Download Geometry—Mrs. Dubler Chapter Four—Congruent Triangles Section

Geometry—Mrs. Dubler
Chapter Four—Congruent Triangles
Section 4.1
Triangle
Triangles and Angles
Classification of Triangles By Sides
2.
1.
3.
Classification of Triangles by
Angles
1.
3.
4.
2.
Example 1: When
classifying triangles
you need to be as
specific as possible.
Vertex
Adjacent sides
Legs
a. ΔABC has three acute angles
and no congruent sides.
Classify the triangle.
b. ΔDEF has one
obtuse angle and two
congruent sides.
Classify the triangle.
hypotenuse
base
Example 2:
Draw an isosceles obtuse triangle.
Draw a scalene acute
triangle.
Example 3:
Classify ΔPOQ by its
sides, then
determine if it is a
right triangle or not.
Interior angles
Exterior Angles
Triangle Sum
Theorem
Exterior Angle
Theorem
The __________ of the
measures of the
____________________
angles of a triangle is
______________.
The measure of an ________________
angle of a triangle is ____________ to
the ___________ of the measures of
the two _________________________
_________________ angles.
Example 4:
Find the value of x.
Corollary
Corollary to the
Triangle Sum
Theorem
Example 5:
The ____________ angles of a
______________ triangle are
______________________.
Find the value of x.
Section 4.2
Congruent figures
Corresponding
angles and sides
Example 1:
ΔABC≅ΔFED. List
all congruent sides
and angles.
Congruence and Triangles
Example 2:
In the diagram
DEFG≅SPQR.
a. Find the
value of x.
b. Find the
value of y.
Third Angles
Theorem
Example 3:
Find m<BDC.
Example 4:
Prove triangle
ABC≅CDA
If two ______________ of one triangle
are ______________ to two ____________
of another triangle, then the
___________ angles are also
____________________.
Example 5:
Decide whether or
not the triangles are
congruent. Justify
your reasoning.
Properties of Congruent Triangles
Reflexive Property
Symmetric Property
Section 4.3
Proving Triangles Congruent
SSS and SAS
If __________________ sides of one
triangle are __________________ to
________________ sides of a second
triangle, then the two triangles are
____________________.
Side-Side-Side
Congruent Postulate
Example 1: Prove
that ΔLMK≅ΔLMN
Transitive
Property
Example 2:
Which are the
coordinates of the
vertices of a triangle
congruent to ΔPQR?
a.
b.
c.
d.
(-1,1)(-1,4)(-4,5)
(-2,4)(-7,4)(-4,6)
(-3,2)(-1,3)(-3,1)
(-7,7)(-7,9)(-3,7)
Side-Angle-Side
Congruence
Postulate
If two _____________ and the
_________________ angle of one triangle
are _______________ to two _____________
and the __________________ angle of a
second triangle, then the two
triangles are ___________________.
Example 3: Prove
that ΔABC≅ΔCDA.
Example 4:
In the diagram QS
and RP pass through
the center of the
circle. What can you
conclude about
ΔMRS and ΔMPQ?
Section 4.4
Angle-Side-Angle
Congruence
Postulate
Proving Triangles are Congruent
ASA and AAS
If two _________________ and the
________________ side of one triangle
are __________________ to two
_______________ and the
_________________ side of second
triangle, then the two triangles are
______________________.
Example 1:
Example 1: Tell if
the triangles can be
proven congruent.
Angle-Angle-Side
If two _____________ and a
_____________________ side of one
triangle are _________________ to two
angles and the ______________________
____________________ side of another
triangle, then the two triangles are
__________________.
Example 2:
Prove ΔBAC≅ΔEDF
without using AAS.
Hypotenuse Leg
Congruence
Theorem
If the ____________________ and a
________ of a __________ triangle are
_________________ to the
_________________ and _________ of a
second right triangle, then the two
triangles are ______________________.
Example 3:
̅̅̅̅̅ ≅ ̅̅̅̅
Given:𝑊𝑌
𝑋𝑍,
̅̅̅̅̅
̅̅̅̅
𝑊𝑍 ⊥ 𝑍𝑌, 𝑎𝑛𝑑
̅̅̅̅ ⊥ 𝑍𝑌.
̅̅̅̅
𝑋𝑌
Prove: ΔWYZ≅ΔXZY
Example 4:
Are the pairs of
triangles congruent?
If so, justify your
answer.
Example 5:
In the diagram, ̅̅̅̅
𝐶𝐸 ⊥
̅̅̅̅
𝐵𝐷
∠𝐶𝐴𝐵 ≅ ∠𝐶𝐴𝐷
Show ΔABE≅ΔABD
Section 4.6
Base angles of an
isosceles triangle
Vertex Angle of an
isosceles triangle
Base Angles
Theorem
Converse of the
Base Angles
Theorem
Example 1:
In ΔDEF,
̅̅̅̅
𝐹𝐷 ≅ ̅̅̅̅
𝐸𝐷
Name two
congruent angles.
Isosceles, Equilateral, and Right Triangles
If two ________ of a triangles are ______________,
then the ___________ ______________ them are
________________.
If two ______________ of a triangle are
____________, then the _______________
_____________ them are _______________
Example 2:
a. If ̅̅̅̅
𝐻𝐺 ≅
̅̅̅̅,then
𝐻𝐾
_____≅_____
b. If ∠𝐾𝐻𝐽 ≅
∠𝐾𝐽𝐻, then
_____≅_____.
Corollary to the
base angles
theorem
Converse corollary
to the base angles
theorem
If a triangle is _____________, then it is
_______________
If a triangle is ____________, then it is
__________________.
Example 3:
Find the measures
of ∠𝑃, ∠𝑄, 𝑎𝑛𝑑 ∠𝑅.
Example 4:
Find the values of x
and y in the
diagram.
Example 5:
In the lifeguard
̅̅̅̅,
tower ̅̅̅̅
𝑃𝑆 ≅ 𝑄𝑅
𝑎𝑛𝑑 ∠𝑄𝑃𝑆 ≅
∠𝑃𝑄𝑅.
a. What
congruence
postulate
can you use
to prove
ΔQPS≅ΔPQ
R?
b. Explain
why ΔPQT
is isosceles.
c. Show
ΔPTS≅ΔQT
R.
Section 4.7
Triangles and Coordinate Proof
Coordinate Proof
Example 1:
Place a 2 unit by 6
unit rectangle in a
coordinate plane.
Example 2:
A right triangle has
legs of 5 units and
12 units. Place the
triangle in the
coordinate plane.
Label the
coordinates of the
vertices and find
the length of the
hypotenuse.
Example 3:
Place an isosceles
triangle in a
coordinate plane.
Find the length of
the hypotenuse,
and the midpoint,
M.
Example 4:
Right ΔOBC has leg
lengths of h units
and k units. You
can find the
coordinates of
points B and C by
considering how
the triangle is
placed in the
coordinate plane.
Find the length of
the hypotenuse
̅̅̅̅ .
𝑂𝐶
C (h,k)
h units
O
k units
B (h,0)