- Congruence of Figures - Corresponding Parts - Third Angle Theorem

1. Geometry - 4.2
Congruence & Triangles

2. Congruent, Corresponding Angles/Sides
Two figures are congruent when their corresponding sides
and corresponding angles are congruent.
Corresponding Angles
< A ≅< P
< B ≅< Q
ΔABC ≅ ΔPQR
< C ≅< R
Corresponding Sides
AB ≅ PQ
There is more than one way to write a
BC ≅ QR congruence statement, but the you must list
CA ≅ RP the corresponding angles in the same order.

4. Naming Congruent Parts
Write a congruence statement for the triangles below. Identify
all pairs of congruent parts.
ΔABC ≅ ΔZXY
Corresponding Angles Corresponding Sides
< A ≅< Z XY ≅ BC
< B ≅< X YZ ≅ AC
< C ≅< Y XZ ≅ AB

5. Identify Corresponding Congruent Parts
Show that the polygons are
congruent by identifying all of
the congruent corresponding
parts. Then write a congruence
statement.
Angles:
Sides:
Answer: All corresponding parts of the two polygons are
congruent. Therefore, ABCDE ≅ RTPSQ.

6. Third Angle Thm
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 < A ≅< D and < B ≅< E then, < C ≅< F

7. Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
ΔABC ≅ ΔABC
Symmetric Property of Congruent Triangles
If ΔABC ≅ ΔDEF, then ΔDEF ≅ ΔABC
Transitive Property of Congruent Triangles
If ΔABC ≅ ΔDEF and ΔGHI ≅ ΔDEF,
then ΔABC ≅ ΔGHI

8. Proof of Third Angle Thm
Given: <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 theorem
5. 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

9. Using the Third Angle Thm.
Find the value of x.
22 + 87 + m < A = 180
109 + m < A = 180
m < A = 71
m<D=m< A
4 x + 15 = 71
4 x = 56
x = 14

10. 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.
Since all of the corresponding
sides and angles are congruent,
<E and <J are congruent because of
the third angle theorem
ΔEFG ≅ ΔHJG

11. Using Properties of Congruent Figures
In the diagram, ABCD ≅ KJHL 4x − 3 = 9
a) Find the value of x.
4 x = 12
b) Find the value of y.
x=3
5 y − 12 = 113
5 y = 125
y = 25

12. Use Corresponding Parts of Congruent Triangles
In the diagram, ΔITP ≅ ΔNGO. Find the values of x and y.
x – 2y = 7.5
x – 2(9) = 7.5
x – 18 = 7.5
∠O ≅ ∠P x = 25.5
6y – 14 = 40
Answer: x = 25.5, y = 9
6y = 54
y= 9

13. In the diagram, ΔFHJ ≅ ΔHFG. Find the values of x and y.
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

14. Prove: ΔLMN ≅ ΔPON
Proof:
Statements Reasons
1. 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Theorem
3. ∠M ≅ 3. Third Angles Theorem
∠O
4. ΔLMN 4. Def of Congruent Triangles
≅

15. Proving Two Triangles Congruent
Given: MN ≅ QP, MN || PQ
O is the midpt of MQ and PN
Prove: ΔMNO ≅ ΔQPO
• 1) O is the midpoint of MQ and PN • 1) Given
MN ≅ QP, MN || PQ
• 2) < OMN ≅< OQP, < MNO ≅< QPO • 2) Alt. Int. <‘s Thm.
• 3) < MON ≅< QOP • 3) Vertical <‘s
• 4) MO ≅ QO, PO ≅ NO • 4) Def of Midpoint
• 5) ΔMNO ≅ ΔQPO • 5) Def of Congruent Tri<‘s

16. Practice Problems
• Textbook p206: 14-32 even, 35