Name __________________________________________________ Date __________
Ms. Moorer
Geometry
Solving for Unknown Angles – Transversals
Opening Exercise
⃡
Use the diagram at the right to determine 𝑥and 𝑦. ⃡𝐴𝐵 and𝐶𝐷
are straight lines.
𝑥 = ________
𝑦 = ________
Name a pair of vertical angles:
_____________________
Find the measure of ∠𝐵𝑂𝐹. Justify your calculation.
________________________________________________
________________________________________________
Relevant Vocabulary
Alternate Interior Angles: Let line 𝑡 be a transversal to
lines 𝑙 and 𝑚 such that 𝑡 intersects 𝑙 at point 𝑃 and intersects 𝑚 at point 𝑄. Let 𝑅 be a point on 𝑙, and 𝑆
be a point on 𝑚 such that the points 𝑅 and 𝑆 lie in opposite half-planes of 𝑡. Then the angle ∠𝑅𝑃𝑄 and the
angle ∠𝑃𝑄𝑆 are called alternate interior angles of the transversal 𝑡 with respect to 𝑚and 𝑙.
Corresponding Angles:Let line 𝑡 be a transversal to lines 𝑙and 𝑚.If ∠𝑥 and ∠𝑦 are alternate interior
angles, and ∠𝑦 and ∠𝑧 are vertical angles, then ∠𝑥 and ∠𝑧 are corresponding angles.
Example 1
Given a pair of lines ⃡𝐴𝐵 and ⃡𝐶𝐷 in a plane (see the diagram below), a third line 𝐸𝐹is called a transversal
if it intersects 𝐴𝐵 at a single point and intersects ⃡𝐶𝐷 at a single but different point. The two lines 𝐴𝐵 and
𝐶𝐷 are parallel if and only if the following types of angle pairs are congruent or supplementary:

Corresponding Angles are equal in measure
_____________________________________

Alternate Interior Angles are equal in measure
_____________________________________

Same Side Interior Angles are supplementary
_____________________________________
a.
b.
𝑚∠𝑎 = ________
𝑚∠𝑏 = ________
c.
d.
𝑚∠𝑐 = ________
𝑚∠𝑑 = ________
Exercises
In each exercise below, find the unknown (labeled) angles. Give reasons for your solutions.
1.
𝑚∠𝑎 = ________________________
𝑚∠𝑏 = ________________________
𝑚∠𝑐 = ________________________
2.
𝑚∠𝑒 = ________________________
𝑚∠𝑓 = _______________________
2
1
a
3
5
4
6
b
#1-8: Using the figure at the right,
find the measure of the angle if a b :
1) m1  45, m6  ?
2) m3  65, m6  ?
3) m3  35, m5  ?
4) m4  130, m7  ?
5) m1  45, m8  ?
6) m4  126, m6  ?
7) m2  115, m6  ?
8) m4  115, m5  ?
8
7
m
7
#9-12: Using the figure at the right, find the value of x
and the indicated angle if m n :
6
n
4
5
3
2
1
8
9) m4  3x  10, m2  x  80, m4  ?
10) m1  3x  10, m2  2 x  40, m3  ?
11) m7  5 x  20, m5  4 x  57, m7  ?
12) m1  5x  40, m3  3x, m2  ?
#13-18) If a b , and lines a and b are cut by transversal c,
a) Identify the relationship between the angles
b) Find x:
13)
14)
a
(3x + 30)°
a
(x + 30)°
70°
b
c
b
(x + 50)°
c
15)
16)
c
a
(3x + 20)°
(3x + 32)°
a
(x + 40)°
b
(8x + 12)°
b
c
17)
c
18)
(5x + 20)°
a
a
(5x - 10)°
b
b
(2x + 20)°
(2x + 50)°
c