Download Vertical Angles

Special Pairs of Angles
Lesson 2.4
Geometry Honors
Objective: Apply the definitions of
complementary and supplementary angles.
State and apply the theorem about vertical
angles.
Page 50
Lesson Focus
Pairs of angles whose measures have the sum
of 90 or 180 appear frequently in geometric
situations. For this reason, they are given
special names. This lesson studies these
special angles and solves problems involving
them.
Special Pairs of Angles
Complementary angles (comp. s)
Two angles whose measures have the sum 90.
Each angle is called the complement of the other.
Example:
Given: 1 and 2 are complements.
If m1 = 42, then m2 = 48.
Special Pairs of Angles
Supplementary angles (supp. s)
Two angles whose measures have the sum of 180.
Each angle is called the supplement of the other.
Example:
Given: 1 and 2 are supplements.
If m1 = 109, then m2 = 71.
EXAMPLE 1
Identify complements and supplements
In the figure, name a pair of
complementary angles, a pair
of supplementary angles, and
a pair of adjacent angles.
SOLUTION
Because 32°+ 58° = 90°, BAC and
complementary angles.
Because 122° + 58° = 180°,
supplementary angles.
RST are
CAD and
RST are
Because BAC and CAD share a common vertex and
side, they are adjacent.
GUIDED PRACTICE
1.
for Example 1
In the figure, name a pair of complementary
angles, a pair of supplementary angles, and a
pair of adjacent angles.
Because 41° + 49° = 90°, FGK
and GKL are complementary
angles.
Because 49° + 131° = 180°,
supplementary angles.
HGK and
GKL are
Because FGK and HGK share a common vertex and
side, they are adjacent.
Special Pairs of Angles
Vertical Angles (Vert. s)
Two angles such that the sides of one angle are opposite rays
to the other sides of the other angle. When two lines
intersect, they form two pairs of vertical angles.
Special Pairs of Angles
Vertical Angle Theorem
Vertical angles are congruent.
Proof:
1 and 2 form a linear pair, so by the Definition of Supplementary
Angles, they are supplementary. That is, m1 + m2 = 180°. (also, Angle
Addition Postulate)
2 and 3 form a linear pair also, so m2 + m3 = 180°. Subtracting
m2 from both sides of both equations, we get m1 = 180° − m2 =
m3. Therefore, 1  3.
You can use a similar argument to prove that 2  4.
EXAMPLE 3 Find angle measures
Sports
When viewed from the side, the frame of a ballreturn net forms a pair of supplementary angles with
the ground. Find m BCE and m ECD.
Practice Quiz
Complete with always, sometimes, or never.
1.
2.
3.
4.
5.
Vertical angles _____ have a common vertex.
Two right angles are _____ complementary.
Right angles are _____ vertical angles.
Angles A, B, and C are _____ complementary.
Vertical angles _____ have a common supplement.
Practice Quiz
Complete with always, sometimes, or never.
1.
2.
3.
4.
5.
Vertical angles always have a common vertex.
Two right angles are never complementary.
Right angles are sometimes vertical angles.
Angles A, B, and C are never complementary.
Vertical angles always have a common supplement.
Homework Assignment
Page 53 – 54
Problems 19 – 31 odd, 32 – 35.