Download Triangles and Congruence

Chapter 4
Triangles and Congruence
4.1 Apply Triangle Sum Properties
4.2 Apply Congruence and Triangles
4.3 Prove Triangles Congruent by SSS
4.4 Prove Triangles Congruent by SAS and HL
4.5 Prove Triangles Congruent by ASA and AAS
4.6 Use Congruent Triangles
4.7 Isosceles and Equilateral Triangles
SOL G.6
The student, given the information in the form of a figure or statement, will prove two
triangles are congruent, using algebraic and coordinate methods as well as
deductive proofs.
SOL G.7
The student, given the information in the form of a figure or statement, will prove two
triangles are similar, using algebraic and coordinate methods as well as deductive
proofs.
Name _________________________________ Block _______
4.1 Apply Triangle Sum Properties
Classifying triangles by their …
ANGLES
Name
Acute
Words
_________ acute angles.
Obtuse
_________ obtuse angle.
Right
_________ right angle.
Equiangular
Picture
_________ angles
congruent.
SIDES
Name
Equilateral
Isosceles
Scalene
Words
_________ sides congruent.
_________ congruent sides.
_________ congruent sides.
Picture
Classify a triangle in a coordinate plane.
Classify ∆RST by its sides. Then determine if the triangle is a right triangle.
m A + m B + m C = ___________
The sum of the measures of the interior angles of a triangle is ______.
m 1=m
_________ + m
_________
The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two ___________________ angles.
Finding an angle measure.
Use the diagram at the right to find the measure of
DCB.
m A + m B = _____
The acute angles of a right triangle are _____________.
Finding angle measures from a verbal description.
The front face of the wheelchair ramp shown forms a right
triangle. The measure of one acute angle in the triangle is
eight times the measure of the other.
Find the measure of each acute angle.
You try!
1. Triangle JKL has vertices J( 2, 1), K(l, 3), and L(5, 0). Classify it by its sides. Then determine if it
is a right triangle.
2. Find the measure of
l in the diagram shown.
3. Find the value of x.
4. Find the value of x.
63˚
88˚
x˚
2x
122˚
56˚
5. Use the diagram to find the measures of the following angles.
m∠1 =
m∠7 =
m∠2 =
m∠8 =
m∠3 =
m∠9 =
m∠4 =
m∠10 =
m∠5 =
m∠11 =
m∠6 =
m∠12 =
6. The ladder is leaning on the ground at a 75º angle. At
what angle is the top of the ladder touching the
building?
4.2 Apply Congruence and Triangles
Same size and shape
Different size or shape
ALWAYS list the corresponding vertices in the same order.
Congruence statement:
Corresponding angles:
Corresponding sides:
Two polygons are congruent if
and only if their corresponding parts are congruent.
Example: Write a congruence statement for the following triangles.
Example: In the diagram, DEFG ≌ SPQR.
a. Find the value of x.
E
D
Q
R
F
b. Find the value of y.
P
S
G
If two angles of one triangle are congruent to two angles of
another triangle, then the third angles are congruent.
If ____________________ and ____________________,
then ____________________.
Example: Find m∠ BDC.
Properties of Congruent Triangles
For any triangle ABC, _____________________
If _____________________then__________________
If _________________________ and
____________________________ then _________________________
Try it!
Write a congruence statement for any figures that can be proved congruent.
Explain your reasoning.
1.
2.
3.
Find the value of x.
4.
5.
In problems 6 and 7, use the given information to find the indicated values.
6. Given ∆ABC ≌ ∆DEF, find the values of x and y.
7. Given ∆HJK ≌ ∆TRS, find the values of a and b.
4.3 Prove Triangles Congruent by
SSS
If three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
B
A
E
C
D
F
If Side ________________ and Side _________________ and Side _______________, then ___________________
HINT: Always look for shared sides!!
Decide whether the congruence statement is true. Explain your reasoning.
1. ∆ABD ≌ ∆CDB
2. ∆RST ≌ ∆RQT
Use the given coordinates to determine if ∆ABC ≌ ∆DEF .
4. A(1, 2) B(4, -3) C(2, 5) D(4, 7) E(7, 2) F(5, 10)
5. A(1, 1) B(4, 0) C(7, 5) D(4, -5) E(6, -6) F(9, -1)
3. ∆ABC ≌ ∆DEF
Determine whether ∆ABC ≌ ∆DEF . Explain your reasoning.
6.
7.
8.
Given:
AB ≌ CB, D is the midpoint of AC
Prove:
∆ABD ≌ ∆CBD
Statements
Reasons
1.
2.
3.
4.
5.
You try!
1. Is it possible to prove that the two triangles below are congruent using the SSS postulate?
Explain your reasoning in words or by formulating a geometric proof.
Given:
AF ≌ CD, AB ≌ EF, BC ≌ ED
4.4 Prove Triangles Congruent by
SAS, HL
If two sides and the included angle of one triangle are congruent to two
sides and the included angle of a second triangle, then the two triangles are
congruent.
If Side ________________ and Angle _________________ and Side _______________, then ___________________
HINT: Always look for vertical angles!!
If the hypotenuse and a leg of a RIGHT TRIANGLE are congruent to the
hypotenuse and a leg of a second right triangle, then the two triangles are
congruent.
If Hypotenuse ____________________ and Leg ____________________, then _______________________
HINT: All right angles are congruent!!
Example: How could we prove these triangles congruent?
Use the diagram to name the included angle between the given pair of sides.
1. AB and BC
2. BD and DA
3. CD and DB
Decide whether enough information is given to prove that the triangles are congruent. If there is
enough information, state the congruence postulate or theorem you would use.
4.
5.
6.
A
7.
Given:
B is the midpoint of AE
B is the midpoint of CD
Prove:
D
B
C
∆ABD ≌ ∆EBC
E
Statements
Reasons
1.
1. Given
2.
2. Definition of midpoint
3.
3. Given
4.
4. Definition of Midpoint
5.
5. Vertical Angles Theorem
6.
6. SAS Congruence Postulate
4.5 Prove Triangles Congruent by
ASA
AAS
If two angles and the included side of one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles are
congruent.
If Angle ________________ and Side _________________ and Angle _______________, then ___________________
HINT: If lines are parallel, look for alternate interior angles!!
If two angles and a non-included side of one triangle are congruent to two
angles and the corresponding non-included side of a second triangle, then
the two triangles are congruent.
If Angle ________________ and Angle _________________ and Side _______________, then ___________________
HINT: Look for shared angles!
Explain how you can prove that the indicated triangles are congruent using the given theorem or postulate.
1. ∆BEF ≌ ∆BED by SAS
2. ∆ADB ≌ ∆CFB by ASA
3. ∆AFB ≌ ∆CDB by AAS
4.6 Use Congruent Triangles
By definition, congruent triangles have congruent corresponding parts. So, if you can
prove that two triangles are congruent, you know that their corresponding parts must
be congruent as well.
You MUST prove triangles are congruent (using SSS, SAS, HL, ASA, or AAS) before you
can use CPCTC!
Example 1
To ensure that sailboat races are fair, the boats and their sails are required to be the same size and
shape.
A. Write a congruence statement relating the triangles in the photo.
B. Name six pairs of congruent segments.
C. Name six pairs of congruent angles.
Example 2
In an umbrella frame, the stretchers are congruent and they open to angles of equal measures. Prove
that the angles formed by the shaft and the ribs are congruent (in other words, prove ∠3 ≌ ∠4).
Statements
Reasons
Example 3
According to legend, one of Napoleon’s officers used congruent triangles to
estimate the width of the river. On the riverbank, the officer stood up straight and lowered the visor of his
cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted
the spot on his side of the river that’s was in line with his eye and the tip of his visor.
Given: ∠DEF and ∠DEG are right angles; ∠EDG ≌ ∠EDF
The officer then paced off the distance to his spot and declared that distance be the width of the river!
Use congruent triangles to prove he is correct!
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
EF ≌ EG
6.
You try!
The object of the two shown is to make the two spheres meet and strike each other repeatedly
on one side of the wand and then again on the other side.
If ∠JKL ≌ ∠MLK and ∠JLK ≌ ∠MKL, prove that JK ≌ ML.
Statements
Reasons
4.7 Use Isosceles and Equilateral Triangles
Isosceles Triangle Vocabulary:
Legs, Vertex Angle, Base, Base Angles
them are congruent.
If two sides of a triangle are congruent, then the angles opposite
B
If ___________________ then _____________________
opposite them are congruent.
A
C
If two angles of a triangle are congruent, then the sides
B
If ___________________ then _____________________
A
C
If a triangle is equilateral, then it is equiangular.
If there are _____________________________ then there are
______________________________________.
If a triangle is equiangular, then it is
equilateral.
If there are _____________________________ then there are
______________________________________.
Find the values of x and y.
1.
2.
x˚
17
9y˚
Find the values of x and y continued...
(3x + 8)˚
3.
4.
98˚
(5x + 6)˚
(y – 9)˚
(8y + 10)˚
53˚
You try!
Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle.
1.
2.
3.
4. The length of YX is 20 feet. Explain why the length of YZ is the same.
Practice with SSS, SA, HL, AAS, and ASA
Determine which postulate can be used to prove that the triangles are congruent. If it is not
possible to prove congruence, write not possible.
Sample SSS Proofs
P
1
Q
S
T
M
J
2
K
L
C
3
R
B
A
D
N
Sample SAS Proofs
1
J
N
M
L
A
2
D
B
C
H
3
G
I
J
Sample HL Proofs
1
2
Sample ASA Proofs
1
2
3
Sample AAS Proofs
1
2
3
Triangle Congruence Practice
Compare the triangles and determine whether they can be proven congruent, if possible, by
SSS, SAS, ASA, AAS, or HL. Write your answer the box.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.