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INVESTIGATION 1
TRANSVERSAL AND ANGLE RELATIONSHIPS
OF PARALLEL & NON-PARALLEL LINES
TRANSVERSAL
A transversal is a line that intersects 2 or
more coplanar lines at different points
In the diagram, which line is the
transversal?
line t
How many angles are formed?
8
CLASSIFYING THE ANGLES
How could you describe the position of
∠1 in the top four angles?
Upper left side of transversal
What angle is in the upper left of the
bottom four angles?
∠5
∠1 & ∠5 are called corresponding angles
CORRESPONDING ANGLES
A pair of corresponding angles is any pair of angles
that lie on the same side of a transversal and on
the same side of the other 2 lines
Besides ∠1 & ∠5, what are some other
corresponding angles?
 ∠2 & ∠6
 ∠3 & ∠7
 ∠4 & ∠8
CLASSIFYING OTHER PAIRS OF ANGLES
∠3, ∠4, ∠5, & ∠6 are between lines m & n
and are called interior angles
What would be a good name for ∠1, ∠2,
∠7, & ∠8?
exterior angles
Angles on opposite sides of the
transversal are called alternate angles
ALTERNATE INTERIOR ANGLES
A pair of alternate interior angles is any
pair of nonadjacent angles that lie on
opposite sides of the transversal and
between the other 2 lines
 ∠3 & ∠6
 ∠4 & ∠5
ALTERNATE EXTERIOR ANGLES
A pair of alternate exterior angles is any
pair of nonadjacent angles that lie on
opposite sides of the transversal and
outside the other 2 lines
 ∠1 & ∠8
 ∠2 & ∠7
SAME-SIDE INTERIOR ANGLES (CONSECUTIVE INTERIOR)
A pair of same-side interior angles is any
pair of angles that lie on same side of the
transversal and between the other 2 lines
 ∠3 & ∠5
 ∠4 & ∠6
SAME-SIDE EXTERIOR ANGLES (CONSECUTIVE EXTERIOR)
A pair of same-side exterior angles is any
pair of angles that lie on same side of the
transversal and outside the other 2 lines
 ∠1 & ∠7
 ∠2 & ∠8
WHAT CAN BE SAID ABOUT THE LINES 𝑚 & 𝑛 IN FIGURE A THAT IS
NOT TRUE IN FIGURE B?
Figure A
Figure B
They are parallel
IN THIS FIGURE, DO YOU THINK ∠1 ≅ ∠5?
Yes
What other angles appear to be congruent to ∠1?
∠4 & ∠8
Why is ∠1 ≅ ∠4?
Vertical Angle Theorem
Do lines m & n need to be parallel for that to be
true?
No
WHAT ANGLES APPEAR TO BE CONGRUENT TO ∠7?
∠6, ∠3, & ∠2
Why is ∠7 ≅ ∠6?
Vertical Angle Theorem
Do lines m & n need to be parallel for
that to be true?
No
IN THIS FIGURE, DO YOU THINK ∠1 ≅ ∠5?
No
Why or what is different from the
previous example?
Because the lines are not parallel
What angle is congruent to ∠5 and why?
∠8
Vertical Angle Theorem
POSTULATE 11: CORRESPONDING ANGLES POSTULATE
If two parallel lines are cut by a
transversal, then the corresponding angles
are congruent.
 ∠1 ≅ ∠5
 ∠2 ≅ ∠6
 ∠3 ≅ ∠7
 ∠4 ≅ ∠8
THEOREM 10-1: ALTERNATE INTERIOR ANGLES THEOREM
If two parallel lines are cut by a
transversal, then the alternate interior
angles are congruent.
 ∠3 ≅ ∠6
 ∠4 ≅ ∠5
THEOREM 10-2: ALTERNATE EXTERIOR ANGLES THEOREM
If two parallel lines are cut by a
transversal, then the alternate exterior
angles are congruent.
 ∠1 ≅ ∠8
 ∠2 ≅ ∠7
OTHER RELATIONSHIPS OF ANGLES
What is the relationship between ∠3 & ∠4? Why?
They are supplementary
Linear Pair Theorem
If ∠4 ≅ ∠5 & ∠3 is supplementary to ∠4, then
what is the relationship between ∠3 & ∠5?
They are supplementary
THEOREM 10-3: SAME-SIDE INTERIOR ANGLES THEOREM
If two parallel lines are cut by a
transversal, then the same-side interior
angles are supplementary.
 ∠3 & ∠5 are supplementary
 ∠4 & ∠6 are supplementary
THEOREM 10-4: SAME-SIDE EXTERIOR ANGLES THEOREM
If two parallel lines are cut by a
transversal, then the same-side exterior
angles are supplementary.
 ∠1 & ∠7 are supplementary
 ∠2 & ∠8 are supplementary
EXAMPLE: 𝑚∠8 = 110º FIND THE MEASURE OF THE OTHER ANGLES
WHEN THE LINES ARE PARALLEL. JUSTIFY YOUR ANSWERS,
1. m∠7 = 70º, Linear Pair Theorem
2. m∠5 = 110º,Vertical Angle Theorem
3. m∠6 = 70º, Linear Pair Theorem
4. m∠4= 110º, Corresponding Angles Postulate
5. m∠2 = 70º, Same-side Exterior Angles Theorem
6. m∠1= 110º, Alternate Exterior Angles Theorem
USE THE DIAGRAM TO IDENTIFY THE FOLLOWING
1. The alternate interior angles with transversal
m.
∠2 & ∠9 or ∠4 & ∠10
2. ∠5 & ∠10 are same-side exterior angles for
which transversal?
transversal n
3. The corresponding angles with transversal p.
∠1 & ∠5 or ∠3 & ∠7 or ∠2 & ∠6 or ∠4 & ∠8
QUESTIONS/REVIEW
Lines do not need to be parallel to
identify the type of angle pairs
However Postulate 11 and Theorems 10-1
to 10-4 are only true when the lines are
parallel
Future lessons we will use the converse of
these postulate and theorems to
show/prove lines are parallel
So it is important to learn these concepts
now
Imagine how hard it would have been to
learn division if you did not know how to
multiply first