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Columbus, Ohio 43240
Lesson 4-1 Classifying Triangles
Lesson 4-2 Angles of Triangles
Lesson 4-3 Congruent Triangles
Lesson 4-4 Proving Congruence–SSS, SAS
Lesson 4-5 Proving Congruence–ASA, AAS
Lesson 4-6 Isosceles Triangles
Lesson 4-7 Triangles and Coordinate Proof
Example 1 Classify Triangles by Angles
Example 2 Classify Triangles by Sides
Example 3 Find Missing Values
Example 4 Use the Distance Formula
ARCHITECTURE The triangular truss below is
modeled for steel construction. Classify
JMN, JKO, and OLN as acute, equiangular, obtuse,
or right.
Answer:
JMN has one angle with measure greater than 90, so it
is an obtuse triangle.
JKO has one angle with measure equal to 90, so it is a
right triangle.
OLN is an acute triangle with all angles congruent, so it
is an equiangular triangle.
ARCHITECTURE The frame of this window design is
made up of many triangles. Classify ABC, ACD, and
ADE as acute, equiangular, obtuse, or right.
Answer: ABC is acute.
ACD is obtuse.
ADE is right.
Identify the isosceles triangles in the figure if
Isosceles triangles have at least two sides congruent.
Answer: UTX and UVX are isosceles.
Identify the scalene triangles in the figure if
Scalene triangles have no congruent sides.
Answer: VYX, ZTX, VZU, YTU, VWX,
ZUX, and YXU are scalene.
Identify the indicated triangles in the figure.
a. isosceles triangles
Answer: ADE, ABE
b. scalene triangles
Answer: ABC, EBC, DEB, DCE, ADC, ABD
ALGEBRA Find d and the measure of each side of
equilateral triangle KLM if
and
Since KLM is equilateral,
each side has the same
length. So
Substitution
Subtract d from each side.
Add 13 to each side.
Divide each side by 3.
Next, substitute to find the length of each side.
Answer: For KLM,
and the measure of
each side is 7.
ALGEBRA Find d and the measure of each side of
equilateral triangle
if
and
Answer:
COORDINATE GEOMETRY Find the measures of the
sides of RST. Classify the triangle by sides.
Use the distance formula to find the lengths of each side.
Answer:
; since all 3 sides
have different lengths, RST is scalene.
Find the measures of the sides of ABC. Classify the
triangle by sides.
Answer:
; since all 3 sides
have different lengths, ABC is scalene.
Example 1 Interior Angles
Example 2 Exterior Angles
Example 3 Right Angles
Find the missing angle measures.
Find
first because the
measure of two angles of
the triangle are known.
Angle Sum Theorem
Simplify.
Subtract 117 from each side.
Angle Sum Theorem
Simplify.
Subtract 142 from each side.
Answer:
Find the missing angle measures.
Answer:
Find the measure of each numbered angle in the figure.
Exterior Angle Theorem
Simplify.
If 2 s form a linear pair, they
are supplementary.
Substitution
Subtract 70 from each side.
Exterior Angle Theorem
Substitution
Subtract 64 from each side.
If 2 s form a linear pair,
they are supplementary.
Substitution
Simplify.
Subtract 78 from each side.
Angle Sum Theorem
Substitution
Simplify.
Subtract 143 from each side.
Answer:
Find the measure of each numbered angle in the figure.
Answer:
GARDENING The flower bed shown is in the shape of
a right triangle. Find
if
is 20.
Corollary 4.1
Substitution
Subtract 20 from each side.
Answer:
The piece of quilt fabric is in the shape of a
right triangle. Find
if
is 32.
Answer:
Example 1 Corresponding Congruent Parts
Example 2 Transformations in the Coordinate Plane
ARCHITECTURE A tower roof is composed of
congruent triangles all converging
toward a point at the top. Name the
corresponding congruent angles
and sides of HIJ and LIK.
Answer: Since corresponding parts of congruent triangles
are congruent,
ARCHITECTURE A tower roof is composed of
congruent triangles all converging
toward a point at the top.
Name the congruent triangles.
Answer: HIJ LIK
The support beams on the fence form congruent
triangles.
a. Name the corresponding
congruent angles and sides of
ABC and DEF.
Answer:
b. Name the congruent triangles.
Answer: ABC DEF
COORDINATE GEOMETRY The vertices of RST are
R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST
are R(3, 0), S(0, ─5), and T(─1, ─1). Verify that
RST RST.
Use the Distance Formula to find the length of each side of
the triangles.
Use the Distance Formula to find the length of each side of
the triangles.
Use the Distance Formula to find the length of each side of
the triangles.
Answer: The lengths of the corresponding sides of two
triangles are equal. Therefore, by the definition
of congruence,
Use a protractor to measure the angles of the triangles. You
will find that the measures are the same.
In conclusion, because
,
COORDINATE GEOMETRY The vertices of RST are
R(─3, 0), S(0, 5), and T(1, 1). The vertices of RST 
are R(3, 0), S(0, ─5), and T(─1, ─1). Name the
congruence transformation for RST and RST.
Answer: RST is a
turn of RST.
COORDINATE GEOMETRY The vertices of ABC are
A(–5, 5), B(0, 3), and C(–4, 1). The vertices of ABC
are A(5, –5), B(0, –3), and C(4, –1).
a. Verify that ABC ABC.
Answer:
Use a protractor to verify that
corresponding angles are
congruent.
b. Name the congruence transformation for ABC
and ABC.
Answer: turn
Example 1 Use SSS in Proofs
Example 2 SSS on the Coordinate Plane
Example 3 Use SAS in Proofs
Example 4 Identify Congruent Triangles
ENTOMOLOGY The wings of one type of moth form
two triangles. Write a two-column proof to prove that
FEG HIG
and G is the midpoint
of both
Given:
G is the midpoint of both
Prove: FEG
HIG
Proof:
Statements
Reasons
1.
1. Given
2.
2. Midpoint Theorem
3. FEG
HIG
3. SSS
Write a two-column proof to prove that ABC
Proof:
Statements
Reasons
1.
2.
3. ABC GBC
1. Given
2. Reflexive
3. SSS
GBC if
COORDINATE GEOMETRY Determine whether
WDV MLP for D(–5, –1), V(–1, –2), W(–7, –4),
L(1, –5), P(2, –1), and M(4, –7). Explain.
Use the Distance
Formula to show that
the corresponding
sides are congruent.
Answer:
By
definition of congruent segments, all
corresponding segments are congruent.
Therefore, WDV MLP by SSS.
Determine whether ABC DEF for A(5, 5), B(0, 3),
C(–4, 1), D(6, –3), E(1, –1), and F(5, 1). Explain.
Answer:
By definition
of congruent segments, all corresponding
segments are congruent. Therefore,
ABC DEF by SSS.
Write a flow proof.
Given:
Prove: QRT
STR
Answer:
Write a flow proof.
Given:
Prove: ABC ADC
.
Proof:
Determine which postulate can be used to prove that
the triangles are congruent. If it is not possible to
prove that they are congruent, write not possible.
Two sides and the
included angle of one
triangle are congruent
to two sides and the
included angle of the
other triangle. The
triangles are
congruent by SAS.
Answer: SAS
Determine which postulate can be used to prove that
the triangles are congruent. If it is not possible to
prove that they are congruent, write not possible.
Each pair of corresponding
sides are congruent. Two
are given and the third is
congruent by Reflexive
Property. So the triangles
are congruent by SSS.
Answer: SSS
Determine which postulate can be used to prove that
the triangles are congruent. If it is not possible to
prove that they are congruent, write not possible.
a.
Answer: SAS
b.
Answer: not possible
Example 1 Use ASA in Proofs
Example 2 Use AAS in Proofs
Example 3 Determine if Triangles Are Congruent
Write a paragraph proof.
Given: L is the midpoint of
Prove: WRL EDL
Proof:
because alternate interior angles are
congruent. By the Midpoint Theorem,
Since vertical angles are congruent,
WRL EDL by ASA.
Write a paragraph proof.
Given:
Prove: ABD CDB
Proof:
because alternate interior
angles are congruent.
because alternate interior angles are congruent.
by Reflexive Property. ABD CDB by
ASA.
Write a flow proof.
Given:
Prove:
Proof:
Write a flow proof.
Given:
Prove:
Proof:
STANCES When Ms. Gomez puts her hands on her hips,
she forms two triangles with
her upper body and arms.
Suppose her arm lengths AB
and DE measure 9 inches, and
AC and EF measure 11 inches.
Also suppose that you are
given that
Determine
whether ABC EDF.
Justify your answer.
Explore We are given measurements of two sides of
each triangle. We need to determine whether the
two triangles are congruent.
Plan
Since
Likewise,
We are given
Check each possibility using the five methods
you know.
Solve
We are given information about three sides. Since
all three pairs of corresponding sides of the
triangles are congruent, ABC EDF by SSS.
Examine You can measure each angle in ABC and EDF
to verify that
Answer: ABC EDF by SSS
The curtain decorating the window forms 2 triangles
at the top. B is the midpoint of AC.
inches and
inches. BE and BD each use the same amount
of material, 17 inches. Determine whether ABE CBD
Justify your answer.
Answer: ABE CBD by SSS
Example 1 Proof of Theorem
Example 2 Find the Measure of a Missing Angle
Example 3 Congruent Segments and Angles
Example 4 Use Properties of Equilateral Triangles
Write a two-column proof.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Def. of
3. ABC and BCD are
isosceles
4.
3. Def. of isosceles 
5.
6.
5. Given
6. Substitution
segments
4. Isosceles  Theorem
Write a two-column proof.
Given:
Prove:
.
Proof:
Statements
1.
Reasons
1. Given
2. ADB is isosceles.
2. Def. of isosceles triangles
3.
4.
3. Isosceles  Theorem
4. Given
5.
6. ABC ADC
7.
5. Def. of midpoint
6. SAS
7. CPCTC
Multiple-Choice Test Item
If
and
measure of
A. 45.5
B. 57.5
Read the Test Item
CDE is isosceles with base
isosceles with
what is the
C. 68.5
D. 75
Likewise, CBA is
Solve the Test Item
Step 1 The base angles of CDE are congruent. Let
Angle Sum Theorem
Substitution
Add.
Subtract 120 from
each side.
Divide each side by 2.
Step 2
are vertical angles so they have
equal measures.
Def. of vertical angles
Substitution
Step 3 The base angles of CBA are congruent.
Angle Sum Theorem
Substitution
Add.
Subtract 30 from each
side.
Divide each side by 2.
Answer: D
Multiple-Choice Test Item
If
and
measure of
A. 25
Answer: A
B. 35
what is the
C. 50
D. 130
Name two congruent angles.
Answer:
Name two congruent segments.
By the converse of the Isosceles Triangle Theorem, the
sides opposite congruent angles are congruent. So,
Answer:
a. Name two congruent angles.
Answer:
b. Name two congruent
segments.
Answer:
EFG is equilateral, and
Find
and
bisects
bisects
Each angle of an equilateral triangle measures 60°.
Since the angle was bisected,
is an exterior angle of EGJ.
Exterior Angle Theorem
Substitution
Add.
Answer:
EFG is equilateral, and
Find
bisects
bisects
Linear pairs are supplementary.
Substitution
Subtract 75 from each side.
Answer: 105
ABC is an equilateral triangle.
a. Find x.
Answer: 30
b.
Answer: 90
bisects
Example 1 Position and Label a Triangle
Example 2 Find the Missing Coordinates
Example 3 Coordinate Proof
Example 4 Classify Triangles
Position and label right triangle XYZ with leg
long on the coordinate plane.
d units
Use the origin as vertex X of the triangle.
Place the base of the
triangle along the positive
x-axis.
Position the triangle in the
first quadrant.
Since Z is on the x-axis, its
y-coordinate is 0. Its
X (0, 0)
x-coordinate is d because
the base is d units long.
Z (d, 0)
Since triangle XYZ is a right triangle the
x-coordinate of Y is 0. We cannot determine the
y-coordinate so call it b.
Answer:
Y (0, b)
X (0, 0)
Z (d, 0)
Position and label equilateral triangle ABC with side
w units long on the coordinate plane.
Answer:
Name the missing coordinates of isosceles right
triangle QRS.
Q is on the origin, so its coordinates
are (0, 0).
The x-coordinate of S is the same
as the x-coordinate for R, (c, ?).
The y-coordinate for S is the distance
from R to S. Since QRS is an
isosceles right triangle,
The distance from Q to R is c units.
The distance from R to S must be
the same. So, the coordinates of S
are (c, c).
Answer: Q(0, 0); S(c, c)
Name the missing coordinates of isosceles right ABC.
Answer: C(0, 0); A(0, d)
Write a coordinate proof to prove that the segment that
joins the vertex angle of an isosceles triangle to the
midpoint of its base is perpendicular to the base.
The first step is to position and label a right triangle on the
coordinate plane. Place the base of the isosceles triangle
along the x-axis. Draw a line segment from the vertex of
the triangle to its base. Label the origin and label the
coordinates, using multiples of 2 since the Midpoint
Formula takes half the sum of the coordinates.
Given: XYZ is isosceles.
Prove:
Proof: By the Midpoint Formula, the coordinates of W,
the midpoint of
, is
The slope of
or undefined. The
slope of
therefore,
is
.
Write a coordinate proof to prove that the segment
drawn from the right angle to the midpoint of the
hypotenuse of an isosceles right triangle is
perpendicular to the hypotenuse.
Proof: The coordinates of the midpoint D are
The slope of
or 1. The slope of
therefore
.
is
or –1,
DRAFTING Write a coordinate proof to prove that the
outside of this drafter’s tool is shaped like a right
triangle. The length of one side is 10 inches and the
length of another side is 5.75 inches.
Proof: The slope of
or undefined. The slope of
or 0, therefore
DEF is a right triangle.
The drafter’s tool is shaped like a
right triangle.
FLAGS Write a coordinate proof to prove this flag is
shaped like an isosceles triangle. The length is 16
inches and the height is 10 inches.
C
Proof: Vertex A is at the origin and B is at (0, 10). The
x-coordinate of C is 16. The y-coordinate is halfway
between 0 and 10 or 5. So, the coordinates of C are
(16, 5).
Determine the lengths of CA and CB.
Since each leg is the same length, ABC is isosceles. The
flag is shaped like an isosceles triangle.
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