3.1 Classifying Triangles
Name ______________________________ Period __________
Vocabulary
Triangle
• Vertex
• Interior Angles
• Exterior Angles
CLASSIFYING TRIANGLES BY ANGLES
Acute Triangle
Right Triangle
Obtuse Triangle
Equiangular Triangle
____ acute angles
____ right angle
____ obtuse angle
____ congruent angles
CLASSIFYING TRIANGLES BY SIDES
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
____ congruent sides
At least ____ congruent sides
____ congruent sides
Example 1: Classifying triangles by the angles and by the sides (Classify Angles b4 Sides!)
a.
b.
c.
12
121˚
3
75˚
85˚
d.
13
20˚
e.
5
f.
11˚
45˚
124˚
*Interior Angles Activity!!!
1st: Tear the 3 angles off the triangle
2nd: Draw a line on the page.
3rd: Put the vertices of all three together on the line. What’s the measure?
1
TRIANGLE SUM THEOREM
Equilateral Triangles
B
The sum of the measures of
the interior angles of a
triangle is ______.
Each angle is _____
C
𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = ________°
A
Example 2:
1. Find the value of x and
explain.
2. Solve for 𝑥. Explain.
𝐴
3. Find the measure of the smallest
angle. Explain.
(3x+10)º
71º
67º
8𝑥 − 2
xº
(6x – 28)º
2xº
𝑇
𝑀
4. Find the value of x.
5. Which of the following sets of angle
measures would not form a triangle?
A. 34°, 62°, 84°
B. 45°, 63°, 62°
C. 27°, 51°, 102°
• Why does Triangle Sum TH
not work here?
D. 35°, 71°, 74°
EXTERIOR ANGLE THEOREM
The measure of an exterior angle of a triangle is equal
to the sum of the measures of the two
_________________________ angles.
2
𝑚∠1= ______ + ______
3
Example 1: Explain your reasoning for each question.
1. Find the value of y.
2. Find the measure of ∠A.
2
1
Reviewing Congruent Triangles…
̅̅̅̅ ≅ 𝑇𝐴
̅̅̅̅, 𝐴𝑀
̅̅̅̅̅ ⊥ 𝐻𝑇
̅̅̅̅
Given: 𝐻𝐴
Statement
̅̅̅̅ ≅ 𝑇𝐴
̅̅̅̅, 𝐴𝑀
̅̅̅̅̅ ⊥ 𝐻𝑇
̅̅̅̅
Prove: ∠𝐻 ≅ ∠𝑇
1. 𝐻𝐴
𝐴
𝑇
𝑀
𝐻
Reason
1. Given
2.
2. Definition of ⊥
3.
3.
4. ̅̅̅̅̅
𝐴𝑀 ≅ ̅̅̅̅̅
𝐴𝑀
4.
5. ∆𝐻𝐴𝑀 ≅ ∆𝑇𝐴𝑀
5.
6.
6.
BASE ANGLES THEOREM
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ ⟹ _____ ≅____
𝐴𝐵
If two sides of a triangle are congruent, then
the angles opposite them are congruent.
The CONVERSE is also true.
∠𝐵 ≅ ∠𝐶 ⟹ ______ ≅_____
• If NO angles are congruent ⇔
Important!
sides congruent
• If 2 angles are congruent ⇔
sides congruent
• If 3 angles are congruent ⇔
sides congruent
2 and 2!
3 and 3!
Example 4: Find the angle measure.
̅̅̅̅̅ and 𝑚∠𝐺 = 63°, then
1. If ̅̅̅̅̅
𝑈𝑀 ≅ 𝐺𝑀
find the measure of ∠𝑀. Explain.
2. Solve for x and explain.
3. Solve for x and explain.
N
U
3x°
57°
O
H
(4x +
M
G
M
G
4. If 𝑚∠𝐸 = (4𝑥 + 8)°, then find the value of 𝑥. Explain.
5. Solve for x and explain.
C
E
D
F
6. Which of the following sets of angle measures would form an isosceles triangle.
A. 50°, 60°, 70°
B. 30°, 30°, 120°
C. 100°, 30°, 50°
3
D. 20°, 20°, 60°
D
3.2 Triangle Inequality
1st: Label the SHORTEST side S
2nd: Label the LONGEST side L
3rd: Using a protractor to measure each angle.
4th: Place an S on the smallest angle and L on the largest angle
What do you notice?
The largest angle is ____________ the ____________ side
The smallest angle is ____________ the ____________ side
SMALL/MEDIUM/LARGE…
Largest angle is opposite the ___________ side
L
S
M
Medium angle is opposite the ___________ side
M
S
Smallest angle is opposite the ___________ side
L
Example 1: Comparing measures
Q
1. Name the largest angle.
2. Name the shortest side.
D
53°
14
63°
10
S
R
18
E
3. List the angles in order from largest to smallest.
4. List the sides in order from shortest to longest.
𝑚∠_____ < 𝑚∠_____ < 𝑚∠_____
________ < ________ < ________
**Toothpick Activity: Will any three sized segments form a triangle? **
TRIANGLE INEQUALITY THEOREM
The sum of the lengths of any two
sides of a triangle is greater than the
length of the third side.
Shortest
Side
Medium
Side
>
Longest
Side
____ + ____ > ____
rd
Question: If two sides lengths of a triangle are 4 and 6, can the 3 side be 11?
So basically…
Given 2 SIDE lengths, you can form a triangle if the
third side meets the following criteria…
Forms a Triangle IF…
#−#< 𝑥 <#+#
#
𝑥
MIN
MAX
X
#
What are the possible lengths for 𝑥? Explain.
7
x
13
Can you think of a shortcut for this problem?
4
F
Example 3: Using the Triangle Inequality TH
1. Which three segment lengths could not form a triangle?
2. The lengths of two sides of a triangle are 16 and
8. The length of the third side is between:
A. 13, 18, 25
B. 21, 9, 12
3. In the figure below, z is a whole number. What is
the smallest possible value for z? Explain.
4. A triangle has one side of length 13 and another side of
length 22. Describe the possible lengths of the third side.
A. 3
8
z
B. 16
C. 4
z
D. 5
Review
1. Solve for 𝑥. Explain.
2. Solve for 𝑥. Explain.
𝑥
𝑥°
3. Classify the triangle by its angles and its
sides.
67°
4. Find the measure to the smallest angle. Explain.
5𝑥 + 10
24°
5
and Midsegment
TH
3.3 Constructions
DEMO
VERSION
EMO
VERSION
DEMO
VERSION
CONSTRUCTION: Equilateral Triangle
1. Place the compass needle on one endpoint and
measure the segment. Use this measurement
to draw a large arc.
A
B
2. Without changing your compass, draw the
same large arc from the other endoint
3. X marks the spot! Lastly, Draw lines
connecting the endpoints of the segment to
the intersection of the arcs.
A
B
B
A
Example 1: Using a straightedge and a compass, construct an equilateral triangle.
1.
2.
CONSTRUCTION: Regular Hexagon (a polygon with _____ congruent sides)
1. Use your compass to draw a full
circle, and label a point on your
circle A.
2. Using the same compass length,
place the needle on point A and
draw an arc that intersects the
circle.
3. From the intersection point
draw another arc that intersects
the circle using the same compass
length. Do this all the way
around the circle.
4. Draw segments connecting all
the intersection points.
Example 2: Using a straightedge and a compass, construct a regular hexagon.
6
Median -
Construction: Median of a triangle
1. Place your compass needle on
one endpoint of the segment that
you want to draw a median to
(𝑨). Open up your compass
more than 1/2 the length of the
segment and draw 2 arcs.
2. Without changing the compass
radius, place the needle on the
other endpoint (𝑩) and repeat
step
B
A
3. X marks the spots! Draw a line
connecting the 2 X’s. The point
where the line crosses the
segment is its midpoint (𝑪).
B
A
B
C
A
4. Draw a line from the midpoint
(𝑪) to the opposite vertex of the
triangle.
C
A
B
̅̅̅̅.
Example 3: Using a straightedge and a compass, construct a median to 𝑨𝑪
1.
2.
B
C
B
C
A
A
MIDSEGMENT THEOREM
A midsegment of a triangle is a segment that connects the midpoints
of two sides of a triangle.
A midsegment connecting two sides of a triangle is parallel to
the third side and is half as long.
_____ ∥ _____ and 2(______) = ______
Example 4: Solve for x.
̅̅̅̅
̅̅̅̅̅ is the midsegment of ∆𝐿𝑁𝑂. Find the length of 𝑃𝑂
1. 𝑀𝑃
̅̅̅̅
and 𝑂𝑁.
2. Circle the statements that are true about the diagram
shown below.
• ̅̅̅̅
𝑄𝑃 is the midsegment of
O
F
∆𝑀𝑂𝑁.
5
• ̅̅̅̅
𝑂𝑁 ∥ ̅̅̅̅
𝑄𝑃
Q
P
3y + 14
• 𝑀𝑁 = 2(𝑄𝑃)
M
C
N
D
• ∠𝑀 ≅ ∠𝑁
6y - 4 G
̅̅̅̅
• ̅̅̅̅̅
𝑀𝑁 ∥ 𝑄𝑃
• ∠𝑂𝑄𝑃 ≅ ∠𝑂𝑀𝑁
7
5y
E
̅̅̅ , and 𝐴𝐺
̅̅̅̅ are midsegments of ∆𝑅𝑉𝐿. Find the
3. ̅̅̅
𝐽𝐴, 𝐽𝐺
value of x. Explain.
R
J
3𝑥 + 5
4. Is ̅̅̅̅
𝐴𝑇 a midsegment? Explain.
H
4
A
4
A
T
6
V
L
G
52
6
M
P
K
̅̅̅̅ is a midsegment of ∆𝐵𝐴𝐶. Find the values of
6. 𝐷𝐸
𝑥, 𝑦, and 𝑧.
5. If 𝑚∠𝐷𝐻𝐺 = 42° and 𝑚∠𝐷𝐹𝐸 = 46° then find….
D
a. 𝑚∠𝐻𝐸𝐼 =_______°
because…
C
67°
A
𝑥°
G
H
b. 𝑚∠𝐷𝐺𝐻 =_______°
because….
D 65°
E
𝑧°
B
F
I
𝑦° E
8. 𝐺 is the midpoint of ̅̅̅̅
𝐷𝐸 and 𝐻 is the midpoint of ̅̅̅̅
𝐸𝐹 . If
𝐷𝐹 = 15𝑥 + 8 and 𝐺𝐻 = 12𝑥 − 14, solve for 𝑥.
Explain.
7. If 𝐷𝐸 = 4𝑥 + 5, 𝐸𝐹 = 2𝑥 + 7, and 𝐺𝐽 = 3𝑥 +
25, solve for 𝑥.
D
G
67°
F
24°
H
E
Review:
1. Which of the following are true statements regarding the
angles in the diagram below. Choose all that apply.
2. Which side is the longest side in the triangle below?
Note: Picture is not drawn to scale.
𝑈
𝑏
A. 𝑎 + 𝑏 + 𝑐 = 180°
B. 𝑑 = 𝑎 + 𝑐
54°
𝑎
𝑐
𝑑
62° 𝑅
C. 𝑎 = 𝑐
𝐹
D. 𝑏 = 𝑑
E. 𝑎 + 𝑏 = 𝑑
F. 𝑐 + 𝑑 = 180°
8
3.4 Intro to Quadrilaterals
What do you think makes a figure a quadrilateral? What types of quadrilaterals have you
seen in real life?
Vocabulary
Quadrilateral
1. Use a protractor to measure the angles below.
Diagonal of a Polygon
2. Draw a diagonal in the quadrilateral below. Label the angles in
one triangle, ∠1, ∠2, and ∠3. In the other triangle, label the
angles ∠4, ∠5, and ∠6.
Find…
a. 𝑚∠1 + 𝑚∠2 + 𝑚∠3 = _______°
b. . 𝑚∠4 + 𝑚∠5 + 𝑚∠6 = _______°
What is their sum?
c. 𝑚∠1 + 𝑚∠2 + 𝑚∠3 + 𝑚∠4 + 𝑚∠5 + 𝑚∠6 = _______°
INTERIOR ANGLES OF A QUADRILATERAL
D
A
The sum of the measures of the interior angles of a
quadrilateral is ______.
B
Example 1
1. Find the value of x.
C
2. For the quadrilateral shown below, what is
𝑚∠𝑥 + 𝑚 ∠𝑦?
𝒙˚
94˚
84˚
3. Which of the following sets of angles could form a quadrilateral? Choose all that apply.
A. 68°, 87°, 94°, 111°
B. 48°, 81°, 106°, 120°
C. 90°, 90°,90°,90°
D. 81°, 85°, 92°, 102°
9
𝒚˚
4. The measures of the angles of quadrilateral are (4𝑥 + 17)˚,
(3𝑥 − 5)˚, (2𝑥)˚, and (6𝑥 + 18)˚. What is the measure of the
largest angle?
4. For the quadrilateral shown below, what is
𝑚∠𝑎 + 𝑚∠𝑐?
𝒂˚
𝟏𝟎𝟎˚
𝒄˚
Properties of Parallelograms
T
A
In a parallelogram opposite
sides are _______.
T
A
In a parallelogram opposite
angles are _______.
M
M
H
_____≅ _____ and _____≅ _____
H
_____≅ _____ and _____≅ _____
In a parallelogram
y°
consecutive ∠s are _______.
T
A
In a parallelogram diagonals
x°
_________ each other.
x°
I
y°
M
_______ + _______ = ________
H
_____≅ _____ and _____≅ _____
Example 2: Find the missing values. Explain with the correct property used.
1. Label the missing sides.
2. Label the missing angles.
Property:
Properties:
1.
2.
3. Find the values of x and y.
4. Is the shape below a parallelogram? Explain why or why
not.
x°F
x
y
F
Property:
Example 3: Solve for the variable. Justify your reasoning.
1. ABCD is a parallelogram.
16
A
2. BRAT is a parallelogram.
T
D
A
4(p + 3)
2x
10
B
y+2
G
5
12
8
10
E
E
136°
C
B
10
R
6
10
D
3.
4.
T
y°
42°
S
2x°
U
R
x°
Example 4 :Proofs involving Parallelograms.
1. Given: ROPE is a parallelogram;E 𝑅𝑂 = 6
Prove: 𝑃𝐸 = 6
1.
O
Statements
Reasons
1.
P
R
E
2.
2.
3.
3.
2. Given: GENT is a parallelogram;
𝑚∠𝐸 = (3𝑥 + 17)°;
𝑚∠𝑇 = (5𝑥 − 17)°,
Prove: 𝑥 = 17
E
𝑚∠𝑇 = (5𝑥 − 17)°,
N
E
G
Statements
1. GENT is a ; 𝑚∠𝐸 = (3𝑥 + 17)°;
2.
2.
3.
3.
4.
4.
5.
5.
6.
3x°
H
W
T
6.
57°
(7x - 7)
(4x + 17)
Statements
(10𝑥 + 25)°;
1. WHAT is a ; 𝑚∠𝐻 = G
1.
𝑚∠𝐴 = (6𝑥 + 11)°
2.
3.
A
1.
x°F
T
2. Given: WHAT is a parallelogram;
𝑚∠𝐻 = (10𝑥 + 25)°;
𝑚∠𝐴 = (6𝑥 + 11)°
Prove: 𝑥 = 9
Reasons
2.
C
D
3.
4.
4.
5.
5.
6.
6.
7.
7.
11
Reasons
3.5
Properties of Rhombuses, Rectangles, and Squares
Vocabulary
Quadrilateral:
Parallelogram:
Rectangle:
Rhombus:
Square:
Example: Classify quadrilaterals (be as specific as possible).
a.
b
c.
d.
e.
f.
̅̅̅̅ . Draw segment
Reflect the triangle across 𝐴𝐵
connecting 𝐶 and 𝐶 ′ .
Rotate the ∆ in 90° increments.
Special Parallelograms
Rotate the ∆ 180° about 𝐴. Then reflect the
image across a horizontal line through 𝐴. Then
rotate 180° again.
45°
5
A
A
A
D
120°
30°
B
45°
C
Describe the triangle:
Describe the triangle:
Describe the triangle:
Name the parallelogram:
Name the parallelogram:
Name the parallelogram:
What do you notice about the lengths
of the diagonals?
What do you notice about ∠𝐶𝐵𝐴 and
∠𝐶′𝐵𝐴
What do you notice about this figure
and the first two. Are there
similarities?
What is the 𝑚∠𝐶′𝐷𝐵?
12
RECAP: What did we learn about a parallelogram?
1.
2.
3.
4.
5.
Opposite sides are ________________.
D
Opposite sides are ________________.
Opposite angles are ________________.
Consecutive angles are ________________.
Diagonals ___________ each other.
D
Rhombus Properties
In a rhombus, the diagonals are ____________.
B
In a rhombus, the diagonals __________ each angle.
B
C
C
______≅______
______≅______
______≅______
______≅______
______ ⊥ ______
A
A
D
D
Rectangle Properties
F
G
In a rectangle, the diagonals are
____≅____
____________.
D
( _____≅_____≅_____≅_____ )
E
H
Examples: Use properties of special quadrilaterals
1. Given 𝐴𝐵𝐶𝐷 is a rectangle, find the following lengths.
A
B
a) 𝐴𝐵
10
12
b) 𝐵𝐶
E
D
2. Given 𝐴𝐷𝐶𝐵 the rhombus if 𝐴𝐵 = 13, 𝐴𝐶 = 10, and
D
𝐷𝐸 = 12.
A
a) 𝐴𝐸
E
c) 𝐷𝐸
b) 𝐵𝐷
d) 𝐴𝐶
c) 𝐵𝐶
C
16
3. Find all missing values in rhombus 𝑀𝑁𝑂.
N
4. Find all missing measures in the square.
O
x°
M
𝐹
𝑆
7 2
𝐵
A
5. Classify the figure below. Explain.
5
9
b) 𝐵𝐴
𝐷
B
6. SQUR is a square. Find:
Q
S
12
a) 𝑚∠𝑅𝑆𝐸 =
E
b) 𝑚∠𝑄𝐸𝑈 =
E
5
a) 𝑆𝐴
c) 𝐵𝐷
60°
P
9
𝐴
7
z°
y°
B
D
16
R
13
C
c) 𝑚∠𝑆𝐸𝑅 =
U
C
y ° 60°
7. InMthe rectangle below,
𝑁𝑄 = 4𝑥 + 5, 𝑂𝑄 = 6𝑥 − 15,
P
𝑀𝑄 = 5𝑥 − 5, and 𝑃𝑄 = 3𝑥 + 15. Solve for x and
explain.
N
O
8. In the figure below, ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷 and ̅̅̅̅
𝐴𝐷 ∥ ̅̅̅̅
𝐶𝐵.
A
What is the value of x?
B
(3x + 19)º
Q
148º
P
M
D
C
What is 𝑚∠𝐴?
9. Use the quadrilateral to answer the question below.
10. 𝑅𝐻𝑂𝑀 is a rhombus. Find the value of 𝑥 and explain.
H
9
8
9
R
114°
8
(12𝑥 + 6)°
O
M
Explain why the quadrilateral is NOT a rectangle.
IDENTIFY WHICH SHAPE IT IS…
1. If quadrilateral ABCD has diagonals that bisect each
other, then what shape best describes ABCD?
2. Quadrilateral JIMY is a parallelogram. If adjacent
angles are congruent, what shape best describes JIMY?
3. If parallelogram BURT has diagonals that are
perpendicular, what shape best describes BURT?
4. If parallelogram MAGY has adjacent angles that are
congruent and adjacent sides that are congruent, what
shape best describes MAGY?
14
3.6 Algebraic Problems with Quadrilaterals
Draw an example of each parallelogram we have learned with diagonals. Put values in for sides and
angles to represent each quadrilateral.
Parallelogram
Rectangle
Rhombus
Square
Example 1: Identify the shape and justify your reasoning.
1. Plot the points and identify the name that best represents the
shape 𝐾𝐼𝑁𝐷.
2. Plot the points and identify the name that best represents the
shape 𝑃𝐴𝑇𝐻.
𝐾(−1, 2), 𝐼(5, 2), 𝑁(3, −2), 𝐷(−3, −2)
𝑃(−2, 2), 𝐴(−2, − 2), 𝑇(3, −2), 𝐻(3,2)
Example 2: Use properties of special quadrilaterals to answer the following.
4. Find the length of a diagonal of the rectangle.
3. If FROG is a rhombus, find m∠FOG.
R
3x – 7
F
8x – 22
(12x + 17)˚
O
G
(5x + 38)˚
5. If WXYZ is a parallelogram, find m∠W.
6. JKLM is a square. If KU = 3x + 3 and UM = 4x – 4,
find the length of JL.
15
Example 3: More practice
1. Rhombus 𝑃𝑄𝑅𝑆 has diagonals 𝑃𝑅 and 𝑄𝑆 that intersect
at T. 𝑃𝑇 = 3𝑥 – 10, 𝑇𝑅 = 𝑥 + 2. Find the value of 𝑥
and 𝑃𝑅.
2. 𝐴𝐵𝐶𝐷 is a parallelogram, if 𝐴𝑂 = 𝑥 + 4, 𝐵𝑂 = 2𝑦 −
6, 𝐶𝑂 = 3𝑥 − 4, and 𝐷𝑂 = 𝑦 + 2, solve for 𝑥 and 𝑦.
A
B
O
D
3. 𝐴𝐵𝐶𝐷 is a parallelogram. If 𝐴𝑂 = 3, 𝐵𝑂 = 4, and
𝐴𝐵 = 6, then 𝐴𝐶 + 𝐵𝐷 =
A
C
4. Rectangle 𝐶𝑈𝑅𝐷 has diagonals 𝐶𝑅 and 𝑈𝐷 that intersect
at 𝐴. 𝐷𝐴 = 4𝑥 – 7, 𝐶𝑅 = 58, find the value of 𝑥.
B
O
D
C
5. Use the quadrilateral to answer the question below.
4
5
6
4
Explain why the quadrilateral is NOT a parallelogram..
6. Given the quadrilateral is a rhombus, which of the
following are true statements regarding the rhombus?
Choose all that apply.
b i
g
h
c
a
d
A. 𝑎 = ℎ
f
B. 𝑑 = 90
e j
C. 𝑎 + 𝑐 + 𝑑 = 180°
D. 𝑎 + 𝑓 = 𝑔 + ℎ
E. 𝑎 + 𝑏 + 𝑖 + 𝑔 + ℎ + 𝑗 + 𝑒 + 𝑓 = 360°
SPIRAL REVIEW – 5 ways to Prove Triangles are Congruent
5 Ways to Prove Triangles Congruent: _________, _________, ________, ________, and ________.
Mark up the given information then determine which theorem of postulate can be used to prove the triangles are
congruent.
a. ̅̅̅̅
𝐽𝑀 ⊥ ̅̅̅
𝐼𝑌, ̅̅̅
𝐼𝑌 bisects ̅̅̅̅
𝐽𝑀
b. ̅̅̅̅
𝐸𝑉 ∥ ̅̅̅̅
𝐿𝑂; ∠𝐸 ≅ ∠𝑂
c. 𝑈 is the midpoint of ̅̅̅̅
𝑃𝐺 ; ∠𝐾 and
∠𝐻
are
right
∠s.
E
J
Y
M
V
K
L
P
O
I
16
H
U
G
3.7 Proofs with Parallelograms
RECALL – 5 ways to Prove Triangles are Congruent
Complete each proof.
Given: 𝑃𝑄𝑅𝑆 is a parallelogram
Prove: ∆𝑃𝑅𝑆 ≅ ∆𝑅𝑃𝑄
Statement
Q
P
R
S
Given: 𝐺𝐻𝐼𝐽 is a parallelogram
Prove: ∆𝐺𝐾𝐻 ≅ ∆𝐼𝐾𝐽
H
1. Given
̅̅̅̅ , ̅̅̅̅
2. ̅̅̅̅
𝑃𝑄 ≅ 𝑆𝑅
𝑃𝑆 ≅ ̅̅̅̅
𝑄𝑅
2.
3.
3. Reflexive prop.
4. ∆𝑃𝑅𝑆 ≅ ∆𝑅𝑃𝑄
4.
Statement
1. 𝐺𝐻𝐼𝐽 is a parallelogram
1. Given
̅̅̅̅ ∥ 𝐽𝐼
̅
2. 𝐺𝐻
G
Reason
1.
Reason
2.
E
(9y - 25)
K
I
J
3. ∠𝐾𝐺𝐻 ≅ ∠𝐾𝐼𝐽, ∠𝐾𝐻𝐺 ≅ ∠𝐾𝐽𝐼
3.
4.
4. In a parallelogram, opposite sides are ≅
(11x + 4)
Given: 𝐴𝐷𝐶𝐵 is a parallelogram
Prove: ∆𝐴𝐸𝐵 ≅ ∆𝐶𝐸𝐷
A
D
E
B
C
F
(8x + 22)
5.
5.
Statement
1. 𝐴𝐷𝐶𝐵 is a parallelogram
1. Given
Reason
2. ̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐶𝐸 and ̅̅̅̅
𝐵𝐸 ≅ ̅̅̅̅
𝐷𝐸
2.
3.
3. Vertical Angles Theorem
4. ∆𝐴𝐸𝐵 ≅ ∆𝐶𝐸𝐷
4.
**When to use CPCTC** Can only be used after you know (or prove) that the triangles are congruent.
50°
*To Find 𝑥 you first need
to check if ∆EFG≌∆HIJ?
𝑥°
Given: 𝑀𝐴𝐺𝑁 is a parallelogram
̅̅̅̅̅ ≅ ̅̅̅̅
Prove: 𝑀𝑁
𝐴𝐺
N
M
Statement
G
A
Ex:
Statements
1. ∆EFG≌∆HIJ
2. ∠F≌∠I
Reasons1. (SSS, ASA, etc.)
2. CPCTC
Reason
1. 𝐺𝐻𝐼𝐽 is a parallelogram
̅̅̅̅̅ ∥ ̅̅̅̅
2. 𝑀𝑁
𝐴𝐺
1. Given
2.
3. ∠_______ ≅ ∠________
3.
4.
4. In a parallelogram, opposite ∠s ≅
5. ̅̅̅̅̅
𝑀𝐺 ≅ ̅̅̅̅̅
𝑀𝐺
5.
6.
6.
7.
7.
17
*We can use triangles in parallelograms to find congruent parts.
1. Parallelogram MTVH is shown below. Which pair of triangles can be established to be congruent to prove that
̅̅̅̅
𝑇𝐷 ≅ ̅̅̅̅
𝐻𝐷 ? Choose all that apply.
T
A. ∆𝑀𝑇𝐻 and ∆𝑉𝑇𝐻
B. ∆𝑀𝐷𝐻 and ∆𝑉𝐷𝑇
C. ∆𝑀𝑇𝐷 and ∆𝑉𝐻𝐷
D. ∆𝑇𝐷𝑉 and ∆𝐻𝐷𝑉
V
D
M
H
2. Parallelogram TNML is shown below. Which pair of triangles can be established to be congruent to prove that
∠𝐿𝑇𝑁 ≅ ∠𝑁𝑀𝐿? Choose all that apply.
T
A. ∆𝐿𝑇𝑁 and ∆𝑁𝑀𝐿
B. ∆𝑇𝑁𝑀 and ∆𝑀𝐿𝑇
C. ∆𝑁𝑂𝑇 and ∆𝑀𝑂𝐿
D. ∆𝐿𝑇𝑁 and ∆𝑀𝑁𝑇
N
O
L
M
3. Parallelogram TRHL is shown below. Which pair of triangles can be established to be congruent to prove that
̅̅̅̅
̅̅̅ ? Choose all that apply.
𝐻𝑅 ≅ ̅𝐿𝑇
H
A. ∆𝑅𝑆𝑇 and ∆𝐿𝑆𝐻
B. ∆𝑅𝐻𝐿 and ∆𝐿𝑇𝑅
C. ∆𝐻𝑅𝑇 and ∆𝐿𝑇𝑅
D. ∆𝐻𝑆𝑅 and ∆𝑇𝑆𝐿
L
S
R
T
Example 2: Drawing your way to an answer.
̅̅̅̅ and 𝐻𝑀
̅̅̅̅̅
1. 𝐻𝐴𝑀𝑇 is a parallelogram with diagonals 𝐴𝑇
that intersect at point 𝑅. Which of the following
statement(s) must be true? CHOOSE ALL THAT
APPLY.
A.
B.
C.
D.
̅̅̅̅ and ̅̅̅̅
2. JANE is a rectangle with diagonals 𝐽𝑁
𝐸𝐴 that
intersect at point T. Which of the following statements
must be true?
A.
B.
C.
D.
̅̅̅̅
̅̅̅̅ ≅ 𝐴𝑅
𝐻𝑅
∠𝑇𝐻𝑀 ≅ ∠𝐴𝑀𝐻
∠𝐴𝑇𝐻 ≅ ∠𝑇𝐴𝑀
∠𝐴𝐻𝑇 ≅ ∠𝐴𝑀𝑇
3. FOXY is a square with diagonals ̅̅̅̅
𝐹𝑋 and ̅̅̅̅
𝑌𝑂 that
intersect at point P. Which of the following statements
must be true?
A.
B.
C.
D.
∆𝐽𝑇𝐴 must be a right triangle
∆𝐽𝑁𝐸 must be an acute triangle
∆𝐽𝑇𝐸 must be an isosceles triangle
∆𝐽𝑇𝐴 ≅ ∆𝑁𝑇𝐴
̅̅̅̅ and 𝑅𝑇
̅̅̅̅ that
4. STOR is a parallelogram with diagonals 𝑆𝑂
intersect at point Y. Which of the following statements
must be true?
ΔYPX must be an obtuse triangle
ΔFPO ≅ ΔXPY
ΔPOX must be an acute triangle
ΔFPO ≅ ΔFYX
A.
B.
C.
D.
18
ΔSTY must be an obtuse triangle
ΔSYR must be an acute triangle
ΔSTR ≅ ΔORT
ΔROY ≅ ΔTOY
Example 4: What is the most specific name for the quadrilateral?
1.
2.
3.
4.
5.
7.
8.
6.
9.
10
6
6
10
Constructions review: use a compass and a straightedge
1. Construct an equilateral
triangle using the segment
below as one side of the ∆.
2. Construct a median of the triangle.
19
3. Construct a regular hexagon inside the
circle below.