Trigonometry Lecture Notes Section 1.2 Page 4 of 6
Section 1.2: Angle Relationships and Similar Triangles
Big Idea: We continue with our review of angles and triangles, which form the foundation of trigonometry. Similar triangles are especially important.
Big Skill: You should be able to compute angles from given information and triangle side lengths for similar triangles.
Geometric Properties of Angles
Vertical angles are on opposite sides of the intersection point of two lines.
· Picture:
· Vertical angels have equal measures.
· Proof:
Parallel lines lie in the same plane and do not intersect.
· Picture:
A transversal is a line that intersects two parallel lines.
· Picture:
· Different angles formed by the transversal are given special names to reflect special relationships between their measures:
o Corresponding angles have equal measures (this is a postulate (i.e., a fact that is accepted without proof) which is used to prove the following relationships).
o Picture:
o Alternate interior angles have equal measures.
o Picture:
o Proof:
o Alternate exterior angles have equal measures.
o Picture:
o Proof:
o Interior angles on the same side of the transversal are supplementary (add to 180°).
o Picture:
o Proof:
Practice:
- Find all angle measures on the diagram below.
Triangles
The sum of the measures of the angles of any triangle is 180°.
Proof:
è Begin by drawing a parallel to any one side, and then think of the other two sides as transversals.
Practice:
- Find all angle measures on the diagram below.
- Find all angle measures on the diagram below.
Types of Triangles
Triangles are classified according to their angles and sides.
· Angle classifications:
o Acute triangles have all angles less than 90°.
o Right triangles have one angle of 90°.
o Obtuse triangles have one angle greater than 90°.
· Side classifications:
o Equilateral triangles have all sides the same length.
o Isosceles triangles have two sides of the same length.
o Scalene triangles have no sides of the same length.
Congruent triangles have all the same side lengths and all the same angles.
Congruent is the proper way to say that two triangles are “equal” or “the same”.
Pictures:
Similar triangles have the same shape (i.e., all the same angles), but not necessarily the same size.
The larger triangle is like a “magnification” of the smaller one.
Pictures:
Conditions for Similar Triangles:
If triangle ABC is similar to triangle DEF, then the following conditions must hold:
· Corresponding angles must have the same measure (a postulate).
· Corresponding sides must be proportional.
· Pictures:
Practice:
- Find the remaining sides:
- Find the remaining sides:
- If a tree casts a shadow that is 100 feet long, and a 6 foot tall person casts a shadow that is 5 feet long, what is the height of the tree?