“Life is what you make it.” –Mr. H’s dad
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“Life is what you make it.” –Mr. H’s dad
You can classify triangles by if their sides are congruent.
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
This triangle has no congruent sides.
This triangle has at least 2 congruent sides
This triangle has all sides being congruent.
You can classify a triangle by the angles that form the triangle.
Acute Triangle
Right Triangle
Obtuse Triangle
Equiangular Triangle
All angles of the triangle are acute.
One angle of the triangle is a right angle.
One angle of the triangle is an obtuse angle.
All angles of the triangle are congruent.
A triangle has the given vertices. Graph the triangle and classify it by its sides. Then determine if it is a right triangle.
1) A(2, 3), B(6, 3), C(2, 7) ◦ Isosceles right triangle
2) A(1,9), B(4, 8), C(2, 5) ◦ Scalene triangle
Triangle Sum Theorem: ◦ The sum of the measure of the interior angles of a
triangle is 180°.
Interior Angles
A
B C
m∠A+m∠B+m∠C=180°
Find the measure of the interior angles of the given triangles.
55°
A
B C
D
E F
G
H I
(3x)°
(2x+10)°
(x-10)°
(2x)°
(3x)°
(x)°
1) 2)
3)
Exterior angleTheorem: ◦ The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two nonadjacent interior angles.
Exterior Angles
A
B C
m∠A+m∠C=m∠1
No special name
1
Find the value of the numbered angles.
1 2
3 4
5 6
10°
50°
Find the value of the numbered angles.
1 2
3 4
5 6
10°
50°
Find the value of the numbered angles.
1 2
3 4
5 6
10°
50°
Find the value of x.
(x+80)°
40°
(3x)°
“Bad is never good until worse happens.” –Danish Proverb
Congruent-same size and shape.
Figures-two dimensional shapes.
Congruent Figures-Two shapes that are the same size. ◦ All parts of one figure are congruent to the
corresponding parts of the other figure.
A
B C
D
E
F
G H
J
K
ABCDE≅FGHJK
This is a congruence statement!
A
B C
D
E
F
G H
J
K
ABCDE≅FGHJK
Corresponding Angles ∠A ≅∠F ∠B ≅∠G ∠C ≅∠H ∠D ≅∠J ∠E ≅∠K
Corresponding Sides 𝐴𝐸 ≅ 𝐹𝐾 𝐸𝐷 ≅ 𝐾𝐽 𝐷𝐶 ≅ 𝐽𝐻 𝐶𝐵 ≅ 𝐻𝐺 𝐴𝐵 ≅ 𝐹𝐺
A
B C
D
E
F
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
Corresponding Angles ∠A ≅∠D ∠B ≅∠E ∠C ≅∠F
Corresponding Sides 𝐴𝐵 ≅ 𝐷𝐸 𝐵𝐶 ≅ 𝐸𝐹 𝐴𝐶 ≅ 𝐷𝐹
Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.
∆𝑀𝑁𝑃 ≅ ∆𝑄𝑅𝑆
Corresponding Sides 𝑀𝑁 ≅ 𝑄𝑅 𝑁𝑃 ≅ 𝑅𝑆 𝑀𝑃 ≅ 𝑄𝑆
Corresponding Angles ∠M ≅∠Q ∠N ≅∠R ∠P ≅∠S
Third Angles Theorem ◦ If two angles of one triangle are congruent to two
angles of another triangle, then the third angles are also congruent.
A
B C
D
E F
If ∠A≅∠D and ∠B≅∠E, then ∠C≅∠F .
Find the value of x.
50° 50° (3x)° (x+30)°
1)
2)
(2x)° 50°
Reflexive Property ◦ For any triangle ABC, ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐶.
◦ A triangle is congruent to itself!
Symmetric Property ◦ If ∆𝐴𝐵𝐶 ≅ ∆DEF, then ∆𝐷𝐸𝐹 ≅ ∆𝐴𝐵𝐶.
Transitive Property ◦ If ∆𝐴𝐵𝐶 ≅ ∆DEF and ∆𝐷𝐸𝐹 ≅ ∆𝐺𝐻𝐼, then ∆𝐴𝐵𝐶 ≅
∆𝐺𝐻𝐼.
◦ Two triangles congruent to the same triangle are congruent to each other.
Name the property illustrated. ◦ If ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐻𝐼, 𝑡ℎ𝑒𝑛 ∆𝐺𝐻𝐼 ≅ ∆𝐴𝐵𝐶.
Symmetric Property
◦ If ∆𝐶𝐵𝐴 ≅ ∆𝐻𝐼𝐹 and ∆𝐻𝐼𝐹 ≅ ∆𝑇𝑂𝑅, then ∆𝐶𝐵𝐴 ≅ ∆𝑇𝑂𝑅.
Transitive Property
◦ ∆𝐷𝐸𝐹 ≅ ∆𝐷𝐸𝐹.
Reflexive Property
Step 1: Write given. ◦ Make a two column proof.
Step 2: Show all sides are congruent to corresponding sides. ◦ Use Reflexive property, definition of midpoint, etc.
Step 3: Show all angles are congruent to corresponding angles. ◦ Use AIA, AEA, CA, CIA, Third angle theorem, etc.
Step 4: Write congruence statement. ◦ Ex. ∆𝐴𝐵𝐶 ≅ ∆DEF
Given:𝐴𝐶 ≅ 𝐶𝐸, 𝐵𝐶 ≅ 𝐶𝐷, 𝐴𝐵 ≅ 𝐷𝐸, 𝐴𝐵||𝐷𝐸
Prove:∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶
A
B C D
E
“Things could be worse. Suppose your errors were counted and published every day, like those
of a baseball player.” –Anon.
Flow Proof-uses a flow chart to show reasons and statements of proofs.
Main reason to use this is because it shows cause and effect.
Flow Proof-uses a flow chart to show reasons and statements of proofs.
Statement 1
Reason 1
Statement 2
Reason 2
Statement 3
Reason 3
Statement 4
Reason 4
These are usually from the diagram/given of the problem.
These are the effect of the statement before.
Prove the Alternate Interior Angles Converse Theorem.
Given: ∠1≅∠2
Prove: mlln
1
2
3
m
n
Statements Reasons
1. ∠1≅∠2 1. Given
2. ∠3≅∠2 2. Vertical Angles Congruence Theorem
3. ∠3≅∠1 3. Transitive Property
4. mlln 4. Corresponding Angles Converse Postulate
Statements Reasons
1. ∠1≅∠2 1. Given
2. ∠3≅∠2 2. Vertical Angles Congruence Theorem
3. ∠3≅∠1 3. Transitive Property
4. mlln 4. Corresponding Angles Converse Postulate
∠1≅∠2
Given
∠3≅∠2
Vertical Angles Congruence Theorem
∠3≅∠1
Transitive Property
m||n
Corresponding Angles Converse Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Corresponding Sides 𝑀𝑁 ≅ 𝑄𝑅 𝑁𝑃 ≅ 𝑅𝑆 𝑀𝑃 ≅ 𝑄𝑆
This causes
∆𝑀𝑁𝑃 ≅ ∆𝑄𝑅𝑆
Given:C is the midpoint of 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐷, 𝐴𝐵 ≅ 𝐷𝐸,
Prove:∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶 A
B
C
D
E 𝐴𝐵 ≅ 𝐷𝐸
Given
𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐸 𝑎𝑛𝑑 𝐵𝐷
Given
𝐴𝐶 ≅ 𝐶𝐸
Definition of Midpoint
𝐵𝐶 ≅ 𝐶𝐷
∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶
SSS Congruence Postulate
Definition of Midpoint
Is ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹? The coordinates of the vertices are A(1, 4), B(4, 4), C(1, -1), D(-2, -2), E(-5, -2), and F(-2, 3). Explain.
Use distance formula to check to see if AB=DE, BC=EF, and AC=DF.
If yes, then triangles are congruent.
If no, then triangles are not congruent
Because there is only one possible triangle with three given sides, they are the most stable shape.
Two shapes that use the same four lengths.
Makes it stable because now we have two triangles that don’t change.
Are the shapes below stable?
1) 2)
3)
“The mind of man(kind) is capable of anything – because everything is in it, all the past as well as
all the future.” –Joseph Conrad
Side-Angle-Side (SAS) Congruence Postulate: ◦ 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.
A
B C
D
E F
Corresponding Sides 𝐴𝐵 ≅ 𝐷𝐸 𝐴𝐶 ≅ 𝐷𝐹
∠A≅ ∠𝐷
This causes
∆𝐵𝐴𝐶 ≅ ∆𝐸𝐷𝐹
Included angle because it is between two congruent sides.
Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate.
1) 2)
3)
What is the third piece of information to show the triangle is congruent using the SAS Congruence Postulate.
1. 𝐴𝐵 ≅ 𝐷𝐸, 𝐴𝐶 ≅ 𝐷𝐹, __________
2. ∠A≅ ∠D, 𝐴𝐶 ≅ 𝐷𝐹, __________
3.∠F≅ ∠C, 𝐴𝐶 ≅ 𝐷𝐹, __________
4. 𝐴𝐵 ≅ 𝐷𝐸, 𝐵𝐶 ≅ 𝐸𝐹, __________
A
B C
D
E F
SSA and the triangles are not congruent. There is no SSA Congruence Postulate because the triangles don’t have to be congruent. One exception (HL)
Sides adjacent to the right angle are legs
Side opposite (across) from the right angle is the hypotenuse.
Hypotenuse
Leg
Leg
What are the parts of the triangle called.
a is the/a __________
b is the/a ___________
c is the/a ___________
C is the/a ___________ a
c
b
C
Hypotenuse-Leg (HL) Congruence Theorem
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. A
B
D
C
E F
Corresponding Sides 𝐴𝐵 ≅ 𝐷𝐸 𝐴𝐶 ≅ 𝐷𝐹
∠C and ∠𝐹 are right angles
This causes
∆𝐵𝐴𝐶 ≅ ∆𝐸𝐷𝐹
Decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem.
1) 2)
Given: 𝐴𝐵 ≅ 𝐵𝐷
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶
B
D
C
A
𝐴𝐵 ≅ 𝐵𝐷
Given
∠𝐴 𝑎𝑛𝑑 ∠𝐷 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠.
Given (Diagram) 𝐵𝐶 ≅ 𝐵𝐶
Reflexive Property
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶
HL Congruence Theorem
∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆𝐷𝐵𝐶 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
Definition of Right Triangle
“That which is bitter to endure may be sweet to remember.”
–Thomas Fuller
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.
A
B C
D
E F
Corresponding Parts
𝐴𝐵 ≅ 𝐷𝐸 ∠𝐵 ≅∠E
∠A≅ ∠𝐷
This causes
∆𝐵𝐴𝐶 ≅ ∆𝐸𝐷𝐹
Included side because it is between two congruent 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.
A
B C
D
E F
Corresponding Sides/Angles
𝐵𝐶 ≅ 𝐸𝐹 ∠𝐵 ≅∠E ∠A≅ ∠𝐷
This causes
∆𝐵𝐴𝐶 ≅ ∆𝐸𝐷𝐹
Explain how you can prove the triangles congruent. If it is not possible, state it.
1) 2)
3) 4)
A
B C
D
E F
A
B C
D
E
A
B C
D
E
A
B C
D
E
F
F
F
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
Is ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹? Explain how you know.
1. ∠A≅ ∠D, 𝐴𝐶 ≅ 𝐷𝐹, and ∠B≅ ∠E
2. ∠C≅ ∠F, 𝐴𝐶 ≅ 𝐷𝐹, and ∠B≅ ∠E
3. ∠A≅ ∠D, 𝐴𝐶 ≅ 𝐷𝐹, and ∠C≅ ∠F
4. 𝐴𝐵 ≅ 𝐸𝐹, 𝐴𝐶 ≅ 𝐷𝐹, and ∠B≅ ∠E
5. ∠A≅ ∠D, ∠C ≅ ∠F, and ∠B≅ ∠E
A
B C
D
E F
SSS Congruence Postulate
SAS Congruence Postulate
HL Congruence Theorem
ASA Congruence Postulate
AAS (SAA) Congruence Theorem
Doesn’t work to show triangles congruent:
SSA
AAA
Given: 𝐹𝐸 ≅ 𝐷𝐸, ∠A≅∠C
Proven: ∆𝐴𝐷𝐸 ≅ ∆𝐶𝐹𝐸
A B C
D
E
F
𝐹𝐸 ≅ 𝐷𝐸
Given
∠A≅∠C
Given ∠𝐸 ≅ ∠E
Reflexive Property
∆𝐴𝐷𝐸 ≅ ∆𝐶𝐹𝐸
AAS Congruence Postulate
A
D
E
C
E
F
“Memory is the thing you forget with.” –Alexander Chase
By definition, if congruent triangles have congruent corresponding parts. We can use this idea to find distances across rivers and other hard to measure distances.
Explain how you use the given information to prove that the parts are congruent.
Given: ∠CAB≅∠CAD, ∠ACB≅∠ACD
Prove: 𝐵𝐶 ≅ 𝐶𝐷
The first step is to show the two triangles are congruent. ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶 are congruent because ASA congruence postulate because of
the two angles given and 𝐴𝐶 ≅ 𝐴𝐶.
Now that I have the triangles being congruent,
𝐵𝐶 ≅ 𝐶𝐷 by Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
A
B
C
D
Use this information to write a plan for a proof.
Given: ∠1≅∠2, ∠3≅∠4
Prove: ∆𝐵𝐶𝐸 ≅ ∆𝐷𝐶𝐸
In ∆𝐵𝐶𝐸 𝑎𝑛𝑑 ∆𝐷𝐶𝐸 we know ∠1≅∠2 and 𝐶𝐸 ≅ 𝐶𝐸. If we can show that 𝐵𝐶 ≅ 𝐶𝐷, then ∆𝐵𝐶𝐸 ≅ ∆𝐷𝐶𝐸.
In ∆𝐵𝐶𝐴 𝑎𝑛𝑑 ∆𝐷𝐶A we know ∠1≅∠2, ∠3≅∠4, and 𝐶𝐴 ≅ 𝐶𝐴. Therefore ∆𝐵𝐶𝐴 ≅ ∆𝐷𝐶A by ASA congruence postulate.
Therefore 𝐵𝐶 ≅ 𝐶𝐷 by CPCTC and consequently ∆𝐵𝐶𝐸 ≅ ∆𝐷𝐶𝐸 by SAS congruence postulate.
A
B
C
D
E 1
2
3
4
Prove the distance across the river is 1 mile. Where segment AB is across the river and segment BE is along the riverside.
You also know 𝐵𝐶 ≅ 𝐶𝐸 and DE=1 mile.
A B
C
D E
A B
C
D E
𝐵𝐶 ≅ 𝐶𝐸
Given
∠E≅∠B
All right ∠’s are ≅
∠𝐵𝐶𝐴 ≅ ∠ECD
Vertical Angles are Congruent
∆𝐴𝐷𝐸 ≅ ∆𝐶𝐹𝐸
AAS Congruence Postulate
∠E and ∠B are right angles
Given(Diagram)
𝐵𝐴 ≅ 𝐷𝐸
CPCTC
BA=DE
Definition of Congruence
DE=1 mile 𝐴𝐵 = 1 𝑚𝑖𝑙𝑒
Given Substitution Property
“Memory is the thing you forget with.” –Chinese proverb
Congruent sides are called legs.
Non-congruent side is called the base.
Angle opposite the base is called the vertex.
Angles adjacent to the base are called the base angle(s).
Leg Leg
Base
Vertex
Base angles
What are the parts of the triangle.
a is the/a __________
b is the/a __________
c is the/a __________
A is the/a __________
B is the/a __________
C is the/a __________
A
B
C
a
b
c
Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
Converse to the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
Find the value of x in each problem.
1) 2)
3) 4)
(5x)° (65)° (45)° (5x)°
x ft 15 ft
(3x)°
6m
Corollary to the Base Angles Theorem
If a triangle is equilateral, then it is equiangular.
Corollary to the Converse of the Base Angles Theorem
If a triangle is equiangular, then it is equilateral.
Find the value of x and y.
5 x
y
(6x)°
x 16
18°
y°
1) 2)
3)
“How is it that we remember the least triviality that happens to us, and yet not remember how often
we have recounted it to the same person?” –La Rochefoucauld
Transformation is an operation that moves or changes a geometric figure in some way to produce a new figure called an image.
Translation moves every point of a figure the same distance in the same direction.
Reflection uses a line of reflection to create a mirror image of the original figure.
Rotation turns a figure about a fixed point, called the center of rotation.
Line of reflection
Center of rotation
Name the type of transformation demonstrated in each picture.
1) 2)
3)
This notation can describe any translation.
(x,y) (x+a, y+b)
a is how far the image has moved horizontally (positive is to the right and negative is to the left)
b is how far the image has moved vertically (positive is upward and negative is downward).
Use coordinate notation to describe the translation.
1) 4 units to the right and 3 units up.
(x,y)(x+4,y+3)
2) 6 units to the left and 2 units down.
(x,y)(x-6,y-2)
3) 5 units to the left and 7 units up.
(x,y)(x-5,y+7)
Draw the figure after the given translation.
(x,y)(x+11,y+8)
A
B C
D(6,7)
E(3,1)
F(9,1)
Reflection over the x-axis.
(x, y) (x, -y)
Reflection over the y-axis.
(x, y) (-x, y)
A point on the original figure and the transformation is given. Find the corresponding point on the image.
1) Point on original shape: (5,4); Transformation: (x,y)(x+2, y-1)
Point on image: (5+2, 4-1) or (7, 3)
2) Point on original shape: (-3,2); Transformation: (x,y)(x,-y)
Point on image: (-3, -2)
A point on the image and the transformation is given. Find the corresponding point on the original figure.
1) Point on image: (7,6); Transformation: (x,y)(x+2, y-1)
Point on original shape: (7-2, 6+1) or (5, 7)
2) Point on image: (-2,5); Transformation: (x,y)(x,-y)
Point on original shape: (-2, -5)
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