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TEKS Focus:
 (6)(A) Verify theorems about angles
formed by the intersection of lines and
line segments, including vertical angles,
and angle formed by parallel lines cut
by a transversal and prove equidistance
between the endpoints of a segment
and points on its perpendicular bisector
and apply these relationships to solve
problems.


What is an Angle?
An angle is a figure formed by two
rays, or sides, with the same endpoint
called the vertex (plural: vertices).
You can name an angle several ways:
by its vertex, by a point on each ray
and the vertex, or by a number.
A
side
B
1
side
C
Names:
B
ABC
CBA
1
 The
measure of an angle is
the absolute value of the
difference of the real
numbers paired with the
sides of the angle.

What are Acute, Obtuse, Right, and Straight
Angles?
Congruent angles are angles that have the same
measure.
In the diagram, mABC = mDEF, so you can write
ABC  DEF. This is read as “angle ABC is
congruent to angle DEF.”
Arc marks are used to show that the two angles are
congruent.
The set of all points between the sides of the
angle is the interior of an angle. The exterior
of an angle is the set of all points outside the
angle.
There is more than one
way to name an angle:
Angle Name could be
R, SRT, TRS, or 1
Example: 1
Write the different ways you can name the angles in
the diagram.
QTS, QTR, STR, 1, 2
Example: 2
Use the diagram to find the measure of each angle.
Then classify each as acute, right, or obtuse.
a. BOA
b. DOB
c. DOC
mDOB = 125°
mDOC = 90°
mBOA = 40°
DOB is obtuse.
EOC is right.
BOA is acute.
Example: 3
mDEG = 115°, and mDEF = 48°. Find mFEG.
mDEG = mDEF + mFEG  Add. Post.
115 = 48 + mFEG
Substitute the given values.
–48° –48°
Subtract 48 from both sides.
67 = mFEG
Simplify.
Example: 4
mXWZ = 121°.
m XWY = (4x + 6)°.
mYWZ = (6x – 5)°.
Find mXWY and mYWZ.
mXWZ = mXWY + mYWZ  Add. Post.
121 = (4x + 6) + (6x – 5)
121 = 10x + 1
120 = 10x
x = 12
Substitute the given values.
Combine like terms
Subtraction Prop. of =
Division Prop. of =
mXWY = 54 and mYWZ = 67
Example: 5
JKM  MKL, mJKM = (4x + 6)°, and
mMKL = (7x – 12)°. Find mJKM.
Example: 5 continued
Step 1 Find x.
mJKM = mMKL
(4x + 6)° = (7x – 12)°
+12
4x + 18
–4x
+12
= 7x
–4x
18 = 3x
6=x
Definition of congruent angles.
Substitute the given values.
Add 12 to both sides.
Simplify.
Subtract 4x from both sides.
Divide both sides by 3.
Simplify.
Step 2 Find mJKM.
mJKM = 4x + 6
= 4(6) + 6
= 30
Substitute 6 for x.
Simplify.



RQT is a straight angle.
What are mTQS and
mRQS?
6x + 20 + 2x + 4 = 180
8x + 24 = 180
8x = 156
x = 19.5
Def. of straight ;  Add. Post.
Combine like terms.
Subtraction Property of Equality
Division Property of Equality
mTQS = 6(19.5) + 20 = 137 and mRQS = 2(19.5) + 4 = 43
Substitution Property of Equality
Example: 7
A surveyor recorded the angles formed by a transit
(point A) and three distant points, B, C, and D.
Name three of the angles.
Possible answer:
BAC
CAD
BAD
Can you name it A? Why or why not?
No; adjacent angles cannot be
named by just the vertex.
Example: 8
Find the measure of each angle. Then classify each
as acute, right, or obtuse.
A. WXV
B. ZXW
mWXV = 30°
mZXW = |130° - 30°| = 100°
WXV is acute.
ZXW = is obtuse.
Basic Terms
An angle bisector is a ray that divides an angle
into two congruent angles.
JK bisects LJM; thus LJK  KJM.
Example: 9
Find the measure of each angle.
JK bisects LJM, mLJK = (-10x + 3)°, and
mKJM = (–x + 21)°. Find mLJM.
Step 1 Find x.
mLJK = mKJM
(–10x + 3)° = (–x + 21)°
+x
+x
–9x + 3 = 21
–3
–3
–9x = 18
x = –2
Substitute the given values.
Add x to both sides.
Simplify.
Subtract 3 from both sides.
Divide both sides by –9.
Simplify.
Example: 9 continued
Step 2 Find mLJM.
mLJM = mLJK + mKJM
mLJM = (–10x + 3)° + (–x + 21)°
mLJM = –10(–2) + 3 – (–2) + 21 Substitute –2 for x.
mLJM = 20 + 3 + 2 + 21
mLJM = 46°
Simplify.