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Chapter 4
Congruent Triangles
Chapter Objectives
• Classification of Triangles by Sides• Classification of Triangles by Angles• Exterior Angle Theorem• Triangle Sum Theorem• Adjacent Sides and Angles• Parts of Specific Triangles• 5 Congruence Theorems for Triangles
Lesson 4.1
Triangles and Angles
Lesson 4.1 Objectives
• Identify the parts of a triangle• Classify triangles according to their sides• Classify triangles according to their angles• Calculate angle measures in triangles
Classification of Triangles by Sides
Name Equilateral Isosceles Scalene
Looks Like
Characteristics 3 congruent sides At least 2 congruent sides
No Congruent Sides
Classification of Triangles by Angles
Name Acute Equiangular Right Obtuse
Looks Like
Characteristics 3 acute angles
3 congruent angles
1 right angles
1 obtuse angle
Example 1
• You must classify the triangle as specific as you possibly can.
• That means you must name– Classification according to angles– Classification according to sides
• In that order!
• Example
Obtuse isosceles
Vertex
• The vertex of a triangle is any point at which two sides are joined.– It is a corner of a triangle.– There are 3 in every triangle
Adjacent Sides and Adjacent Angles
• Adjacent sides are those sides that intersect at a common vertex of a polygon.– These are said to be adjacent
to an angle.
• Adjacent angles are those angles that are right next to each other as you move inside a polygon.– These are said to be adjacent
to a specific side.
Special Parts in a Right Triangle
• Right triangles have special names that go with it parts.
• For instance:– The two sides that form the right angle are called the
legs of the right triangle.– The side opposite the right angle is called the
hypotenuse.• The hypotenuse is always the longest side of a right triangle.
legs
hypotenuse
Special Parts of an Isosceles Triangle
• An isosceles triangle has only two congruent sides– Those two congruent sides are called legs.– The third side is called the base.
legs
base
More Parts of Triangles
• If you were to extend the sides you will see that more angles would be formed.
• So we need to keep them separate– The three original angles are called interior angles because they
are inside the triangle.– The three new angles are called exterior angles because they lie
outside the triangle.
Example 2
Classify the following triangles by their sides and their angles.
Scalene
ObtuseScalene
RightIsosceles
Acute
Theorem 4.1:Triangle Sum Theorem
• The sum of the measures of the interior angles of a triangle is 180o.
A
B
C
mA + mB + mC = 180o
Example 3
Solve for x and then classify the triangle based on its angles.
3x + 2x + 55 = 180 Triangle Sum Theorem
5x + 55 = 180 Simplify
5x = 125 SPOE
x = 25 DPOE
75
50
Acute
Theorem 4.2:Exterior Angle Theorem
• The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
m A +m B = m C
A
B
C
Example 4
Solve for x
x = 50 + 70 Exterior Angle Theorem
x = 120 Simplify
Corollary to theTriangle Sum Theorem
• A corollary to a theorem is a statement that can be proved easily using the original theorem itself.– This is treated just like a theorem or a postulate in proofs.
• The acute angles in a right triangle are complementary.
C
A
BmA + mB = 90o
Homework 4.1
• In Class– 1-9
• p199-201
• In HW– 10-26, 31-39, 41-47, 49, 50, 52-68
• Due Tomorrow
Lesson 4.2
Congruence and Triangles
Lesson 4.2 Objectives
• Identify congruent figures and their corresponding parts.
• Prove two triangles are congruent.• Apply the properties of congruence to
triangles.
Congruent Triangles
• When two triangles are congruent, then – Corresponding angles are congruent.– Corresponding sides are congruent.
• Corresponding, remember, means that objects are in the same location.– So you must verify that when the triangles are drawn
in the same way, what pieces match up?
Naming Congruent Parts• Be sure to pay attention to the proper notation when
naming parts. ABC DEF
• This is called a congruence statement.
A
B
C
D
E
F A D B E C F
and
AB DEBC EFAC DF
Theorem 4.3:Third Angles Theorem
• If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Prove Triangles are Congruent
• In order to prove that two triangles are congruent, we must– Show that ALL corresponding angles are
congruent, and– Show that ALL corresponding sides are congruent.
• We must show all 6 are congruent!
Example 5Complete the following statements.a) Segment EF ___________
a) segment OPb) P ________
b) Fc) G ________
c) Qd) mO = ________
d) 110o
e) QO = ________e) 7 km
f) GFE __________f) QPO
• Yes, the order is important!
Theorem 4.4:Properties of Congruent Triangles
• Reflexive Property of Congruent Triangles ABC ABC
• Reflexive Property of
• Symmetric Property of Congruent Triangles– If ABC DEF, then DEF ABC.
• Symmetric Property of
• Transitive Property of Congruent Triangles– If ABC DEF and DEF JKL, then
ABC JKL.• Transitive Property of
Homework 4.2
• None!
Lesson 4.3
Proving Triangles are Congruent:SSS&
SAS
Lesson 4.3 Objectives
• Prove triangles are congruent using the SSS Congruence Postulate
• Prove triangles are congruent using the SAS Congruence Postulate
Postulate 19:Side-Side-Side Congruence Postulate
• If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Postulate 20:Side-Angle-Side 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.
Which One Do I Use?• Remember there are 6 parts to every triangle.
– Identify which parts of the triangle do you know (100% sure) are congruent.
– Rotate around the triangle keeping one thing in mind.• Cannot rotate so that 2 parts in a row are missed!• That means as you rotate by counting angle, then side, then angle,
then side, then angle, and then side you cannot miss two pieces in a row!
– You can skip 1, but not 2!!– Be sure the pattern that you find fits the same pattern in
the same way from the other triangle.• If it fits, they are congruent.
Example 6Decide whether or not the congruence statement is true.Explain your reasoning.
Reflexive Property of Congruence
The statement is true because ofSSS Congruence
Reflexive Property of Congruence
The statement is not true because the vertices areout of order.
The statement is not true because the vertices areout of order.Because the segment is shared between
two triangles, and yet it is the same segment
Example 7
Decide whether or not there is enough information to conclude SAS Congruence.
Reflexive Property of Congruence
Yes!
Yes!
No
Homework 4.3
• In Class– 1-5
• p216-218
• HW– 6-20
• Due Tomorrow
Lesson 4.4
Proving Triangles are Congruent:ASA
&AAS
Lesson 4.4 Objectives
• Prove that triangles are congruent using the ASA Congruence Postulate
• Prove that triangles are congruent using the AAS Congruence Theorem
Postulate 21:Angle-Side-Angle Congruence
• 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.
Theorem 4.5:Angle-Angle-Side Congruence
• If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.
Example 8
Complete the proof
Given
Given
Reflexive POC
SSS Congruence
Homework 4.4
• In Class– 1-7
• p223-227
• HW– 8-18 evens
• Due Tomorrow
Lesson 4.5
Using Congruent Triangles
Lesson 4.5 Objectives
• Observe that corresponding parts of congruent triangles are congruent
Showing Triangles are Congruent
• You only have 4 shortcuts right now to show that two triangles are congruent to each other.
1. SSS Congruence2. SAS Congruence3. ASA Congruence4. AAS Congruence
• Otherwise you need to show all 6 parts of a triangle have matching congruent parts to another triangle.
• If you can use one of the above 4 shortcuts to show triangle congruency, then we can assume that all corresponding parts of the triangles are congruent as well.
Surveying
MNP MKL– Given
• Segment NM Segment KM– Definition of a midpoint
LMK PMN– Vertical Angles Theorem
KLM NPM– ASA Congruence
• Segment LK Segment PN– Corresponding Parts of Congruent Triangles
Example 9Tell which triangles you show to be congruent in order to prove the
statement is true. What postulate or theorem would help you show the triangles are congruent.
Show:STV UTV
Reflexive Property of Congruence
STV UTV
SSS Congruence
Corresponding Parts of Congruent Triangles
Show:Segment XY Segment ZW
Reflexive Property of Congruence
Alternate Interior Angles Theorem (Parallel Lines)
WXZ YZX
ASA Congruence
Corresponding Parts of Congruent Triangles
Homework 4.5
• In Class– 1-3
• p232-235
• HW– 4-18, 25-36
• Due Tomorrow
Lesson 4.6
Isosceles,Equilateral,
andRight Triangles
Lesson 4.6 Objectives
• Use properties of isosceles and equilateral triangles.
• Identify more properties based on the definitions of isosceles and equilateral triangles.
• Use properties of right triangles.
Isosceles Triangle Theorems
• Theorem 4.6: Base Angles Theorem– If two sides of a triangle
are congruent, then the angles opposite them are congruent.
• Theorem 4.7: Converse of Base Angles Theorem– If two angles of a
triangle are congruent, then the sides opposite them are congruent.
Example 10
Solve for x
Theorem 4.7
4x + 3 = 15
4x = 12
x = 3
Theorem 4.6
7x + 5 = x + 47
6x + 5 = 47
6x = 42
x = 7
Equilateral Triangles
• Corollary to Theorem 4.6– If a triangle is
equilateral, then it is equiangular.
• Corollary to Theorem 4.7– If a triangle is
equiangular, then it is equilateral.
Example 11Solve for x
Corollary to Theorem 4.6
In order for a triangle to be equiangular, all angles must equal…
5x = 60
x = 12
Corollary to Theorem 4.6
It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest!
2x + 3 = 4x - 5
3 = 2x - 5
8 = 2x
x = 4
Theorem 4.8:Hypotenuse-Leg 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.– Abbreviate using
• HL
Example 12
Determine if enough information is given to conclude the triangles are congruent using HL Congruence
Reflexive Property of Congruence
Yes they are congruent!
Reflexive Property of Congruence
Neither triangle is a right triangle, so…
Not congruent
Homework 4.6
• In Class– 1-7
• p239-242
• HW– 8-28 even, 33, 34
• Due Tomorrow
Lesson 4.7
TrianglesAnd
Coordinate Proof
Lesson 4.7 Objectives
• Place geometric figures in a coordinate plane.• Use the Distance Formula to verify congruent
triangles.
Coordinate Proof• A coordinate proof involves placing geometric figures in a
coordinate plane.• Then you employ the following tools to prove concepts from
your picture– Distance Formula
– Midpoint Formula(x2 – x1)2 + (y2 – y1)2
(x1 + x2)
(y1 + y2)( ),2 2
Homework 4.7
• WS