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- Congruence of Figures - Corresponding Parts - Third Angle Theorem
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Geometry - 4.2Congruence &
Triangles
Congruent, Corresponding Angles/Sides
A P
B Q
C R
AB PQ
BC QR
CA RP
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.
ΔABC ≅ ΔPQR
Naming Congruent Parts
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
ΔABC ≅ ΔZXY
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 Theorem. - 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
Transitive Property of Congruent Triangles
Reflexive Property of Congruent Triangles
Symmetric Property of Congruent Triangles
ΔABC ≅ ΔABC
If ΔABC ≅ ΔDEF, then ΔDEF ≅ ΔABC
If ΔABC ≅ ΔDEF and ΔGHI ≅ ΔDEF, then ΔABC ≅ ΔGHI
Proof of Third Angle ThmGiven: <A ≅ <D, <B ≅ <E
Prove: <C ≅ <F
1. <A ≅ <D, <B ≅ <E 1. Given
2. m<A = m<D, m<B = m<E 2. Def’n of Congruent Angles
3. m<A + m<B + m<C = 180 3. Triangle Sum Theorem
4. m<D + m<E + m<F = 180 4. Triangle Sum theorem5. m<A + m<B + m<C = m<D + m<E + m<F 5. Transitive Property
6. m<D + m<E + m<C = m<D + m<E + m<F 6. Substitution Property
7. m<C = m<F 7. Subtraction Property
8. <C ≅ <F 8. Def’n of Congruent Angles
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
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,
ΔEFG ≅ ΔHJG
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 midpoint 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
Given:
O is the midpt of MQ and PN
Prove:
, ||MN QP MN PQ
ΔMNO ≅ ΔQPO
ΔMNO ≅ ΔQPO
Practice Problems
•Textbook p206: 14-32 even, 35