Congruence is...
4.1 Definition of Congruent Triangles
when two figures have the exact same shape and size.
I CAN...
Corresponding Parts are...
Prove triangles and their corresponding parts congruent by definition of congruence. matching parts of two figures, including corresponding angles and corresponding sides.
Jul 30­10:36 AM
Jul 30­10:36 AM
A
H
Definition of Congruent Polygons
J
Two polygons are congruent if and only if their corresponding parts are congruent.
B
**For triangles remember:
Corresponding Parts of Congruent Triangles are Congruent.
H
K
∠ A ∠B ∠C ∠H ∠J ∠K AB JK BC AC HJ HK
≅
≅
A
J
≅
≅
≅
≅
Congruence Statement
C
B
∠B ∠C
∠A
K
Jul 30­10:36 AM
≅
∠J
∠H
∠K
Jul 30­10:36 AM
Corresponding Parts Example 1
Corresponding Parts Example 2
Show that the triangles are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.
In the diagram, ΔITP ≅ ΔNGO. Find the values of x and y. Q
C
Corresponding Parts Practice
Angles
Sides
X
Angles
S
R
Sides
Y
Z
Which corresponding parts are congruent? Where should we start with this problem?
Congruence Statement
Jul 30­10:36 AM
Jul 30­10:36 AM
1
Corresponding Parts Example 2
Corresponding Parts Example 2
Step 2) (Corresponding parts of congruent triangles are congruent)
Step 1) ∠O ≅ ∠P
(Corresponding parts of congruent triangles are congruent)
NG = IT
m∠O = m∠P
x – 2y = 7.5
6y – 14 = 40
x – 2(9) = 7.5
6y = 54
x – 18 = 7.5
y = 9
x = 25.5
Jul 30­10:36 AM
Jul 30­10:36 AM
Corresponding Parts Example 3
Corresponding Parts Example 3
In the diagram, ΔFHJ ≅ ΔHFG. Find the values of x and y.
In the diagram, ΔFHJ ≅ ΔHFG. Find the values of x and y.
Which corresponding parts are congruent? Do the two triangles have any sides/angles in common (Reflexive Property)?
Where should we start with this problem?
Jul 30­10:36 AM
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent.
Example: If ∠C ≅ ∠K, and ∠B ≅ ∠J, then ∠A ≅ ∠L.
J
Jul 30­10:36 AM
Congruent Triangles Example 3
A drawing of a tile contains a series
of triangles, rectangles, squares, and a circle. If ΔKLM ≅ ΔNJL, ∠KLM ≅ ∠KML and m∠KML = 47.5, find m∠LNJ.
K
B
Which corresponding parts are congruent?
A
L
What is a plan to solve the problem? C
Jul 30­10:36 AM
Jul 30­10:36 AM
2
Congruent Triangles Example 3
Step 1) ∠KLM ≅ ∠NJL and ∠KML ≅ ∠NLJ
(Corresponding parts in congruent triangles are congruent)
Congruent Triangles Example 3
Step 2) m∠LNJ + m∠NLJ + m∠NJL = 180
(Triangle Angle Sum Thm)
m∠LNJ + m∠NLJ + m∠NJL = 180
m∠LNJ + 47.5 + 47.5 = 180
m∠KLM = m∠NJL and m∠KML = m∠NLJ
m∠LNJ + 95 = 180
If ∠KLM ≅ ∠KML, then ∠NJL ≅ ∠NLJ
m∠LNJ = 85
If m∠KML = 47.5, then m∠KML = m∠NLJ = m∠NJL = 47.5.
Jul 30­10:36 AM
Jul 30­10:36 AM
Properties of Triangle Congruence
Reflexive Property of Triangle Congruence
ΔABC ≅ ΔABC
Symmetic Property of Triangle Congruence
If ΔABC ≅ ΔEFG, then ΔEFG ≅ ΔABC
Transitive Property of Triangle Congruence
If ΔABC ≅ ΔEFG and ΔEFG ≅ ΔJKL, then ΔABC≅ ΔJKL.
Jul 30­10:36 AM
3