Geometry – Section 4.1 – Notes and Examples – Classifying Triangles
A triangle is a _________________ with __________ sides. Triangles can be classified ______ ways: by
their ___________ measures or their _________ lengths.
A
̅̅̅̅
̅̅̅̅ , and ̅̅̅̅
𝐴𝐵 , 𝐵𝐶
𝐴𝐶 are sides of triangle ABC.
B
C
Points 𝐴, 𝐵, and 𝐶 are the triangle’s vertices.
Acute
Equilateral
Classified by Angle Measure
Equiangular
Right
Classified by Side Length
Isosceles
Obtuse
Scalene
When you look at a figure, you cannot assume _________________ are _________________ based on
appearance. They must be _______________ as congruent.
Classify the following triangles by their angle measures.
Problem 1
Problem 2
Classify the following triangles by their sides lengths.
Problem 3
Problem 4
Problem 5
Find the side lengths of triangle JKL.
Problem 6
Find the side lengths of triangle FGH.
Geometry – Section 4.2 – Notes and Examples – Angle Relationships in Triangles
The __________ of the Triangle Sum Theorem uses an
_______________ line. An auxiliary _________ is a line
that is added to a figure to aid in a ____________.
A _______________ is a theorem whose ___________ follows directly from another _____________.
The ______________ is the set of all points ___________ the ______________. The _______________
is the set of all points ______________ the figure. An ______________ angle is formed by ______
sides of a triangle. An ________________ angle is formed by one _________ of the triangle and the
_________________ of an ________________ side.
∠1, ∠2, and ∠3 are interior angles
Exterior
∠4 is an exterior angle
Each ___________________ angle has two _____________ interior angles. A _________________
interior angle is an ________________ angle that is not ___________________ to the exterior angle.
Problem 1
Use the diagram to find 𝒎∠𝑿𝒀𝒁, 𝒎∠𝒀𝑿𝑾, and
𝒎∠𝒀𝑾𝑿.
Problem 2
The measure of one of the acute angles in a right
triangle is 𝟔𝟑. 𝟕°. What is the measure of the
other acute angle?
Problem 3
Find the 𝒎∠𝑩.
Problem 4
Find the 𝒎∠𝑨𝑪𝑫.
Problem 5
Find the
𝒎∠𝑱 𝐚𝐧𝐝
𝒎∠𝑲.
Problem 6
Find the
𝒎∠𝑷 𝐚𝐧𝐝
𝒎∠𝑻.
Geometry – Section 4.8 – Notes and Examples – Isosceles and Equilateral Triangles
Recall that an _________________ triangle has at least ________ congruent ___________. The
congruent sides are called the ________. The ____________ __________ is the angle formed by the
________. The side ______________ the vertex angle is called the _________, and the _________
__________ are the two angles that have the _________ as a side.
∠3 is the vertex angle.
∠1 and ∠2 are the base angles.
The Isosceles Triangle Theorem is sometimes stated as “Base angles of an _________________ triangle
are _________________.”
The following _________________ and its _______________ show the connection between
__________________ triangles and ____________________ triangles.
Corollary:
Converse:
Later on in the course we will work with coordinate proofs. A _________________ proof may be easier
if you place ______ side of the ______________ along the ___-_______ and locate a _____________ at
the ___________ or on the ___-________.
Problem 1
Find the 𝒎∠𝑭.
Problem 2
Find the 𝒎∠𝑮.
Problem 3
Find the 𝒎∠𝑯.
Problem 4
Find the 𝒎∠𝑵.
Problem 5
∆𝑳𝑴𝑲 is equilateral. Find the value of 𝒙.
Problem 6
∆𝑵𝑶𝑷 is equiangular. Find the value of 𝒚.
Problem 7
∆𝑱𝑳𝑲 is equiangular. Find the length of 𝑱𝑳.
Geometry – Section 4.3 – Notes and Examples – Congruent Triangles
Geometric figures are _________________ if they are the same _________ and same __________.
________________________ angles and corresponding _________ are in the same _____________ in
polygons with an ____________ number of _________. Two polygons are _____________________
polygons if and only if their corresponding __________ and sides are ________________. Thus
triangles that are the same ________ and __________ are congruent.
Two ____________ that are the _______________ of a side are called __________________ vertices.
To name a ______________, write the vertices in __________________ order. For example, you can
name polygon PQRS as QRSP or SRQP, but not as PRQS.
In a congruence statement, the order of the ____________ indicates the ________________________
parts.
When you write a statement such as 𝐴𝐵𝐶 ≅ 𝐷𝐸𝐹, you are also stating which parts are congruent.
Identify all pairs of corresponding congruent parts.
Problem 1
Problem 2
Given: ∆𝑷𝑸𝑹 ≅ ∆𝑺𝑻𝑾.
Given: polygon 𝑳𝑴𝑵𝑷 ≅ polygon 𝑬𝑭𝑮𝑯
Problem 3
Given: ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑩𝑪. Find the value of 𝒙 and 𝒎∠𝑫𝑩𝑪. Include
justifications.
Problem 4
Given: ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭. Find the value of 𝒙 and 𝒎∠𝑭.
Include justifications.
Problem 5
Given: ∠𝒀𝑾𝑿 and ∠𝒀𝑾𝒁 are right angles. ̅̅̅̅̅
𝒀𝑾 bisects ∠𝑿𝒀𝒁.
̅̅̅̅
̅̅̅̅
̅̅̅̅
W is the midpoint of 𝑿𝒁. 𝑿𝒀 ≅ 𝒀𝒁.
Prove: ∆𝑿𝒀𝑾 ≅ ∆𝒁𝒀𝑾
Statements
1. ∠𝑌𝑊𝑋 and ∠𝑌𝑊𝑍 are right angles
1.
2. ∠𝑌𝑊𝑋 ≅ ∠𝑌𝑊𝑍
2.
3. YW bisects ∠𝑋𝑌𝑍
3.
4. ∠𝑋𝑌𝑊 ≅ ∠𝑋𝑌𝑍
4.
5. W is midpoint of ̅̅̅̅
𝑋𝑍
5.
̅̅̅̅̅ ≅ 𝑍𝑊
̅̅̅̅̅
6. 𝑋𝑊
6.
7. ̅̅̅̅̅
𝑌𝑊 ≅ ̅̅̅̅̅
𝑌𝑊
7.
8. ∠𝑋 ≅ ∠𝑍
8
̅̅̅̅ ≅ 𝑌𝑍
̅̅̅̅
9. 𝑋𝑌
9.
10. ∆𝑋𝑌𝑊 ≅ ∆𝑍𝑌𝑊
10.
Reasons
Problem 6
̅̅̅̅ bisects 𝑩𝑬
̅̅̅̅. 𝑩𝑬
̅̅̅̅ bisects 𝑨𝑫
̅̅̅̅. 𝑨𝑩
̅̅̅̅ ≅ 𝑫𝑬
̅̅̅̅. ∠𝑨 ≅ ∠𝑫
Given: 𝑨𝑫
Prove: ∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑪
Statements
Reasons
1. ∠𝐴 ≅ ∠𝐷
1.
2. ∠𝐵𝐶𝐴 ≅ ∠𝐷𝐶𝐸
2.
3. ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐶
3.
4. ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐸
4.
̅̅̅̅ . 𝐵𝐸
̅̅̅̅ bisects ̅̅̅̅
5. ̅̅̅̅
𝐴𝐷 bisects 𝐵𝐸
𝐴𝐷.
5.
̅̅̅̅ ≅ ̅̅̅̅
6. 𝐵𝐶
𝐸𝐶 , ̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐷𝐶
6.
7. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶
7.