Download Congruent Figures

Congruent Triangles
Geometry (Holt 4-4)
K. Santos
Congruent Geometric Figures
Congruent Geometric Figures
* have the same size and shape
Congruent Figures/Triangles (Definition)
Congruent Figures----have congruent corresponding sides
and congruent corresponding angles
A
B
R
S
D
C
U
T
A corresponds with S
B corresponds with R
C corresponds with U
D corresponds with T
<A corresponds with <S 𝐴𝐵 corresponds with 𝑆𝑅
<B corresponds with <R 𝐵𝐶 corresponds with 𝑅𝑈
<C corresponds with <U 𝐶𝐷 corresponds with 𝑈𝑇
<D corresponds with <T 𝐷𝐴 corresponds with 𝑇𝑆
all the corrresponding parts are congruent---so ABCD ≅ SRUT
Congruent Corresponding Parts
The statement: ∆ABC ≅ ∆DEF tells you a lot of information.
It tells you about corresponding congruent angles….
<A≅<D
<B≅<E
<C≅<F
It tells you about corresponding congruent sides…
𝐴𝐵 ≅ 𝐷𝐸
𝐵𝐶 ≅ 𝐸𝐹
𝐴𝐶 ≅ 𝐷𝐹
Example—Corresponding parts
Given: ∆PQR ≅ ∆STW. Identify all pairs of congruent
corresponding parts.
<P ≅ <S
<Q ≅ <T
<R ≅ <W
𝑃𝑄 ≅ 𝑆𝑇
𝑄𝑅 ≅ 𝑇𝑊
𝑃𝑅 ≅ 𝑆𝑊
Example—Finding a missing side
8
Given: ∆ABC ≅ ∆DEF.
B
D
F
53
2x – 2
A
Find x.
AB = DE
2x - 2 = 6
2x = 8
x=4
Find m<F.
m<F = m<C
Find m<C first
53 + 90 = 143
180 – 143 = 57°
So m< F = 57°
6
C
E
10
Example—Finding missing angles
Given: ∆ABC ≅ ∆DBC.
A 50°
B
Find x.
2x – 16 = 90
2x = 106
x = 53°
Find m < D
m<D = m<A
m<A = 50°
So m<D = 50°
Find m<DBC.
90+ 50 =140
180-140=40
m<FBC = 40°
2x - 16
C
D
Proving two triangles
congruent (using definition)
Given: 𝐶𝐺 ≅ 𝐷𝐺
N is a midpoint of 𝐶𝐷
< C ≅< D
∆GNC 𝑎𝑛𝑑 ∆GND are right triangles
Prove: ∆GNC ≅ ∆GND
G
C
N
D
Proof
Statements
1. 𝐶𝐺 ≅ 𝐷𝐺
2. N is a midpoint of 𝐶𝐷
3. 𝐶𝑁 ≅ 𝐷𝑁
4. 𝐺𝑁 ≅ 𝐺𝑁
5. < C ≅ < D
6. ∆GNC 𝑎𝑛𝑑 ∆GND
are right triangles
7. <GNC and <GND
are right angles
8. <GNC ≅ < GND
9. <CGN ≅ < DGN
10. ∆GNC ≅ ∆GND
Reasons
1. Given
2. Given
3. definition of a midpoint
4. Reflexive Property
5. Given
6. Given
7. definition of a right
triangle
8. all right angles are congruent
(Right angles congruent Theorem)
9. Third angles theorem
10. Definition of congruent
triangles