1-4 & 1-5 Angles Measures
and Relationships
Objectives:
The student will be able to:
1. Measure and classify angles.
2. Use congruent angles and the bisector of an
angle.
3. Identify and use special pairs of angles.
4. Identify perpendicular lines.
1
Classifying Angles
Acute Angles < 90°
Naming Angles:
Right Angles = 90°
Obtuse Angles > 90°
When naming angles using 3 letters, the vertex
must be the second of the 3 letters. You can
name an angle using a single letter only when
there is exactly one angle located at the vertex.
Naming angles.
3
Congruent Angles
& Angle Bisector:
A ray that divides an angle into two
congruent angles.  PQS ≅  TQS The
bisector of  PQT is QS .
In the figure, QS is the angle bisector of ÐPQT . Point S lies in
the interior of ÐPQT and ÐPQS = ÐTQS . If mÐPQT = 50 and
mÐPQS = 4x +14 , find the value of x.
ÐPQS = ÐTQS
50 = 4x + 14
-14
-14
36 = 4x
9 = x
Example:
In the figure, QS is the angle bisector of ÐPQT .
Point S lies in the interior of ÐPQT
and ÐPQS = ÐTQS . If mÐTQS = 6x - 2 and mÐPQS = 3x +13,
find the value ofÐPQT .
ÐTQS = ÐPQS
6x - 2 = 3x + 13
-3x
+2
3x
=
15
x = 5
Did we answer the question?
NO!
If mÐTQS = 6x - 2 and mÐPQS = 3x +13 , find the value
of ÐPQT .
PQS  TQS  PQT
6(5) – 2 + 3(5) + 13 = PQT
30 – 2 + 15 + 13 = PQT
56° = PQT
Special Angle pairs
1 2
3 4
5 6
7 8
Adjacent Angles:
Two angles that lie in the same plane and
have a common vertex and a common side,
but no common interior points.
 1 & 2,  1 &  3,  2 &  4,  3 &  4,
 5 &  6,  5 &  7,  6 &  8,  7 &  8
Vertical Angles:
Two angles that are opposite angles. Vertical
angles are congruent.
 1   4,  2   3,  5   8,  6   7
Linear Pair:
Supplementary angles that form a line
(sum = 180)
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5
Special Angle pairs
E
Congruent Angles:
Two or more angles that have the same
measure.
 AEB &  BEC,  CED &  DEA,
 AEB &  DEC,  BEC &  AED
Perpendicular angles: Lines, segments, and rays that form
right angles (90 degrees).
 AEC &  BED
Complementary & Supplementary Angles
Complementary Angles: Two angles whose measures have a
sum of 90°.
A + B = 30 + 60 = 90
Supplementary Angles:
Two angles whose measures have a
sum of 180°.
F + G = 120 + 60 = 180
Identify:
Two Obtuse vertical angles:
Two acute adjacent angles:
An angle supplementary to TNU:
Find x so that DZ ^ PZ.
If the two angles are perpendicular they MUST = 90° .
(9x + 5) + (3x + 1) = 90
12x + 6
-6
= 90
-6
12x = 84
x=7
Example:
Find the measures of 2 supplementary angles if the
difference in their measures is 18.
1  2 = 180
x + (x – 18) = 180
2x – 18 = 180
+18 +18
2x
= 198
x = 99
Are we through?
NO!!
If x = 99, what are the
measures of the
supplementary angles?
Ð2 = 99 -18 = 81
Ð1= 99
How can I check to see if
that’s correct?
99 + 81 = 180
Find x and y so that KO and HM are perpendicular.
1. Find x by setting the two angles equal to 90.
2. Vertical angels tell us if ÐKJI +ÐIJH =90,
then ÐMJO=90.
3. Find y by setting ÐMJO=90.
ÐKJI +ÐIJH = 90
(3x + 6) + (9x) = 90
12x + 6
-6
12x
= 90
-6
= 84
x=7
ÐMJO=90
(3y + 6) = 90
- 6 -6
3y
= 84
y = 28
1. Are the angles congruent?
Yes – set the expressions equal to each other.
A=B
2. Do the angles add up to 90°?
Yes – add the expressions and set them equal to 90°.
A + B = 90
3. Do the angles add up to 180°?
Yes – add the expressions and set them equal to 180°.
A + B = 180
4. Do the angles add up to some other value given in
the problem?
Yes – add the expressions and set them equal to the
value.
A + B = other value