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Chapter3
Lesson 1: Triangles
Triangles are used in geometry because they are the simplest polygon. A polygon is a closed, straightsided shape.
Properties of All Triangles
The sum of the interior angles for a triangle is 180°.
a + b + c = 180°
a
b
c
The longest side of a triangle is opposite the widest angle.
The shortest side of a triangle is opposite the smallest angle.
D
If ∠F < ∠D <∠E, then DE < EF < DF.
E
F
The sum of any two sides of a triangle has to be greater than the third side.
Imagine a triangle made up of three boards. If you lay the
longest board on the ground, the other two boards would
have to be longer than this board so that they could make
the peak of the triangle.
If you extend a side of the triangle, it will form an exterior angle. The exterior angle at any vertex is
equal to the sum of the opposite interior angles.
r
∠t is the exterior angle
∠r and ∠s are opposite interior angles
s
t
∠r + ∠s = ∠t
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Classifying Triangles
Triangles can be classified by their sides or their angles.
By Sides:
Scalene
Three different lengths
Isosceles
Two sides are the same length
Property:
The base angles are equal.
base angles
By Angles:
Equilateral
All three sides are the same length
Property:
All three angles are equal.
The angles equal 60°.
Acute
All angles are less than 90°
Right
It contains one right angle
The longest side is called the hypotenuse.
Obtuse
It contains one obtuse angle
hypotenuse
Equiangular All three angles are equal
An equiangular triangle is also acute and equilateral.
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Chapter 3, Lesson 1 Homework:
Find the value of x. Then classify the triangle by its angles.
2.
1.
60°
3x°
3.
64°
x°
x°
60°
60°
70°
x°
4. Find the measure of the exterior angle.
(2x – 2)°
x°
45°
5. Find the measure of the lettered angles.
b
a
c
d
40°
20°
f
e
6. ΔXYZ is a right triangle. ∠Y is the right angle. ∠X is four times as large as ∠Z. Find the measure
of ∠X and ∠Z.
N
61°
7. For ΔMNO, name the:
a. longest side
b. shortest side
59°
60°
M
O
J
8. For ΔJKL, name the:
a. largest angle
b. smallest angle
9 in.
K
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10 in.
8 in.
L
9. Is it possible to form a triangle with each set of sides?
a. 3 feet, 4 feet, 5 feet
b. 5 inches, 10 inches, 15 inches
c. 2 meters, 4 meters, 5 meters
10. Indicate whether each triangle is possible.
a. A scalene right triangle.
b. An obtuse isosceles triangle.
c. An equilateral right triangle.
d. An acute isosceles triangle.
e. A triangle with two obtuse angles.
Find the values of x and y.
11.
102°
12.
y°
y°
x°
(x + 7)°
55°
13. One angle in an isosceles triangle is 40°. What are the measures of the other angles in the
triangle? There are two possible solutions. Find both of them.
Bonus
Along a road, there are five villages. Let’s call them A, B, C, D, and E, for short. The
distance from A to D is 6 miles, from A to E is 16 miles, from D to E is 22 miles, from D to
C is 6 miles, and from A to B is 16 miles. Find the correct order in which the villages are
located along the road.
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Lesson 2: Triangles and Special Lines
Triangles are simple polygons, but there are hundreds of special lines and points associated with them.
Here are descriptions of a few of them.
A median of a triangle is a straight line that connects a vertex to the
midpoint of the opposite side. A median divides the triangle into two
equal areas. The three medians of a triangle intersect at a point called the
centroid. If the triangle were cut out of a uniformly-thick material, the
centroid is the center of mass, or balancing point, for the shape.
An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. The
three altitudes intersect at a point called the orthocenter.
The perpendicular bisectors of the three sides of a triangle intersect at a
called the circumcenter. If you put the needle of a compass at the
circumcenter and the lead of the compass on a vertex, you can draw a
circle that touches all three vertices of the triangle. The circle is
circumscribed around the triangle. When a circle passes through all the
vertices of a shape, the shape called cyclic. All triangles are cyclic.
point
Construction #1: Circumcenter
1. Bisect each side of the triangle with a perpendicular bisector.
2. At the point where the bisectors intersect (the circumcenter), put the needle of your compass.
Open the compass until the lead touches one vertex of the triangle.
3. Draw a circle around the triangle.
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The three angle bisectors of a triangle intersect at a point called the incenter, the center of the triangle’s
incircle. The incircle touches all three of the triangles sides.
Construction #2: Incenter
1. Bisect each angle of the triangle.
2. Put the needle of your compass at the point where all three bisectors intersect (the incenter).
Open your compass until the lead is outside the triangle. Swing an arc across one side. Label the
points where the arc crosses the side A and B.
3. Bisect AB.
4. Put the needle of your compass at the incenter and put the lead on the midpoint of AB.
5. Draw a circle inside the triangle.
The Euler line contains the orthocenter, the circumcenter, and the centroid for a triangle. You know that
two distinct points determine a line, but it is surprising to find that these three points always line in the
same line for any triangle.
A midsegment is a line that connects the midpoints of two adjacent sides of a triangle. A midsegment is
parallel to the third side and half as long.
A
MN is parallel to BC.
MN is ½ BC.
M
B
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N
C
Chapter 3, Lesson 2 Homework:
1. What is a median? Use a ruler to draw a triangle and one of its medians.
2. What is an altitude? Draw a triangle and one of its altitudes.
3. What is a midsegment? Draw a triangle and one of its midsegments.
4. Name the point that each set of lines locates:
a. Angle bisectors
b. Medians
c. Perpendicular bisectors
d. Altitudes
5. All triangles are cyclic. What does “cyclic” mean?
6. Use a straightedge to draw a triangle that has sides between 3 and 6 inches long. Use a ruler to
find the midpoint of each side. Be VERY precise when measuring. Draw the median for each
side. Label the centroid.
7. Draw an acute triangle that has sides between 3 and 6 inches long. Use the square corner of an
index card to draw the altitude to each vertex. Label the orthocenter.
8. Use your protractor and a ruler to construct a right triangle. The sides should be 3” and 4” long.
Use an index card, as you did in the previous problem, to locate the orthocenter of the triangle.
9. Use a straightedge to construct a triangle with sides between 3 and 4 inches long. Carefully
bisect each side using your compass and a straightedge. Label the circumcenter. Draw a circle
that touches each vertex.
10. Draw a triangle. Use a ruler to locate the midpoint of each side. Use your straightedge to draw
the three midsegments of the triangle.
H
11. A, B, and C are the midpoints of the sides of ΔGHJ.
A
B
a. If AC = 3.4 meters, what is the length of HJ?
b. If GH = 35 cm, what is the length of CB?
G
c. If AB = 3x + 8 and GJ = 2x + 24, what is the length of AB?
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C
J
Lesson 3: Triangle Constructions
Construction #3: Copying a Triangle
1. Draw a line on which to construct the triangle.
2. Use your compass to transfer the length of one side of the triangle (AB) on to the line. Label the
endpoints of your new segment D and E.
3. Set the needle of your compass at A and open the compass until the lead is on C.
4. Set the needle of your compass on D and swing an arc above the line.
5. Set the needle of your compass at B and open the compass until the lead is on C.
6. Set the needle of your compass on E and swing and arc above the line.
7. The point where the two arcs intersect is the third vertex of your triangle. Use your straightedge to
connect it to the endpoints of DE.
Construction #4: Equilateral Triangle
1.
2.
3.
4.
5.
Draw a line equal to the length of one side of your triangle.
Put the needle of your compass at A and open your compass until the lead is on B.
Swing an arc above the line.
Put the needle of your compass at B and swing an arc above the line.
The point where the arcs intersect is the third vertex for the triangle. Use your straightedge to
connect it to the endpoints of AB.
Chapter 3, Lesson 3 Homework:
1. Use your straightedge and copy to construct a copy of this triangle on your paper.
D
G
P
2. Construct an equilateral triangle with sides 3 inches long.
3. Use a straightedge to construct a triangle that takes up almost an entire sheet of 8 ½” x 11”
paper. Bisect each angle using a compass and straightedge. Label the incenter. Find the radius
of the incircle, then draw the circle.
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Bonus: Spiral Triangles
Draw a large equilateral triangle. Draw the altitude of the triangle. Use the altitude to draw a
smaller equilateral triangle. Continue drawing altitudes and new triangles until you have made
at least eight equilateral triangles.
Lesson 4: Pythagorean Theorem
One of the most famous theorems of geometry is the Pythagorean Theorem. Pythagoras was a Greek
philosopher and mathematician who lived in the 6th century BC and is given credit for constructing the
first proof of this theorem:
Pythagorean Theorem – The sum of the squares of the sides of a right triangle is equal to the square of
the hypotenuse.
This theorem can also be written as a formula: a2 + b2 = c2.
a and b are the lengths of the sides of a right triangle. c is the length of the hypotenuse.
Pythagoras’ proof of his theorem didn’t look like the 2-column proofs you have been practicing. It was a
visual proof that used the area of shapes. To understand the proof, you need to remember that the
area of a rectangle equals its base times its height and the area of a triangle equals half of its base times
its height.
Start with a triangle with sides a and b and a hypotenuse c, such as this one:
Every yellow triangle on the diagrams below are congruent to this triangle.
The area of the triangle equals ½ ab.
c
a
b
Next construct two large squares with sides equal to a + b. The square on the left is divided into six
regions. Four of them are equal to the area of the yellow triangle ( ½ ab ), one region has the area a2,
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and the last region has an area of b2. The square on the right is divided into five regions – four yellow
triangles and one large triangle with sides equal to the hypotenuse of the yellow triangle. Its area is c2.
b
a
a
b
a
a
a
b
b
c
c
b
c
a
b
a
Area of Square 1
(a + b)2 = 4( ½ ab) + a2 + b2
b
c
b
a
Area of Square 2
(a + b)2 = 4( ½ ab) + c2
Since both large squares have the same area, the regions that make up the squares are equal.
4( ½ ab) + a2 + b2 = 4( ½ ab) + c2
2 ab + a2 + b2 = 2 ab + c2
Multiplication
2
2
2
a +b =c
Subtraction
When graphing points on a coordinate plane, the Pythagorean Theorem can be used to determine the
distance between two points.
Point A has the coordinates (x1, y1) and Point B has the coordinates (x2, y2). The length of one side of the
triangle equals x2 – x1 and the length of the other side is y2 – y1. When you put these lengths in the
Pythagorean Theorem, you get:
( − ) + ( − )= To solve for the length of the hypotheses c, take the square root of both sides.
( − ) + ( − )= c
y2
y1
x1
x2
Distance Formula: Distance = ( − ) + ( − )
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Chapter 3, Lesson 4 Homework:
Find the missing length for each right triangle. Round your answers to 2 decimal places.
1.
15
18
2.
32
16
3.
10
7
4.
22
11
5. What is the length of the hypotenuse of a right triangle with sides of 8 meters and 15
meters?
6. A baseball diamond is a square with sides 90 feet long. A
player got a hit and is running to first base. He is 60 feet
from home plate. How far is he from:
a. first base?
b. second base?
c. third base?
7. If the catcher at home plate throws to the second
baseman, how far does he throw?
60
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In each figure, squares have been drawn on the sides of a right triangle. Given the two areas shown
in each figure, find the third area and the length of each side of the triangle.
8.
9.
?
169
36
?
64
25
10. The given lengths are two sides of right triangle. All three side lengths are integers. Find the
length of the third side and tell whether it is a leg or the hypotenuse.
a. 24 and 51
b. 20 and 48
11. Tell whether the following sets of side lengths can form a right triangle. If the triangle is not
right, indicate whether it is acute or obtuse.
a. 10, 11, 14
b. 5, 6, 9
c. 15, 20, 25
12. Solve for x.
x
6
10
36
9
39
13. Find the distance between the points. Round your answer to 2 decimal places.
a. (8, 2) and (12, -4)
b. (-5, 9) and (-11, -1)
c. (4, -7) and (3, 0)
d. (-54, 26) and (-41, -8)
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14. Calculate the distances between each pair of points. Are these points on the vertices of a
right triangle? A(-2, 4), B(6, 0), C(-5, -2)
15. Surveyors were setting stakes at the four corners of a new building that was going to be
constructed. The coordinates of the stakes were (18, 7), (98, 25), (2, 37), and (82, 55). The
measurements are in feet. What is the perimeter of the building?
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Lesson 5:Congruence Proofs
Congruent triangles have the same exact shape. All of their sides match and all of their angles
match. Another word for matching is corresponding. So we can say that the triangles have
corresponding sides and corresponding angles.
If ∆ABC ≅ ∆LMN, you could say the Angle A corresponds with Angle L. You could also say that side
BC corresponds with side MN. By saying this, you are saying they are the same exact size. Two
congruent triangles will always have six pairs of corresponding parts.
A
L
Angles
Sides
∠A ≅ ∠L
AB ≅ LM
∠B ≅ ∠M
BC ≅ MN
∠C ≅ ∠N
AC ≅ LN
B
C
M
N
There are five ways to prove that two triangles are congruent.
Side-Side-Side Theorem (SSS)
If three sides of one triangle are congruent to the three sides of another triangle, the triangles are
congruent.
Side-Angle Side Postulate (SAS)
If two sides and the included angle of one triangle are congruent to two sides and the included
angle of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA)
If two angles and the included side of a triangle are congruent to two angles and the included side
of another triangle, the triangles are congruent.
Angle-Angle-Side (AAS)
If two angles and the side opposite one of them in one triangle are congruent to the corresponding
parts of another triangle, the triangles are congruent.
The fifth method is for right triangles only.
Hypotenuse-Leg Theorem (HL)
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another
triangle, the triangles are congruent.
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Chapter 3, Lesson 5 Homework:
1. List the six pairs of congruent parts for ΔCAT ≅ ΔDOG.
Tell which triangles you can prove congruent. List the theorem you would use to prove it.
5.
2.
6.
3.
4.
7.
A
8. Copy and complete the proof. Be sure to include the figure.
Given: AC ≅ EC, BC ≅ CD
Prove: ΔACB ≅ ΔECD
B
C
D
Statements
1. AC ≅ EC, BC ≅
CD
2. ∠ACB ≅ ∠ECD
3. ∆ACB ≅ ∆ECD
E
Reasons
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9. Copy and complete the proof. Be sure to include the figure.
K
G
J
H
Given: GH ≅ JK, HJ ≅ KG
Prove: ΔGHJ ≅ ΔJKG
Statements
1. GH ≅ JK, HJ ≅ KG
2. JG ≅JG
3. : ∆GHJ ≅ ∆JKG
Reasons
T
10. Complete a 2-column proof.
U
R
Given: Y is the midpoint of XZ.
ΔRXZ is isosceles.
∠T ≅ ∠U
Prove: ΔUXY ≅ ΔTZY
X
Y
11. Complete a 2-column proof.
Z
B
Given: AB ≅CB, AC ⊥ BD
Prove: ΔADB ≅ ΔCDB
A
D
C
12. Complete a 2-column proof.
Given: PQ bisects ∠SPT, SP ≅ TP
Prove: ΔSPQ ≅ ΔTPQ
P
S
T
Q
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Lesson 6: Congruence Applications
If you know that two triangles are congruent, then you know that their corresponding parts are also
congruent. This is just the converse of the definition of congruent triangles, but it is a useful idea
that you can use in a proof. “Corresponding parts of congruent triangles are congruent” can be
abbreviated as CPCTC.
To use CPCTC in a proof, you have to show that two triangles are congruent first. Then you can
state that individual pairs of their corresponding parts are congruent as well. Follow the proof
shown below to see how useful CPCTC is.
C
Given: M is the midpoint of AB, CM ⊥ AB
Prove: ∠A ≅ ∠B
1.
2.
3.
4.
5.
6.
7.
Statements
M is the midpoint of AB, CM
⊥ AB
AM ≅MB
∠1 and ∠2 are right angles.
∠1 ≅ ∠2
CM ≅ CM
∆ACM ≅ ∆BCM
∠A ≅ ∠B
A
M
B
Reasons
Given
A midpoint divides a segment into two equal parts.
Perpendicular lines form right angles.
All right angles are equal.
Reflexive property
SAS
CPCTC
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Chapter 3, Lesson 6 Homework:
1. What does CPCTC stand for?
2. What has to come before CPCTC in a proof?
3. If you know that ΔMOP ≅ ΔRAG, you could use CPCTC to show which 6 facts about the two
triangles?
4. Give the reasons for each step in the proof.
D
Given: AB ≅CB, AD ≅CD
Prove: DB bisects ∠ADC.
B
A
Statements
8. AB ≅CB, AD ≅CD
9. BD ≅BD
10. ∆ABD ≅∆CBD
11. ∠ADB ≅ ∠CDB
12. DB bisects ∠ADC
C
Reasons
5. Give the reasons for each step of the proof.
E
Given: ∠A ≅ ∠D
AB ≅ CD
∠AEB ≅ ∠ DEC
Prove: ∠1 ≅ ∠2
1.
2.
3.
4.
5.
6.
7.
8.
Statements
∠A ≅ ∠D, AB ≅ CD,
∠AEB ≅ ∠ DEC
∆AEB ≅ ∆DEC
∠ABE ≅ ∠DCE
∠ABE + ∠1 = 180°
∠DCE + ∠2 = 180°
∠ABE + ∠1 =∠DCE +
∠2
∠ABE + ∠1 =∠ABE +
∠2
∠1 =∠2
1
A
B
2
C
D
Reasons
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Write a 2-column proof for each problem. Be sure to include the figure with the congruent parts
marked.
6. Given: AC ≅ BD
AB ⊥ BC, DC ⊥ BC
B
1
C
2
Prove: ∠1 ≅ ∠2
A
D
J
K
7. Given: ∠J ≅ ∠K
PJ ≅ PK
Prove: JY ≅ KX
X
Y
P
8. Given: ∠S ≅ ∠H
SR ⊥RW, HW ⊥RW
ST ≅ HT
Prove: T is the midpoint of RW.
S
H
R
T
9. Given: OV ≅ LV
KO ≅ ZL
W
V
Prove: ΔKVZ is isosceles.
K
3 1
O
2 4
L
Z
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