Lecture 4.3
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Transcript of Lecture 4.3
Geometry - 4.3Congruent Triangles
Congruent, Corresponding Angles/Sides
A P
B Q
C R
AB PQ
BC QR
CA RP
ABC PQR
Two figures are congruent when their corresponding sides and corresponding angles are congruent.
Corresponding Angles
Corresponding Sides
There is more than one way to write a congruence statement, but the you must list the corresponding angles in the same order.
Naming Congruent Parts
ABC ZXY
A Z
B X
C Y
XY BC
YZ AC
XZ AB
Write a congruence statement for the triangles below. Identify all pairs of congruent parts.
Corresponding Angles Corresponding Sides
Identify Corresponding Congruent Parts
Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.
Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.
Sides:
Angles:
Third Angle Thm
A D B E C F
Third Angle Thm. - If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
If and then,
Properties of Congruent Triangles
ABC ABC
,If ABC DEF then DEF ABC
, ,If ABC DEF and DEF HIJ
then ABC HIJ
Transitive Property of Congruent Triangles
Reflexive Property of Congruent Triangles
Symmetric Property of Congruent Triangles
Proof of Third Angle Thm
• 1)• 2) m<A = m<D, m<B = m<E• 3) m<A + m<B + m<C = 180• 4) m<D + m<E + m<F = 180• 5) m<A + m<B + m<C =
m<D + m<E + m<F • 6) m<C = m<F • 7)
• 1) Given• 2) Def of congruent angles• 3) Triangle Sum Thm• 4) Triangle Sum Thm• 5) Substitution
• 6) Subtraction• 7) Def of congruent angles
A D B E and
A D B E
C F
C F
Given:
Prove:
Using the Third Angle Thm.
22 87 180
109 180
71
m A
m A
m A
4 15 71
4 56
14
m D m A
x
x
x
Find the value of x.
Determining Triangle Congruency
EFG HJG
Decide whether the triangles are congruent. Justify your reasoning.
From the diagram all corresponding sides are congruent and that <F and <H are congruent.
<EGF and <HGJ are congruent because of Vertical angles.
<E and <J are congruent because of the third angle theorem
Since all of the corresponding sides and angles are congruent,
Using Properties of Congruent Figures
ABCD KJHL 4 3 9
4 12
3
x
x
x
5 12 113
5 125
25
y
y
y
In the diagram,
a) Find the value of x.
b) Find the value of y.
Use Corresponding Parts of Congruent Triangles
In the diagram, ΔITP ΔNGO. Find the values of x and y.
O P
6y – 14 = 406y = 54
y = 9
x – 2y = 7.5
x – 2(9) = 7.5
x – 18 = 7.5
x = 25.5
Answer: x = 25.5, y = 9
A. x = 4.5, y = 2.75
B. x = 2.75, y = 4.5
C. x = 1.8, y = 19
D. x = 4.5, y = 5.5
In the diagram, ΔFHJ ΔHFG. Find the values of x and y.
2. LNM PNO 2. Vertical Angles Theorem
Proof:
Statements Reasons
3. M O
3. Third Angles Theorem
4. ΔLMN ΔPON
4. Def of Congruent Triangles
1. Given1.
Prove: ΔLMN ΔPON
Proving Two Triangles Congruent
• 1) O is the midpt of MQ and PN
• 2)• 3)• 4)• 5)
• 1) Given
• 2) Alt. Int. <‘s Thm.• 3) Vertical <‘s• 4) Def of Midpoint• 5) Def of Congruent Tri<‘s
, ||MN QP MN PQ
,MO QO PO NO
,OMN OQP MNO QPO MON QOP
MNO QPO
Given:
O is the midpt of MQ and PN
Prove:
, ||MN QP MN PQ
MNO QPO
Practice Problems
•Pg.257 #8,9-15(odds),19-23(odds),24
•HW Check Next Class