Download Lesson 2: Solve for Unknown Angles—Transversals

Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
U2
GEOMETRY
Lesson 2: Solve for Unknown Angles—Transversals
Learning Targets: I can identify all types of angles formed by parallel lines cut by transversal and apply the knowledge of
relationships between angles formed by parallel lines cut by a transversal to find the missing angle.
Opening Activity: Solve the following equations
1. 4(𝑥 − 2) = 8(𝑥 − 3) − 12
2. (𝑥 − 1)(𝑥 + 5) = 𝑥 2 + 4𝑥 − 2
New Vocabulary
A transversal is a line that intersects two or more lines (in the same plane).
Remember that:
- the word INTERIOR means BETWEEN the lines.
- the word EXTERIOR means OUTSIDE the lines.
- the word ALTERNATE means "opposite sides" of the transversal and “on different” lines
When the lines are NOT parallel
When the lines are parallel...
"Names" given to pairs of angles formed by two parallel lines cut by a transversal:




alternate interior angles
alternate exterior angles
corresponding angles
interior angles on the same side of the transversal
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
U2
GEOMETRY
Alternate interior angles are "interior" (between the parallel lines), and on
"alternate" sides of the transversal
Identify the pairs of alternate interior angles
___________
___________
If two parallel lines are cut by a transversal, the alternate interior angles are
congruent.
Converse
If two lines are cut by a transversal and the alternate interior angles are congruent,
the lines are parallel.
Alternate exterior angles are "exterior" (outside the parallel lines), and on
"alternate" sides of the transversal
Identify the pairs of alternate interior angles
___________
___________
If two parallel lines are cut by a transversal, the alternate exterior angles are
congruent.
Converse
If two lines are cut by a transversal and the alternate exterior angles are congruent,
the lines are parallel.
If you copy one of the corresponding angles and you translate (“slide”) it along
the transversal, it will coincide with the other corresponding angle.
Identify all the pairs of coresponding angles
___________ ___________ ___________ ___________
If two parallel lines are cut by a transversal, corresponding angles are congruent.
Converse
If two lines are cut by a transversal and corresponding angles are congruent, the
lines are parallel..
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
U2
GEOMETRY
The "interior" angles (between the parallel lines) on the same side of the
transversal, are called “same-side interior angles”.
Identify all pairs of same-side interior angles
___________
___________
If two parallel lines are cut by a transversal, same-side interior angles are
supplementary.
Converse
If two lines are cut by a transversal and the same-side interior angles are
supplementary, the lines are parallel.
Using the theorems above, what equations can you create from the
diagram at the right?
Congruent: ______ = _____
Type: _______________
Supplementary: ____ + ____ = ____
Type: _______________
Example 1 In the diagram below, find the unknown (labeled) angles. Give reasons for your solutions.
𝑚∠𝑎 = _______
Reason:________________________
𝑚∠𝑏 = _______
Reason:________________________
𝑚∠𝑐 = _______
Reason:________________________
Example 2
Given the diagram at the right with straight lines m, n and t.
Which statement could always be used to prove m || n ?
Choose:
∠2 and ∠6 are supplementary
m∠2 = m∠3
∠3 and ∠5 are supplementary
m∠5 = m∠7
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
U2
GEOMETRY
Lesson 2: Solve for Unknown Angles—Transversals
Classwork
Two lines 𝐴𝐵 and 𝐶𝐷 are parallel if and only if any one of the following conditions are true:

Corresponding Angles are equal in measure.
or

Alternate Interior Angles are equal in measure.
or

Same Side Interior Angles are supplementary:
1. Transversal
intersects
and
, as shown in the diagram below.
Which statement could always be used to prove
a)
b)
c)
and
are supplementary
d)
and
are supplementary
?
2. A transversal intersects two lines. Which condition would always make the two lines parallel?
a) Vertical angles are congruent.
b) Alternate interior angles are congruent.
c) Corresponding angles are supplementary.
d) Same-side interior angles are
complementary.
3. Find m∠ 1 and then m∠ 2. Justify each answer.
m 1  __________because __________________________________
____________________________________________________
m 2  _________because ___________________________________
____________________________________________________
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
In Problems 4 and 5, use the diagram at the right.
4. Given ∠2 ≅ ∠6, what justifies 𝑘 ∥ 𝑚.
a. Converse Alternate Interior Angles Theorem
b. Converse Alternate Exterior Angles Theorem
c. Converse Corresponding Angles Theorem
d. There is not enough info to state parallel
5. Given 𝑛 ∥ 𝑝 , what justifies ∠1 ≅ ∠12
a. Alternate Interior Angles Theorem
b. Alternate Exterior Angles Theorem
c. Corresponding Angles Theorem
d. There is not enough info to make this statement
6. Determine the relationship between ∠1 & ∠10.
a. Alternate Interior
b. Same-side Interior
c. Corresponding Angles
d. None of these
7. Determine the relationship between ∠5 & ∠15.
a. Alternate Exterior
b. Alternate Interior
c. Same-side Interior
d. None of these
8. If 𝑚∠9 = 62°, then find the measure the following angles:
a. 𝑚∠1 = ________
b. 𝑚∠2 = ________
c. 𝑚∠4 = ________
d. 𝑚∠5 = ________
e. 𝑚∠15 = _______
U2
GEOMETRY
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
9.
Period:________ Date:__________
Given straight lines p, q, t, and s and angles as marked.
Which value of x will make lines p and q parallel?
Choose:
73º
87º
107º
113º
10. Given the diagram shown at the right. Assume all lines are
straight. Find the measures of all of the numbered angles 1
through 12.
m∠1 ________________
m∠2 ________________
m∠3 _________________
m∠4 _________________
m∠5 ________________
m∠6 _________________
m∠7 _________________
m∠8 _________________
m∠9 _________________
m∠10 _________________
m∠11 _________________
m∠12 _________________
11.
U2
GEOMETRY
Lesson 2
RCSD Geometry Local MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:__________
GEOMETRY
Lesson 2: Solve for Unknown Angles—Transversals
Homework
1. Find the measure of the unknown angle, and give the name of the theorem used.
A.
B.
ma = ________
mb = ________
Theorem:
____________________
Theorem: ________________________
__________________________________
________________________________
C.
D.
mc = ________
md = ________
Theorem:
____________________
Theorem: _______________________
__________________________________
________________________________
2. Given that 𝑝 ∥ 𝑞 and 𝑙 ∥ 𝑚 , find the measures of all the
numbered angles in the diagram at right, giving reasons for
each measurement. The first one is done for you.
a. 𝑚∠1 = 42
by _corresponding angle theorem__ to __Given Angle___.
b. 𝑚∠2 = _____ by ______________________________ to ________________.
c. 𝑚∠3 = _____ by ______________________________ to ________________.
d. 𝑚∠4 = _____ by ______________________________ to ________________.
e. 𝑚∠5 = _____ by ______________________________ to ________________.
f. 𝑚∠6 = _____ by ______________________________ to ________________.
g. 𝑚∠7 = _____ by ______________________________ to ________________.
h. 𝑚∠8 = _____ by ______________________________ to ________________.
U2