Download Geometry * Chapter 4

GEOMETRY –
CHAPTER 4
Congruent Triangles
4.1 – Apply Angle Sum Properties
• Triangle
• Polygon with three sides & three vertices
• Triangles can be classified by side and angles
Classifying Triangles by Sides
Scalene
Iscoceles
Equilateral
No congruent
sides
Two congruent
sides
All sides congruent
Classifying Triangles by Angles
Acute
Right
Obtuse
Equiangular
3 acute angles
(< 90)
1 right angle (=
90)
1 obtuse angle
(> 90)
3 congruent
angles
Example 2
• Classify ∆PQO by its sides, then determine if the triangle
is right.
• Points are:
• P (-1, 2)
• Q (6, 3)
• O (0, 0)
GP: #1-2
Angles
• Interior Angles
• Angles on the inside of the triangle (there are three)
• Exterior Angles
• Angles that form linear pairs with interior angles (there are 6)
Theorems
• 4.1 – Triangle Sum Theorem
• The sum of the measure of the interior angles of a triangle is 180°
• 4.2 – Exterior Angle Theorem
• The measure of an exterior angle of a triangle is equal to the sum
of the measures of the two nonadjacent interior angles
Example #3 p. 209
Corollary to a theorem
• Corollary to a theorem
• Statement that can be proved easily using the theorem
• Corollary to the Triangle Sum Theorem
• The acute angles of a right triangle are complementary
• Example 4
• A tiled staircase forms a right triangle. The measure of one acute
angle in the triangle is twice the measure of the other. Find the
measure of each acute angle
GP #3 & 5 p. 210
4.2 – Apply congruence & triangles
• Two geometric figures are congruent if they have exactly
the same size and shape
• Congruent figures
• All parts of one figure are congruent to the corresponding parts of
the other figure (corresponding sides & corresponding angles)
• Congruence Statements
• Be sure to name figures by their corresponding vertices!
examples
• Example 1
• Writing a congruence statement and identifying all congruent parts
• Example 2
• Using properties of congruent figures
• 𝐷𝐸𝐹𝐺 ≅ 𝑆𝑃𝑄𝑅
GP #1-3 p. 216
Third angles theorem
• Theorem 4.3 – Third Angles Theorem
• If two angles of one triangle are congruent to two angles of another
triangle, then the third angles are also congruent
Using third angles theorem
• Example 4
• Find m< BDC
A
B
45°
N
30°
C
D
GP #4-5 p. 217
Properties of congruent triangles
• The properties of congruence that are true for segments
and angles are also true for triangles
• Theorem 4.4 – Properties of Congruent Triangles
• Reflexive property
• ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐶
• Symmetric property
• ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹, 𝑡ℎ𝑒𝑛 ∆𝐷𝐸𝐹 ≅ ∆𝐴𝐵𝐶
• Transitive property
• ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 𝑎𝑛𝑑 ∆𝐷𝐸𝐹 ≅ ∆𝑋𝑌𝑍 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍
4.3 – relate transformations & congruence
• Rigid motion
• Transformation that preserves length, angle measure, and area
• Examples of rigid motions (isometry): translations, reflections,
rotations
• Congruent figures and Transformations
• Two figures are congruent if and only if one or more rigid motions
can be used to move one figure onto the other. If any combination
of translations, reflections, and rotations can be used to move one
shape onto the other, the figures are congruent
4.4 – Prove triangles congruent by SSS
• Postulate 19 – Side-Side-Side (SSS) Congruence
Postulate
• If three sides of one triangle are congruent to three sides of a
second triangle, then the two triangles are congruent
Example 1
• Use the SSS congruence postulate
• GP #1-3 p. 232
4.5 – congruence by SAS and HL
• Postulate 20 – Side-Angle-Side (SAS) Congruence
Postulate
• If two sides and the included angle of one triangle are congruent to
two sides and the included angle of a second triangle, then the two
triangles are congruent
Right triangles
• In a right triangle, the sides adjacent to the right angles
are called the legs
• The side opposite the right angle is called the hypotenuse
• Theorem 4.5 – Hypotenuse-Leg (HL) Congruence
Theorem
• If the hypotenuse and a leg of a right triangle are congruent to the
hypotenuse and a leg of a second right triangle, then the two
triangles are congruent
4.6 – Prove using ASA & AAS
• Postulate 21 – Angle-Side-Angle (ASA) Congruence
Postulate
• If two angles and the included side of one triangle are congruent to
two angles and the included side of a second triangle, then the two
triangles are congruent
AAS theorem
• Theorem 4.6 – Angle-Angle-Side (AAS) Congruence
Theorem
• If two angles and a non-included side of one triangle are congruent
to two angles and the corresponding non-included side of a second
triangle, then the two triangles are congruent
Triangles postulates &theorems
• 5 methods for proving that triangles are congruent
SSS
All 3 sides
are
congruent
SAS
Two sides
and the
included
angle are
congruent
HL (right
angles only)
Hypotenuse
and one of
the legs are
congruent
ASA
AAS
Two angles
and the
included side
are
congruent
Two angles
and a nonincluded side
are
congruent
4.7 – use congruent triangles
• Congruent triangles have congruent corresponding parts
• If two triangles are congruent, their corresponding parts
must be congruent as well
Euclid’s river example
4.8 – use isosceles and equilateral
triangles
• Legs
• Two congruent sides of an isosceles triangle
• Vertex angle
• Angle formed by the legs
• Base
• Third side of an isosceles triangle
• Base angles
• Angles adjacent to the base (opposite the legs)
Isosceles triangles theorem
• Theorem 4.7 – Base Angles Theorem
• If two sides of a triangle are congruent, then the angles opposite
them are congruent
• Theorem 4.8 – Converse of Base Angles Theorem
• If two angles of a triangle are congruent, then the sides opposite
them are congruent
Example 1
• Name two congruent angles
F
D
E
GP #1-2 p. 264
Corollaries
• Corollary to the Base Angles Theorem
• If a triangle is equilateral, then it is equiangular
• Corollary to the Converse of Base Angles Theorem
• If a triangle is equiangular, then it is equilateral
• Example 2
• If a triangle is equilateral, what is the measure of each angle?
• Example 3 – on board
GP #5 p. 266