Slide 1 / 209 This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative ® Slide 2 / 209 www.njctl.org Congruent Triangles Geometry 2014-06-03 Slide 3 / 209 Table of Contents Classifying Triangles Interior Angle Theorems Isosceles Triangle Theorem Congruence & Triangles SSS Congruence SAS Congruence ASA Congruence AAS Congruence HL Congruence CPCTC Triangle Coordinate Proofs Triangle Congruence Proofs Exterior Angle Theorems Slide 4 / 209 Return to Table of Contents Classifying Triangles Slide 5 / 209 Parts of a triangle Side Side Vertex Vertex Vertex A B C Side opposite and are adjacent sides interior Vertex (vertices) - points joining the sides of triangles Adjacent Sides - two sides sharing a common vertex Slide 6 / 209 Parts of a triangle (cont'd) hypotenuse leg leg leg leg base In a right triangle, the hypotenuse is the side opposite the right angle. The legs are the 2 sides that form the right angle. In an isosceles triangle, the base is the side that is not congruent to the other two sides (legs). If an isosceles triangle has 3 congruent sides, it is an equilateral triangle.
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Slide 1 / 209
This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.
Click to go to website:www.njctl.org
New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative®
Slide 2 / 209
www.njctl.org
CongruentTriangles
Geometry
2014-06-03
Slide 3 / 209
Table of ContentsClassifying TrianglesInterior Angle Theorems
Polygon - a closed plane figure composed of line segments
Sides - the line segments that make up a polygonVertex (vertices) - the endpoints of the sidesAcute Triangle - all angles < 90°
Obtuse triangle - one angle is between, 90° < angle < 180°
Right Triangle - one 90° angle
Equiangular Triangle - 3 congruent angles
Equilateral Triangle - 3 congruent sides
Scalene triangle - No congruent sides
Isosceles Triangle - 2 congruent sides
Slide 8 / 209A triangle is formed by line segments joining three noncollinear points. A triangle can be classified by its sides and angles.
Equilateral
3 congruent sides
Isosceles
2 congruent sides
Scalene
No congruent sides
Classification by Sides
Classification by Angles
3 acuteangles
Acute Equiangular
3 congruentangles
Right
1 right angle
Obtuse
1 obtuse angle
(also acute)
Slide 9 / 209Classify the triangles by sides and angles
equilateralequiangular
scaleneacute
isoscelesacute
isoscelesobtuse
isoscelesright
click
clickclick
click click
Slide 10 / 209
Measure and Classify the triangles by sides and anglesExample
isosceles, right isosceles, acuteClick for AnswerClick for Answer Click for AnswerClick for Answer scalene, obtuseClick for AnswerClick for Answer
Slide 11 / 209
Measure and Classify the triangles by sides and anglesExample
scalene, obtuse scalene, acuteClick for AnswerClick for Answerequilateral, acute/equiangularClick for AnswerClick for Answer Click for AnswerClick for Answer
Slide 12 / 209
1 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 12 (Answer) / 209
1 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
C Scalene
Bonus: F Right32+42 = 52
9+16 = 25
Slide 13 / 209
2 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 13 (Answer) / 209
2 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
B Isosceles
Bonus: D Acute
3 cm 3 cm
2 cm
Slide 14 / 209
3 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
Ans
wer
Slide 14 (Answer) / 209
3 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm
A EquilateralB IsoscelesC Scalene
D AcuteE EquiangularF RightG Obtuse
[This object is a pull tab]
Ans
wer
A Equilateral
Bonus:E Equiangular (all equilateral triangles are equiangular)D Acute (all angles are 60o )
Slide 15 / 209
4 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 15 (Answer) / 209
4 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
F Right
Bonus: C Scalene (all angles are different, so all sides are different)
Slide 16 / 209
5 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 16 (Answer) / 209
5 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
G Obtuse
Bonus: C Scalene (all angles are different, so all sides are different)
Slide 17 / 209
6 Classify the triangle with the given information: Angle Measures: 60°, 60°, 60°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 17 (Answer) / 209
6 Classify the triangle with the given information: Angle Measures: 60°, 60°, 60°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
D AcuteE Equiangular
Bonus:A Equilateral (if a triangle is equiangular, then it's equilateral)
Slide 18 / 209
7 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 18 (Answer) / 209
7 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer
C ScaleneF Right
Slide 19 / 209
8 Classify the triangle with the given information: Side lengths: 3 cm, 3 cm, 3 cmAngle measures: 60°, 60°, 60°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
Ans
wer
Slide 19 (Answer) / 209
8 Classify the triangle with the given information: Side lengths: 3 cm, 3 cm, 3 cmAngle measures: 60°, 60°, 60°
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
[This object is a pull tab]
Ans
wer A Equilateral
D AcuteE Equiangular
Slide 20 / 209
9 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
A B120°
C
Ans
wer
Slide 20 (Answer) / 209
9 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
A B120°
C
[This object is a pull tab]
Ans
wer
C ScaleneG Obtuse
Slide 21 / 209
10 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
L
MN
Ans
wer
Slide 21 (Answer) / 209
10 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse
L
MN
[This object is a pull tab]
Ans
wer
B IsoscelesF Right
Slide 22 / 209
11 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse H
J
K45°
85°
50° Ans
wer
Slide 22 (Answer) / 209
11 Classify the triangle by sides and angles
A Equilateral
B Isosceles
C Scalene
D Acute
E Equiangular
F Right
G Obtuse H
J
K45°
85°
50°
[This object is a pull tab]
Ans
wer
C ScaleneD Acute
Slide 23 / 209
12 An isosceles triangle is _______________ an equilateral triangle.
A Sometimes
B Always
C Never
Ans
wer
Slide 23 (Answer) / 209
12 An isosceles triangle is _______________ an equilateral triangle.
A Sometimes
B Always
C Never
[This object is a pull tab]
Ans
wer
A Sometimes
Slide 24 / 209
13 An obtuse triangle is _______________ an isosceles triangle.
A Sometimes
B Always
C NeverA
nsw
er
Slide 24 (Answer) / 209
13 An obtuse triangle is _______________ an isosceles triangle.
A Sometimes
B Always
C Never
[This object is a pull tab]
Ans
wer
A Sometimes
Slide 25 / 209
14 A triangle can have more than one obtuse angle.
True
False
Ans
wer
Slide 25 (Answer) / 209
14 A triangle can have more than one obtuse angle.
True
False
[This object is a pull tab]
Ans
wer
False
Slide 26 / 209
15 A triangle can have more than one right angle.
True
False
Ans
wer
Slide 26 (Answer) / 209
15 A triangle can have more than one right angle.
True
False
[This object is a pull tab]
Ans
wer
False
Slide 27 / 209
16 Each angle in an equiangular triangle measures 60°
True
False
Ans
wer
Slide 27 (Answer) / 209
16 Each angle in an equiangular triangle measures 60°
True
False
[This object is a pull tab]
Ans
wer
True
Slide 28 / 209
17 An equilateral triangle is also an isosceles triangle
True
False
Ans
wer
Slide 28 (Answer) / 209
17 An equilateral triangle is also an isosceles triangle
True
False
[This object is a pull tab]
Ans
wer
False
Slide 29 / 209
InteriorAngle
Theorems
Return to Tableof Contents
Slide 30 / 209
T1. Triangle Sum TheoremThe measures of the interior angles of a triangle sum to 180°
A B
C
If you have a triangle, then you know the sum of its three interior angles is 180°
Why is this true? Click here to go to the lab titled, "Triangle Sum Theorem"
Slide 31 / 209
Example: Triangle Sum TheoremFind the measure of the missing angle
J
K L
32°
20°
Theorem T1. The Triangle Sum Theorem says that the interior angles of must sum to 180°.
Interior angles of a triangle add up to 180, so40+60+m<5 = 180100+m<5 = 180m<5 = 80o
Slide 75 / 209
Isosceles TriangleTheorem
Return to Tableof Contents
Slide 76 / 209
Parts of an Isosceles TriangleAn isosceles triangle has at least two congruent sides (an equilateral triangle is an isosceles triangle w/three congruent sides)
If an isosceles triangle has exactly two congruent sides, the: - two congruent sides are called legs, - the noncongruent side is called the base, - the two angles adjacent to the base are the base angles,
leg leg
baseangles
vertex angle
base
The vertex angle is the angle opposite the base ORit is the angle included by the legs
Slide 77 / 209
T3. Base Angles Theorem (BAT)If two sides of a triangle are congruent, the angles opposite them are congruent.
If , then
A
B C
Corollary to BAT (T3)
If a triangle is equilateral, then it is equiangular.
A
B C
Slide 78 / 209
x°
y°
44°
Examples:Find the values of x & y in the isosceles triangle below.
x = 44; Base Angles are Congruent
y + 44 + 44 = 180; Triangle Sum Th.y + 88 = 180y = 92
Find the values of x & y in the isosceles triangle below.
x° y°
52° x = y; Base Angles are Congruent
x + y + 52 = 180; Triangle Sum Th.x + x + 52 = 180; Substitution2x + 52 = 1802x = 128x = 64
Slide 79 / 209
x°
y°
35°
46 Solve for the measurements of the angles x and y
Ans
wer
Slide 79 (Answer) / 209
x°
y°
35°
46 Solve for the measurements of the angles x and y
49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?
A 71° B 38° C 83° D 104°
Ans
wer
Slide 82 (Answer) / 209
49 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?
A 71° B 38° C 83° D 104°
[This object is a pull tab]
Ans
wer
A
Slide 83 / 209
T4. Converse of the Base Angles TheoremIf two angles of a triangle are congruent, then the sides opposite them are congruent.
If , then
A
B C
Corollary to Converse of the BAT (T4)
If a triangle is equiangular, then it is equilateral.
A
B C
Slide 84 / 209
D
EF 4
50 What is the measurement of FD?
Ans
wer
Slide 84 (Answer) / 209
D
EF 4
50 What is the measurement of FD?
[This object is a pull tab]
Ans
wer
FD = 4
Triangle is equiangular, so it's equilateral too
Slide 85 / 209
51 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
A
BC
7
40o
Ans
wer
Slide 85 (Answer) / 209
51 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
A
BC
7
40o
[This object is a pull tab]
Ans
wer
B & E
Slide 86 / 209
52 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G rightA
B
C
4
4
4
Ans
wer
Slide 86 (Answer) / 209
52 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G rightA
B
C
4
4
4[This object is a pull tab]
Ans
wer
A, B, D & E
Slide 87 / 209
A
B C5
3 3113o
53 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
Ans
wer
Slide 87 (Answer) / 209
A
B C5
3 3113o
53 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
[This object is a pull tab]
Ans
wer
B & F
Slide 88 / 209
54 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
12
12
Ans
wer
Slide 88 (Answer) / 209
54 Classify the triangle by sides and angles
A equilateral
B isosceles
C scalene
D equiangular
E acute
F obtuse
G right
12
12[This object is a pull tab]
Ans
wer
A, B, D & E
Slide 89 / 209
ExampleFind the value of x and y
x°
y°
1. First, consider the top triangle. The 3 marks indicate this is an equilateral triangle
2. From the Corollary to the BAT(T3), we know that an equilateral triangle is also equiangular
3. Since the Triangle Sum Theorem (T1) says the interior angles must sum to 180°, y° = 60.
x°
y°
Slide 90 / 209
x°
60°
60°
60°
120°
4. Two adjacent angles whose non-shared sides form a straight line are a linear pair.
5. The supplement to 60° is 120° (60° + 120° = 180°)
6. Using the Base Angles Theorem (T3) and the Triangle Sum theorem (T1), we can determine x°
x°
60°
60°
60°
120°x°
x + x + 120 = 180 2x + 120 = 180 2x = 60 x = 30
Slide 91 / 209
55 What is the value of y?
A 120°
B 70°
C 55°
D 125°
70°
y°
Ans
wer
Slide 91 (Answer) / 209
55 What is the value of y?
A 120°
B 70°
C 55°
D 125°
70°
y°
[This object is a pull tab]
Ans
wer
D
Slide 92 / 209
50° x°
56 What is the value of x?
A 50°
B 25°
C 30°
D 130°
Ans
wer
Slide 92 (Answer) / 209
50° x°
56 What is the value of x?
A 50°
B 25°
C 30°
D 130°
[This object is a pull tab]
Ans
wer
B
Slide 93 / 209
57 Solve for x in the diagram.
A 3 2/3B 14
C 15
D 16
3x - 17
28 Ans
wer
Slide 93 (Answer) / 209
57 Solve for x in the diagram.
A 3 2/3B 14
C 15
D 16
3x - 17
28
[This object is a pull tab]
Ans
wer
C
Slide 94 / 209
Congruence &
Triangles
Return to Tableof Contents
Slide 95 / 209
CongruenceTwo figures are congruent if they have the exact size and shape (They are similar if they have the same shape, but a different size)
Congruent figures have a correspondence between their angles and sides where pairs of corresponding angles are congruent and pairs of corresponding sides are congruent.
A
B
C N
O
P
A
B
C N
O
P
Slide 96 / 209
ExampleThe two triangles are congruent , write: 1) a congruence statement 2) identify all congruent corresponding parts
Have students arrive at the answers as a class, or independently.
Slide 100 / 209
58 What is the corresponding part to J
A R
B K C Q D P
J
K L R Q
P
JKL PQR=~
Ans
wer
Slide 100 (Answer) / 209
58 What is the corresponding part to J
A R
B K C Q D P
J
K L R Q
P
JKL PQR=~
[This object is a pull tab]
Ans
wer
D
Slide 101 / 209
59 What is the corresponding part to Q
A R
B K C Q D P
J
K L R Q
P
JKL PQR=~
Ans
wer
Slide 101 (Answer) / 209
59 What is the corresponding part to Q
A R
B K C Q D P
J
K L R Q
P
JKL PQR=~
[This object is a pull tab]
Ans
wer
B
Slide 102 / 209
60 What is the corresponding part to QP
A JL
B LK C KJ D PQ
J
K L R Q
P
JKL PQR=~
Ans
wer
Slide 102 (Answer) / 209
60 What is the corresponding part to QP
A JL
B LK C KJ D PQ
J
K L R Q
P
JKL PQR=~
[This object is a pull tab]
Ans
wer
C
Slide 103 / 209
61 Write a congruence statement for the two triangles
A
B
C
D
Z
X
CV
B
Ans
wer
BVC XCZ=~
XCB BCX=~
VBC ZXC=~
CBV CZX=~
Slide 103 (Answer) / 209
61 Write a congruence statement for the two triangles
A
B
C
D
Z
X
CV
B
BVC XCZ=~
XCB BCX=~
VBC ZXC=~
CBV CZX=~
[This object is a pull tab]
Ans
wer
C
Slide 104 / 209
Y
ZW
X
What else can be marked congruent?
62 Complete the congruence statement
A
B
C
D
XYZ =~
XWZ
ZWX
WXZ
ZXW
Ans
wer
Slide 104 (Answer) / 209
Y
ZW
X
What else can be marked congruent?
62 Complete the congruence statement
A
B
C
D
XYZ =~
XWZ
ZWX
WXZ
ZXW
[This object is a pull tab]
Ans
wer
B
XZ ZX<W <Y
=~=~
Extension Answer
Slide 105 / 209
T5. Third Angles TheoremIf two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent.
Q
R
S
T
U
V
Can you give a reason for why this might be true?If the sum of the interior angles is 180 o and both sets of angles are the same, then the third angles will have the same measure. Example: m S = m V = 40o & m R = m U = 80o degrees, then m Q = m T = 60o.Click to reveal
Slide 106 / 209
ExampleFind the value of x.
A
B
C45° 75°
(2x+40)°W
X
Y
1) From the Third AngleTheorem (T5), we know m B = m Y
2) The m B is easy to find withthe Triangle Sum Theorem (T1),
3) Substitute to find x
Slide 107 / 209
S
Q
R48°117°
H
I
J
63 What is the measurement of J A
nsw
er
Slide 107 (Answer) / 209
S
Q
R48°117°
H
I
J
63 What is the measurement of J
[This object is a pull tab]
Ans
wer
48 + 117 + m<R = 180165 + m<R = 180m<R = 15o
m<J = m<R = 15o
Slide 108 / 209
I
J
K
80°
32°
P
Q
R(2x+14)°
64 Solve for x
Ans
wer
Slide 108 (Answer) / 209
I
J
K
80°
32°
P
Q
R(2x+14)°
64 Solve for x
[This object is a pull tab]
Ans
wer
80+32+m<I = 180112+m<I = 180m<I=68o
m<I = m<P68 = 2x+1454 = 2x27 = x
Slide 109 / 209
65 Find the value of x.
62° 78°Q
R
S
(3x+10)°
C
B
A Ans
wer
Slide 109 (Answer) / 209
65 Find the value of x.
62° 78°Q
R
S
(3x+10)°
C
B
A
[This object is a pull tab]
Ans
wer
62+78+m<R = 180140 + m<R = 180m<R = 40o
m<B = m<R3x+10 = 403x = 30 x = 10
Slide 110 / 209
T6. Properties of Congruent Triangles
Reflexive Property of Congruent TrianglesEvery triangle is congruent to itself A
B
C A
B
C
Symmetric Properties of Congruent Triangles
A
B
C D
E
F A
B
CD
E
F
Transitive Property of Congruent Triangles
A
B
C D
E
F D
E
F J
K
L A
B
C J
K
L
Slide 111 / 209
SSS Congruence
Return to Tableof Contents
Slide 112 / 209
From the Congruence and Triangles section, you learned that two triangles are congruent if the 3 corresponding pairs of sides and the 3 corresponding pairs of angles are congruent.
However, we do not always need all 6 pieces of information to prove 2 triangles congruent.
Slide 113 / 209
Postulate:
Side-Side-Side (SSS) CongruenceIf three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
A B
C
E
D
F
Click here to go to the lab titled, "Triangle Congruence SSS"
BGSolution:The congruence marks on the sides show that:
Slide 115 / 209
Example
F
GH
K
J
G
F
H
J
K
JJ
Slide 116 / 209
A
B
C H
J
K
You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ
66 The congruence statement is ABC = HJK
True
False
Hint
~A
nsw
er
Slide 116 (Answer) / 209
A
B
C H
J
K
You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ
66 The congruence statement is ABC = HJK
True
False
Hint
~
[This object is a pull tab]
Ans
wer
True
Slide 117 / 209
R
S
T U
67 SRT = SUT
True
False
~
Ans
wer
Slide 117 (Answer) / 209
R
S
T U
67 SRT = SUT
True
False
~
[This object is a pull tab]
Ans
wer True
SSS holds becauseST = ST ; Reflexive Property
Slide 118 / 209
A
B C Q R
S
3
4
5 3
4
5
68 ABC = _____?
A QRS B SRQ
C ACB
D RSQ
Ans
wer
~
Slide 118 (Answer) / 209
A
B C Q R
S
3
4
5 3
4
5
68 ABC = _____?
A QRS B SRQ
C ACB
D RSQ
~
[This object is a pull tab]
Ans
wer
B
Slide 119 / 209
SAS Congruence
Return to Tableof Contents
Slide 120 / 209
Included angle: the angle made by two lines with a common vertex
41 °
A
B
C
Included side: the side between two angles
C
D
E
Slide 121 / 209
Postulate:
Side-Angle-Side (SAS) CongruenceIf 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.
F
P
B
A
C
Q
Click here to go to the lab titled, "Triangle Congruence SAS"
Why Not SSA? Move the side with the length of 2 units and create a triangle.
2 units
200
5 units
Can a different triangle be made than the first one made?
Ans
wer
Slide 123 (Answer) / 209
200
5 units
Why Not SSA? Move the side with the length of 2 units and create a triangle.
2 units
200
5 units
Can a different triangle be made than the first one made?
[This object is a pull tab]
Ans
wer
200
5 units
2 units
Two different shapes with same sides and length
200
5 units2 units
Slide 124 / 209
69 What is the included angle of the given sides of the triangle?
A J
B K
C L
Hint: Draw the triangle!
JKL, sides KL and JK
Ans
wer
Slide 124 (Answer) / 209
69 What is the included angle of the given sides of the triangle?
A J
B K
C L
Hint: Draw the triangle!
JKL, sides KL and JK
[This object is a pull tab]
Ans
wer
B
Slide 125 / 209
P
QR
S
TV4 4
5 5
100° 100°
70 List the congruent parts of the triangles below. Is PQR = STV?
Yes
No
~
Ans
wer
Slide 125 (Answer) / 209
P
QR
S
TV4 4
5 5
100° 100°
70 List the congruent parts of the triangles below. Is PQR = STV?
Yes
No
~
[This object is a pull tab]
Ans
wer
Yes, by SAS
PQ = ST<Q = <TRQ = VT
~~~
Slide 126 / 209
F
GH
X
Y Z46° 46°
1010
77
Why?
71 Is FGH = XYZ by SAS?
Yes
No
Ans
wer
~
Slide 126 (Answer) / 209
F
GH
X
Y Z46° 46°
1010
77
Why?
71 Is FGH = XYZ by SAS?
Yes
No
~
[This object is a pull tab]
Ans
wer
No, the angles marked congruent are not the included angles in the triangles.
FG = XYHF = ZX<H = <Z
~~~
Slide 127 / 209
A B
C D
72 Using SAS, what information do you need to show ABC = DCB
A DBC = ACB B B = C
C ABD = DCA D ABC = DCB
~~
~
~
~
Ans
wer
Slide 127 (Answer) / 209
A B
C D
72 Using SAS, what information do you need to show ABC = DCB
A DBC = ACB B B = C
C ABD = DCA D ABC = DCB
~~
~
~
~
[This object is a pull tab]
Ans
wer
D
Slide 128 / 209
73 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
Ans
wer
Slide 128 (Answer) / 209
73 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
[This object is a pull tab]
Ans
wer
B
Slide 129 / 209
74 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
Ans
wer
Slide 129 (Answer) / 209
74 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
[This object is a pull tab]
Ans
wer
B
Slide 130 / 209
75 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
Ans
wer
Slide 130 (Answer) / 209
75 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
[This object is a pull tab]
Ans
wer
A
Slide 131 / 209
76 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
Ans
wer
Slide 131 (Answer) / 209
76 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
[This object is a pull tab]
Ans
wer
C
Slide 132 / 209
77 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
Ans
wer
Slide 132 (Answer) / 209
77 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent
[This object is a pull tab]
Ans
wer
C
Slide 133 / 209
78 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent45° 45°
12 12
Ans
wer
Slide 133 (Answer) / 209
78 What type of congruence exists between the two triangles?
A SSS
B SAS
C Not congruent45° 45°
12 12
[This object is a pull tab]
Ans
wer
B
Slide 134 / 209
Return to Tableof Contents
ASA Congruence
Slide 135 / 209
Postulate:
Angle-Side-Angle (ASA) CongruenceIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
R
Q S
T
U
V
Click here to go to the lab titled, "Triangle Congruence ASA"
Second: if it is not already marked, check and mark the diagram with that information,
Third: check your congruence postulates - what piece of information are you missing (side/angle) and where does it need to be for your chosen congruence?
Slide 138 / 209
W
X
Y
79 What is the included side for X and W?
A YX
B YW
C XW
Ans
wer
Slide 138 (Answer) / 209
W
X
Y
79 What is the included side for X and W?
A YX
B YW
C XW
[This object is a pull tab]
Ans
wer
C
Slide 139 / 209
W
X
Y
80 What is the included side for X and Y
A XW
B YX
C YW
Ans
wer
Slide 139 (Answer) / 209
W
X
Y
80 What is the included side for X and Y
A XW
B YX
C YW
[This object is a pull tab]
Ans
wer
B
Slide 140 / 209
M
N
O
P
81 What piece of information do we need to have ASA congruence between the two triangles?
A
B
C
D
Ans
wer
Slide 140 (Answer) / 209
M
N
O
P
81 What piece of information do we need to have ASA congruence between the two triangles?
A
B
C
D
[This object is a pull tab]
Ans
wer
C
NP = PN is true by the Reflexive Property. If you mark the diagram to show the corresponding congruent parts, you can see NP = PN is needed
~
~
Slide 141 / 209
A
B
C
D
82 What piece of information do we need to have ASA congruence between the two triangles?
A
B
C
D
Ans
wer
Slide 141 (Answer) / 209
A
B
C
D
82 What piece of information do we need to have ASA congruence between the two triangles?
A
B
C
D
[This object is a pull tab]
Ans
wer
B is true by the Reflexive Property. Since BDC is a right angle, BDA must also be a right angle since they form a linear pair. The included segment we need is
Slide 142 / 209
E
F G
M
H
83 Why is ?
A ASA
B vertical angles
C included angles
D congruent
Ans
wer
Slide 142 (Answer) / 209
E
F G
M
H
83 Why is ?
A ASA
B vertical angles
C included angles
D congruent
[This object is a pull tab]
Ans
wer
B
Mark the congruent vertical angles when you see two lines intersect
Slide 143 / 209
84 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
S
Q R
TU
Ans
wer
Slide 143 (Answer) / 209
84 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
S
Q R
TU[This object is a pull tab]
Ans
wer A
Slide 144 / 209
When you have overlapping figures that share sides and/or angles, marking the diagram with the given information & pulling the triangles apart (when needed) makes it much easier to understand the problem.
Slide 145 / 209
85 What type of congruence exists between the two triangles?A SSS
B SAS
C ASA
D Not congruent
J
L
M
N
K
L
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:
Ans
wer
click to reveal
click to reveal
click to reveal
Slide 145 (Answer) / 209
85 What type of congruence exists between the two triangles?A SSS
B SAS
C ASA
D Not congruent
J
L
M
N
K
L
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
[This object is a pull tab]
Ans
wer B
Slide 146 / 209
A
B
C
Q R
86 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
Mark the diagram with the given information. Be careful you don't always use all information
Hint
Ans
wer
click to reveal
Slide 146 (Answer) / 209
A
B
C
Q R
86 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
Mark the diagram with the given information. Be careful you don't always use all information
Hint
click to reveal
[This object is a pull tab]
Ans
wer C
Slide 147 / 209
C
B
Q R
A
B
Q R
87 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
Ans
wer
Slide 147 (Answer) / 209
C
B
Q R
A
B
Q R
87 What type of congruence exists between the two triangles?
A SSS
B SAS
C ASA
D Not congruent
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal[This object is a pull tab]
Ans
wer A
Slide 148 / 209
vertical
88 What type of congruence exists between the two triangles?
A SSSB SASC ASAD Not congruent
At the intersection of two lines you always have _____ angles.
Hint
ST
N
D
A
Ans
wer
Click to Reveal Click
Slide 148 (Answer) / 209
vertical
88 What type of congruence exists between the two triangles?
A SSSB SASC ASAD Not congruent
At the intersection of two lines you always have _____ angles.
Hint
ST
N
D
A
Click to Reveal Click[This object is a pull tab]
Ans
wer
B - SAS
Slide 149 / 209
89 What type of congruence exists between the two triangles?
A SSS
B SAS C ASA D Not Congruent
Ans
wer
Slide 149 (Answer) / 209
89 What type of congruence exists between the two triangles?
A SSS
B SAS C ASA D Not Congruent
[This object is a pull tab]
Ans
wer
We have two Sides and one Angle congruent, but they are not in the correct order.
D - Not Congruent
Slide 150 / 209
90 What type of congruence exists between the two triangles?
A SSS B SAS C ASA D Not Congruent
C
P M
S
A
Hint:
Mark the given information into your diagram. Identifying vertical angles plays an important part.
Ans
wer
click to reveal
Slide 150 (Answer) / 209
90 What type of congruence exists between the two triangles?
A SSS B SAS C ASA D Not Congruent
C
P M
S
A
Hint:
Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal
[This object is a pull tab]
Ans
wer
C - ASA
Slide 151 / 209
AAS Congruence
Return to Tableof Contents
Slide 152 / 209
Theorem (T7):
Angle-Angle-Side (AAS) CongruenceIf two angles and the nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent.
Q
R
S
T
U
V
Slide 153 / 209
Why is AAS a Theorem ?
Given two triangles:
AB
C
P R
K
75° 75°
65° 65°
12 12
The Triangle Sum Theorem (T1) allows us to find the measurement of the third angle in each triangle. 180°-(65°+75°)= 40°
AB
C
P R
K
75° 75°
65° 65°
12 1240°
Since AAS follows from ASA, AAS is a theorem rather than a postulate
Hypotenuse-Leg (HL) CongruenceIf the hypotenuse and a leg of one right triangle are equal to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent.
J
K L
M
N O
If you have a right triangle, make sure you check if HL applies
Slide 165 / 209
Why does HL Congruence work?Recall another theorem for right triangles:
c2 = a2 + b2 a
bc
Pythagorean Theorem:
If we know the lengths of two sides of a right triangle, we can solve for the length of the third side. HL Congruence theorem applies when the corresponding hypotenuse and one of the legs is congruent. When this is the case, the two right triangles are congruent.
A
B C E
FG
J
K L
M
N O
Slide 166 / 209
Example Are the two triangles congruent?
A B
C
R S
TThese are right triangles so let's try for HL congruence
A B
C
R S
T
Slide 167 / 209
Q
R S
X
Y Z
Mark the given on your diagram. Note that it is a right triangle.
99 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Given: QS = XZ RS = YZ
~~
Ans
wer
HintClick to reveal
Slide 167 (Answer) / 209
Q
R S
X
Y Z
Mark the given on your diagram. Note that it is a right triangle.
99 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Given: QS = XZ RS = YZ
~~
HintClick to reveal
[This object is a pull tab]
Ans
wer
EQ
R S
X
Y Z
QSR = XZYby HL congruence
~
Slide 168 / 209
If they are congruent what is the congruence statement?
100 What type of congruence exists, if any, between the two triangles?
A SSSB SASC ASAD AASE HLF Not congruent
L
M
N O
P
Q
Ans
wer
Slide 168 (Answer) / 209
If they are congruent what is the congruence statement?
100 What type of congruence exists, if any, between the two triangles?
A SSSB SASC ASAD AASE HLF Not congruent
L
M
N O
P
Q
[This object is a pull tab]
Ans
wer C
LMN = OPQ~
Slide 169 / 209
If they are congruent what is the congruence statement?
101 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
A
B
CD
E
F
Ans
wer
Slide 169 (Answer) / 209
If they are congruent what is the congruence statement?
101 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
A
B
CD
E
F
[This object is a pull tab]
Ans
wer F
Slide 170 / 209
If they are congruent what is the congruence statement?
102 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
T
U
V W
X
Y
Ans
wer
Slide 170 (Answer) / 209
If they are congruent what is the congruence statement?
102 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
T
U
V W
X
Y
[This object is a pull tab]
Ans
wer
C
TUV = WXY~
Slide 171 / 209
If they are congruent what is the congruence statement?
103 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Q
W
EY
Ans
wer
Slide 171 (Answer) / 209
If they are congruent what is the congruence statement?
103 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Q
W
EY
[This object is a pull tab]
Ans
wer
D
QWY = EWY~
Slide 172 / 209
If they are congruent what is the congruence statement?
104 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
N
M
O J
K
L
Ans
wer
Slide 172 (Answer) / 209
If they are congruent what is the congruence statement?
104 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
N
M
O J
K
L
[This object is a pull tab]
Ans
wer E
OMN = JKL~
Slide 173 / 209
If they are congruent what is the congruence statement?
105 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
E
F
G
H
Ans
wer
Slide 173 (Answer) / 209
If they are congruent what is the congruence statement?
105 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
E
F
G
H
[This object is a pull tab]
Ans
wer
E
EFH = GFH~
Slide 174 / 209
If they are congruent what is the congruence statement?
106 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
E
F
G
HA
nsw
er
Slide 174 (Answer) / 209
If they are congruent what is the congruence statement?
106 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
E
F
G
H
[This object is a pull tab]
Ans
wer B
EFH = GFH~
Slide 175 / 209
If they are congruent what is the congruence statement?
107 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
K F
B M
Ans
wer
Slide 175 (Answer) / 209
If they are congruent what is the congruence statement?
107 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
K F
B M
[This object is a pull tab]
Ans
wer A
KFB = MBF~
Slide 176 / 209
< POY and < UYO
If they are congruent what is the congruence statement?
P O
UY
What angles are congruent when parallel lines are cut by a transversal?
108 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Ans
wer
Click to Reveal
Slide 176 (Answer) / 209
< POY and < UYO
If they are congruent what is the congruence statement?
P O
UY
What angles are congruent when parallel lines are cut by a transversal?
108 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
Click to Reveal
[This object is a pull tab]
Ans
wer D
POY = UYO~
Slide 177 / 209
If they are congruent what is the congruence statement?
109 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
O K
MJ
Ans
wer
Slide 177 (Answer) / 209
If they are congruent what is the congruence statement?
109 What type of congruence exists, if any, between the two triangles?
A SSS
B SAS
C ASA
D AAS
E HL
F Not congruent
O K
MJ
[This object is a pull tab]
Ans
wer F
Slide 178 / 209
If they are congruent what is the congruence statement?
110 What type of congruence exists, if any, between the two triangles?
A SSSB SASC ASAD AASE HLF Not congruent
A S
XZ Ans
wer
Slide 178 (Answer) / 209
If they are congruent what is the congruence statement?
110 What type of congruence exists, if any, between the two triangles?
A SSSB SASC ASAD AASE HLF Not congruent
A S
XZ
[This object is a pull tab]
Ans
wer E
ASZ = XZS~
Slide 179 / 209
Triangle Congruence Proofs
Return to Tableof Contents
Slide 180 / 209
Congruent Reasons Summary(Drag ones that don't work out of the chart. Then put HL where it would belong.)
Mark the congruent segments from "Def. of midpoint step" in your diagram
G
F
H
J
K
Slide 183 / 209
In two-column proofs, the statements in the left column are justified by the reasons on the right-side column. As we read down the table, we can see the thought process laid out.
A D
ECB
1 2
Example
Statements Reasons
Note: for SAS, corresponding congruent sides and angles are needed, which we have.
Slide 184 / 209
Example
A
B
C
DStatements Reasons
1. Given, AC bisects BCD
click ___________
click ___________
click ___________
click ___________
A
Slide 185 / 209
Problem
Q R
S
T
click click
click ___________
click ___________click ___________
click ___________
Slide 186 / 209
Problem
D
FG
E
Statements Reasons
click ___________
click ___________
click ___________ click ___________
click ___________
Slide 187 / 209
Problem
GHJ
F
H is the midpoint of GJ
click
click ___________
click ___________
click ___________
click ___________
click ___________ click ___________
Slide 188 / 209
Statements Reasons
Problem
A
B
C
DT
click ___________
click ___________click ___________
___________
click ___________
click ___________
click ___________
Slide 189 / 209
Statements Reasons
ProblemD C
A B
lines__ __
click ___________
click ___________
click ___________
click ___________
click ___________
click ___________
click ___________
Slide 190 / 209Problem
P
Q R S
TGiven: R is the midpoint of QS, PQR and TSR are right 's, PR = TR~
__
__
click
click ___________
click ___________
click ___________click ___________
click ___________
click ___________
Slide 191 / 209
Statements Reasons
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Given: AC = BD, E is the midpoint of AB and CD
~
~Prove: AEC = BED
A
B
DC
E
Problem
Def. of midpoint
E is the midpoint of AB and CD
SSS
AC = BD~
Def. of midpoint
AE = BE~
Given~AEC = BED
CE = DE~Given
Teac
her N
otes
Slide 191 (Answer) / 209
Statements Reasons
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Given: AC = BD, E is the midpoint of AB and CD
~
~Prove: AEC = BED
A
B
DC
E
Problem
Def. of midpoint
E is the midpoint of AB and CD
SSS
AC = BD~
Def. of midpoint
AE = BE~
Given~AEC = BED
CE = DE~Given
[This object is a pull tab]
Teac
her N
otes
Given1)
2)
3)
4)
5)
Statements Reasons
1)
2)
3)
4)
5) SSS
Def. of midpoint
Def. of midpoint
AC = BD~
CE = DE~
AE = BE~
~AEC = BED
E is the midpoint of AB and CD Given
This example can be solved by matching the statements & reasons with their appropriate location.
Ans. is given below.
Slide 192 / 209
Return to Tableof Contents
CPCTCCorresponding Parts of Congruent Triangles are Congruent
Slide 193 / 209
CPCTC says that if two or more triangles are congruent by:
SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent.
Corresponding Parts of Congruent Triangles are Congruent
CPCTC
Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true.
Slide 194 / 209
Process for proving that two segments or angles are congruent
1. Find two triangles in which the two sides or two angles are corresponding parts
2. Prove that the two triangles are congruent (SSS, SAS, ASA, AAS, HL)
3. State that the two parts are congruent, using as the reason: "corresponding parts of congruent triangles are congruent"
Slide 195 / 209
MN
O
E L
111 Which two triangles might you try to prove congruent in order to prove
A
B
C
D
Ans
wer
Slide 195 (Answer) / 209
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111 Which two triangles might you try to prove congruent in order to prove
A
B
C
D
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wer B and D
Slide 196 / 209
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112 Which two triangles might you try to prove congruent in order to prove
112 Which two triangles might you try to prove congruent in order to prove
A
B
C
D
[This object is a pull tab]
Ans
wer A and C
Slide 197 / 209
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113 Which two triangles might you try to prove congruent in order to prove
A
B
C
D1 2
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Slide 197 (Answer) / 209
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113 Which two triangles might you try to prove congruent in order to prove
A
B
C
D1 2
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Ans
wer B and D
Slide 198 / 209
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114 Which two triangles might you try to prove congruent in order to prove
A
B
C
D
Ans
wer
Slide 198 (Answer) / 209
MN
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114 Which two triangles might you try to prove congruent in order to prove
A
B
C
D
[This object is a pull tab]
Ans
wer B and D
Slide 199 / 209
3. Given
4.
5.
6.
Statements Reasons
3. C is the midpoint of AD
4.
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6.
Problem
A
B
C D
E
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Problem A
B
C
D
DB bisects ABC ABD = CBD~
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Slide 201 / 209
Problem AB
C
D E
We are given that BCA = DCE, BC = CD, and B and D are right angles. Since all right angles are congruent, B = D. With the congruent angles and segments, we can conclude that ABC = EDC by ASA. Therefore, BA = DE by CPCTC.
A coordinate proof places a triangle, or any other geometric figure, into a coordinate plane.
A coordinate proof combines: - the geometric postulates, theorems, and properties, and - the Distance Formula and Midpoint Formula.
The only thing that changes from the proofs we have done earlier is you will need to use the Distance and/or Midpoint Formula to calculate side and segment lengths.
Slide 205 / 209
Midpoint Formula TheoremThe midpoint of a segment joining points with coordinates and is the point with coordinates
(x1, y1)(x2, y2)
The Distance FormulaThe distance 'd' between any two points with coordinates and is given by the formula:(x1, y1) (x2, y2)