Chapter 4 PPT
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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 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, each side is 7. and the measure of
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.
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 2 SSS on the Coordinate Plane
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
1.
Reasons
1. Given
2.
3.
FEG
HIG
2. Midpoint Theorem
3. SSS
Write a two-column proof to prove that
ABC
GBC if
Proof:
Statements
1.
2.
3.
ABC
GBC
Reasons
1. Given
2. Reflexive
3. SSS
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 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 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
1.
2.
3.
ABC and
BCD are isosceles
4.
5.
6.
Reasons
1.
Given
2.
Def. of segments
3.
Def. of isosceles
4.
Isosceles
Theorem
5.
Given
6.
Substitution
Write a two-column proof.
Given: .
Prove:
Proof:
Statements
1.
2.
ADB is isosceles.
3.
4.
5.
6.
ABC
ADC
7.
Reasons
1.
Given
2.
Def. of isosceles triangles
3.
Isosceles
Theorem
4.
Given
5 . Def. of midpoint
6.
SAS
7.
CPCTC
Multiple-Choice Test Item
If and measure of what is the
A.
45.5 B.
57.5 C.
68.5 D.
75
Read the Test Item
CDE is isosceles with base Likewise,
CBA is isosceles with
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 what is the measure of
A.
25 B.
35 C.
50 D.
130
Answer: A
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 bisects bisects
Find and
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 bisects bisects
Find
Linear pairs are supplementary.
Substitution
Subtract 75 from each side.
Answer: 105
ABC is an equilateral triangle. bisects a. Find x .
Answer: 30 b.
Answer: 90
Example 1 Position and Label a Triangle
Example 2 Find the Missing Coordinates
Position and label right triangle XYZ with leg d units long on the coordinate plane.
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 -coordinate is d because the base is d units long.
X (0, 0) 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 is therefore, .
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 is or –1, or 1. The slope of therefore .
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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