180. The sum of angle x, angle 2 and angle 3 = 180. Substitute and
subtract angle x to both sides.
17. 112
18. AB = HC since ABCH is a parallelogram. HC = GD since HCDG is a parallelogram. GD = FE
since GDEF is a parallelogram. Substitute EF for GD and then substitute GD for FE in the
equations.
19. x = 16
20. Since AY = AY, XAY  ZAY , and YXA  YZA , then XAY  ZAY . Since the triangles are
congruent, their sides are congruent.
NUCC| Secondary II Math 79
Unit 2
Exam
Name: ___________________________________________
1. Which of the following is a pair of vertical angles?
a. AFB and BFC
b. AFB and CFD
c. BFC and CFD
d. AFE and BFC
e. EFD and AFC
Hour: _______
B
A
C
D
F
E
2. Identify which is a pair of alternate interior angles.
a. 1 and 10
b. 4 and 6
c. 2 and 7
d. 1 and 15
e. 10 and 15
1 2
4 3
5 6
7 8
9 10
12 11
13 14
16 15
3. Find the distance between the line y   x  2 and the point (2, 4).
a.
30
b.
32
c.
42
d.
2 14
e.
3 14
NUCC| Secondary II Math 80
Unit 2
4. Find the value of x.
a. 29
b. 36
c. 39
d. 46
e. Not enough information
41
103
x
5. Solve for x.
a. x = 7
b. x = 12
c. x = 21
d. x = 23
e. x = 26
(2x+10)
(3x-7)
6.
Solve for x.
a. x = 3.8
b. x = 4.6
c. x = 4.8
d. x = 5.2
e. x = 5.8
13.5
5x - 11
7. Identify the following bisector of the triangle.
a. centroid/medians
b. incenter/angle bisectors
c. circumcenter/perpendicular bisectors
d. orthocenter/altitudes
NUCC| Secondary II Math 81
Unit 2
8. Solve for x.
a. x = 16
b. x = 18.4
c. x = 20.6
d. x = 22.8
e. x = 23.2
75
32
(2x+8)
(3x-15)
9. In the following parallelogram, find the value of y.
1
4
1
y
3
1
y
2
3
y
4
3
y
2
y
a.
b.
c.
d.
e.
2x+5
14
13
3x+4y
10. What value of x guarantees that ABCD is a parallelogram?
a. 4.666
A
b. 5.6
B
c. 6
2x+11
d. 28
3x-17
e. 187
D
C
NUCC| Secondary II Math 82
Unit 2
11. Solve for x and y.
B
C
(3x+5)
A
(6x-2)
D
(5y+7) F
E
12. Describe the relationship between angles a and b. Describe the relationship between angles b and c.
b
a
c
d
13. If AE = EB =9 ft, ED = 40 ft, and BC = 8 cm, find AC and AD.
C
A
E
B
D
14. Solve for x and y.
x
32
44
27
y
NUCC| Secondary II Math 83
Unit 2
15. Given that triangle XYZ is isosceles and Z is the midpoint of WY, justify mXZW  90 .
X
W
Y
Z
16. Is ED always, sometimes or never the mid-segment of the following sketch? Explain why.
A
E
12
D
C
24
B
17. If point S is the orthocenter of triangle PQR, find the value of x.
P
S
Q
(8x-13)
R
NUCC| Secondary II Math 84
Unit 2
18. Given that PXQR is a parallelogram and XY  YZ , prove that QRP  Z .
X
Q
P
R
Z
Y
19. Given that ABCD and DCFA are both parallelograms, prove AB = EF.
A
B
C
D
F
E
20. Solve for x.
(13x-11)
(6x+5)
(7x+42)
NUCC| Secondary II Math 85
Unit 2
Solutions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
e
c
b
b
d
d
a
a
c
d
x
83
,
9
y
11.
12. a + b = 180,
83
5
b and c have no relationship
AD  41, AC  145
x = 104, y = 49
SSS
Sometimes; AE = EC and AD = DB or ED parallel to BC
103
17. x = 8
18. Since XPRQ is a parallelogram, mX  mQRP . If XY = YZ, the triangle XYZ is an isosceles
mX  mY . By substitution, mQRP  mZ and then QRP  Z .
triangle so
19. AB = DC since ABCD is a parallelogram. DC = EF since DCEF is a parallelogram. Substitute EF for
DC in the first equation.
20. x = 4
13.
14.
15.
16.
NUCC| Secondary II Math 86