ACTIVITY 15 Continued
Lesson 15-2
Lesson 15-2
Two-Column Geometric Proofs
ACTIVITY 15
continued
PLAN
My Notes
Pacing: 1 class period
Chunking the Lesson
Example C
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer,
Think-Pair-Share, Close Reading, Discussion Groups, Self Revision/Peer
Revision, Group Presentation
Try These C
Check Your Understanding
Lesson Practice
When writing two-column proofs to solve algebraic equations, you justify
each statement in these proofs by using an algebraic property. Now you will
use two-column proofs to prove geometric theorems. You must justify each
statement by using a definition, a postulate, a property, or a previously
proven theorem.
TEACH
Bell-Ringer Activity
Read the introduction with students and
remind them of the definition of vertical
angles. Encourage them to add this term
and the Vertical Angles Theorem to
their Vocabulary Organizers and to their
list of theorems.
MATH TERMS
The Vertical Angles Theorem states
that vertical angles are congruent.
Observe the structure of the proof above
the two columns. There is a statement of
the theorem that will be proved, along
with the Given statement. In Example A,
the Given information and the Prove
statement is stated in terms of the
diagram.
After the proof of the Vertical Angles
Theorem in Example A, students are
asked to complete the proof of another
theorem in Example B, namely that all
right angles are congruent. Students
should add both of these theorems as
well as the term theorem to their
Vocabulary Organizers.
260
Recall that vertical angles are opposite angles formed by a pair of intersecting
lines. In the figure below, ∠1 and ∠2 are vertical angles. The following
example illustrates how to prove that vertical angles are congruent.
Example A
Theorem: Vertical angles are congruent.
Example A, Guided Example B
Vocabulary Organizer, Think-PairShare, Close Reading, Discussion
Groups, Construct an Argument,
Critique Reasoning This is the first
opportunity for students to see formal
geometric proofs. The relationship
between statements and reasons in a
two-column proof should be discussed.
Asking students to explain how the
reason justifies the statement in several
steps of the proof is one method of
having students analyze the proofs in
this Example.
Students will prove that vertical angles
have the same measure. They will then
use that conclusion to prove other
theorems. Because this is the first
theorem that students have encountered,
note that the only justifications that can
be used are definitions, properties, and
postulates. Then they will be able to use
the Vertical Angles Theorem to prove
other theorems.
• Complete two-column proofs to prove theorems about segments.
• Complete two-column proofs to prove theorems about angles.
Given: ∠1 and ∠2 are vertical angles.
Prove: ∠1 ≅ ∠2
1
Statements
3. m∠1 + m∠3 = m∠2 + m∠3
4. m∠1 = m∠2
4. Subtraction Property of Equality
5. ∠1 ≅ ∠2
5. Definition of congruent angles
2. m∠2 + m∠3 = 180°
As you listen to the group
discussion, take notes to aid
comprehension and to help you
describe your own ideas to others
in your group. Ask questions to
clarify ideas and to gain further
understanding of key concepts.
Reasons
1. Definition of supplementary
angles
2. Definition of supplementary
angles
3. Substitution Property
1. m∠1 + m∠3 = 180°
DISCUSSION GROUP TIPS
3 2
Guided Example B
Supply the missing statements and reasons.
Theorem: All right angles are congruent.
Given: ∠A and ∠B are right angles.
Prove: ∠A ≅ ∠B
Statements
1. ∠A and ∠B are right angles.
Reasons
2. m∠A = 90°; m∠B = 90°
1. Given
2. Definition of right angles
3. m∠A = m∠B
3. Transitive Property
4. ∠A ≅ ∠B
4. Definition of congruent angles
260 SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
Differentiating Instruction
Support students who struggle to distinguish given
information from what they are to prove by having
them rewrite the theorem as an if-then statement: If
two angles are vertical angles, then they are
congruent. Have students identify the hypothesis as
the Given information, and the conclusion is what
they need to write for the Prove statement.
Extend the lesson and challenge students to rewrite
the theorem All right angles are congruent as an
if-then statement. Ask them to draw a diagram to
accompany the Given and Prove statements.
SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
© 2017 College Board. All rights reserved.
#1–2
Try These A–B
© 2017 College Board. All rights reserved.
Examples A, B
Learning Targets:
Lesson 15-2
Two-Column Geometric Proofs
ACTIVITY 15
continued
My Notes
Try These A–B
a. Complete the proof.
P
Given: Q is the midpoint of PR.
Q
R
QR ≅ RS
S
Prove: PQ ≅ RS
Statements
Reasons
1. Q is the midpoint of PR.
1. Given
2. PQ ≅ QR
2. Definition of midpoint
3. QR ≅ RS
3. Given
4. PQ ≅ RS
4. Transitive Property of Congruence
MATH TIP
More than one statement in a
two-column proof can be given
information.
Prove: m∠2 = 112°
2
1
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
Statements
1. Given
2. m∠1 + m∠2 = 180°
2. Definition of supplementary
angles
3. m∠1 = 68°
3. Given
4. 68° + m∠2 = 180°
4. Substitution Property
5. m∠2 = 112°
5. Subtraction Property of
Equality
Have students discuss these proofs and
monitor discussions to ensure that all
students are participating. Encourage
students to challenge others in their
group to justify or explain the
statements they make in the discussion.
Then have groups together present their
proofs to the rest of the class.
Developing Math Language
Evaluate students’ use of all types of
vocabulary in their written responses to
ensure that they are using everyday
words as well as academic vocabulary
and math terms correctly.
Reasons
1. ∠1 and ∠2 are supplementary.
Try These A–B Vocabulary
Organizer, Think-Pair-Share, Close
Reading, Discussion Groups,
Construct an Argument, Critique
Reasoning In each proof, note that
there are two statements that are given
information and that both are necessary
for proving the conclusion.
Monitor students’ group discussions to
ensure that complex mathematical
concepts are being verbalized precisely
and that all group members are actively
participating in discussions through the
sharing of ideas and through asking and
answering questions appropriately.
b. Complete the proof.
Given: ∠1 and ∠2 are
supplementary.
m∠1 = 68°
ACTIVITY 15 Continued
1–2 Activating Prior Knowledge,
Close Reading In these items, students
build on their knowledge of the nature
of a proof. Without writing full proofs,
they put the essential steps of an
argument in the correct order. Confirm
that students understand that each step
in a proof follows from an earlier step or
steps.
Statements in a proof must be arranged in a logical order.
1. If you are given the information that ∠A and ∠B are straight angles,
what is the logical order for the two statements below? Explain your
reasoning.
m∠A = m∠B
m∠A = 180° and m∠B = 180°
You need to state the measures of the angles before stating that the
measures are equal, so the logical order is the statement m∠A = m∠B
followed by the statement m∠A = 180° and m∠B = 180°.
Activity 15 • Proofs About Line Segments and Angles
261
Activity 15 • Proofs About Line Segments and Angles
261
Students have not used the Transitive
Property with congruence before this
time. Remind students that two angles
are congruent when their measures are
equal and that the Transitive Property of
Equality can be applied to angle
measures. Point out that there are
several algebra properties that can be
used with congruence:
Lesson 15-2
Two-Column Geometric Proofs
ACTIVITY 15
continued
My Notes
2. You are given that ∠X and ∠Y are complementary angles, and that
m∠X = 45°. What is the logical order for the statements below? Explain.
45° + m∠Y = 90°
m∠X + m∠Y = 90°
You need to use the definition of complementary angles to state that
the sum of the measures of ∠X and ∠Y is 90°. Then you can substitute
45° for m∠X. So, the logical order is m∠X + m∠Y = 90°, followed by the
statement 45°+ m∠Y = 90°.
Example C
MATH TIP
The theorem stated in Example C
is called the Congruent
Complements Theorem.
Arrange the statements and reasons below in a logical order to complete
the proof.
Theorem: If two angles are
complementary to the same
angle, then the two angles
A
are congruent.
Given: ∠A and ∠B are each
complementary to ∠C.
• Reflexive Property (a = a) A figure is
always congruent to itself.
Prove: ∠A ≅ ∠B
• Symmetric Property (If a = b, then
b = a.) For example, if AB ≅ CD ,
then CD ≅ AB .
• Transitive Property (If a = b and
b = c, then a = c.) For example, if
∠A ≅ ∠B and ∠B ≅ ∠C , then
∠A ≅ ∠C.
Encourage students to add these
properties to their Vocabulary
Organizers.
Have students add the theorem from
Example C to their Vocabulary
Organizers, and tell them that they will
be able to use this theorem to justify
other statements in future proofs.
B
C
m∠A + m∠C = m∠B + m∠C
Transitive Property
∠A and ∠B are each
complementary to ∠C.
Given
∠A ≅ ∠B
Definition of congruent segments
m∠A = m∠B
Subtraction Property of Equality
m∠A + m∠C = 90°;
m∠B + m∠C = 90°
Definition of complementary
angles
Start the proof with the given information. Then decide which statement
and reason follow logically from the first statement. Continue until you
have proved that ∠A ≅ ∠B.
Statements
262
262
Reasons
1. ∠A and ∠B are each
complementary to ∠C.
1. Given
2. m∠A + m∠C = 90°
m∠B + m∠C = 90°
2. Definition of complementary
angles
3. m∠A + m∠C =m∠B + m∠C
3. Transitive Property
4. m∠A = m∠B
4. Subtraction Property of Equality
5. ∠A ≅ ∠B
5. Definition of congruent angles
SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
© 2017 College Board. All rights reserved.
Example C Vocabulary Organizer,
Think-Pair-Share, Close Reading,
Discussion Groups, Construct an
Argument, Critique Reasoning Ask
students to cover the bottom of the page
with a sheet of paper, and challenge
them to work together to place the
statements and reasons in logical order.
When they have finished, they can
compare their work with the proof at
the bottom of the page.
© 2017 College Board. All rights reserved.
ACTIVITY 15 Continued
Lesson 15-2
Two-Column Geometric Proofs
ACTIVITY 15
continued
My Notes
Try These C
a. Attend to precision. Arrange the statements and reasons below in a
logical order to complete the proof.
Given: ∠1 and ∠2 are vertical angles; ∠1 ≅ ∠3.
Prove: ∠2 ≅ ∠3
∠1 ≅ ∠2
Vertical angles are congruent.
∠2 ≅ ∠3
Transitive Property
∠1 ≅ ∠3
Given
∠1 and ∠2 are vertical angles.
Given
b. Write a two-column proof of the following theorem.
ACTIVITY 15 Continued
Try These C Vocabulary Organizer,
Think-Pair-Share, Close Reading,
Discussion Groups, Construct an
Argument, Critique Reasoning
Students may notice the parallelism
between part b and Example C.
Encourage them to add each new
theorem to their Vocabulary Organizers.
Monitor students’ written work carefully
and note that the structure is important
to help them organize their proofs in a
logical manner. The graphic organizer
Proof Builder may be a helpful tool for
students who struggle with organizing
their proofs. As students use the graphic
organizer, encourage them to draw
diagrams based on the statements they
are given.
Theorem: If two angles are supplementary to the same angle, then the two
angles are congruent.
Answers
Given: ∠R and ∠S are each supplementary to ∠T.
a. Sample proof is shown.
Try These C
Prove: ∠R ≅ ∠S
Statements
Check Your Understanding
1. ∠1 and ∠2
are vertical
angles.
1. Given
2. ∠1 ≅ ∠2
2. Vertical
angles are
congruent.
3. ∠1 ≅ ∠3
3. Given
4. ∠2 ≅ ∠3
4. Transitive
Property
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
3. If you know that ∠D and ∠F are both complementary to ∠J, what
statement could you prove using the Congruent Complements Theorem?
4. What types of information can you list as reasons in a two-column
geometric proof?
5. Kenneth completed this two-column proof.
What mistake did he make? How could
you correct the mistake?
ur
u
Given: JL bisects ∠KJM; m∠KJL = 35°.
K
L
b. Sample proof is shown.
35°
Prove: m∠LJM = 35°
J
Statements
ur
u
1. JL bisects ∠KJM.
Reasons
Statements
M
2. ∠KJL ≅ ∠LJM
2. Definition of congruent
angles
3. m∠KJL = m∠LJM
3. Definition of angle bisector
4. m∠KJL = 35°
4. Given
5. m∠LJM = 35°
5. Transitive Property
3. ∠D ≅ ∠F
4. given information, postulates, properties,
definitions, and previously proven
theorems
5. The reasons for Statements 2 and 3 do not
make sense logically based on the previous
statements of the
proof. Based on
Statement 1 (JL bisects ∠KJM), Kenneth
can conclude that ∠KJL ≅ ∠LJM by the
definition of an angle bisector. Based on
Statement 2 (∠KJL ≅ ∠LJM), he can
conclude that m∠KJL = m∠LJM by the
definition of congruent angles.
Reasons
1. ∠R and ∠S are 1. Given
each
supplementary
to ∠T.
1. Given
Answers
Reasons
2. m∠R + m∠T = 2. Definition of
supplementary
180°; m∠S +
angles
m∠T = 180°
3. m∠R + m∠T = 3. Transitive
Property
m∠S + m∠T
Activity 15 • Proofs About Line Segments and Angles
263
4. m∠R = m∠S
4. Subtraction
Property of
Equality
5. ∠R ≅ ∠S
5. Definition of
congruent
angles
Check Your Understanding
Debrief students’ answers to these items
to ensure that students understand the
structure of a two-column proof.
Discuss the answers as a class and be
sure to address any misconceptions that
students may have. Students who have
difficulty may benefit from additional
practice with putting statements and
reasons for a proof in logical order.
Activity 15 • Proofs About Line Segments and Angles
263
Lesson 15-2
Two-Column Geometric Proofs
ACTIVITY 15
continued
Students’ answers to the Lesson Practice
items will provide a formative assessment
of their understanding of writing
two-column proofs to prove theorems
about segments and angles, and of
students’ ability to apply their learning.
My Notes
LESSON 15-2 PRACTICE
6. Supply the missing statements and reasons.
uuur
Given: ∠1 is complementary to ∠2; BE bisects ∠DBC.
Prove: ∠1 is complementary to ∠3.
Short-cycle formative assessment items
for Lesson 15-2 are also available in the
Assessment section on SpringBoard
Digital.
A
D
Refer back to the graphic organizer the
class created when unpacking Embedded
Assessment 2. Ask students to use the
graphic organizer to identify the concepts
or skills they learned in this lesson.
B
6.
Reasons
1. Given
2. ∠2 ≅ ∠3
2. Def. of ∠
bisector
3. m∠2 = m∠3
3. Def. of ≅ ∠
See the Activity Practice on page 275 and
the Additional Unit Practice in the
Teacher Resources on SpringBoard Digital
for additional problems for this lesson.
You may wish to use the Teacher
Assessment Builder on SpringBoard
Digital to create custom assessments or
additional practice.
2.
3. Definition of congruent angles
4. ∠1 is complementary to ∠2.
4.
5. ∠1 + m∠2 = ____
5.
6.
7. Definition of complementary
angles
Prove: LN = 16
uuur
8. Given: BD bisects ∠ABC; m∠DBC = 90°.
MATH TIP
D
Prove: ∠ABC is a straight angle.
If the given information does not
include a diagram, it may be
helpful to make a sketch to
represent the information.
A
9. Given: PQ ≅ QR , QR = 14, PR = 14
B
C
Q
Prove: PQ ≅ PR
P
R
10. Reason abstractly. What type of triangle is shown in Item 9? Explain
how you know.
264
SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
7. Sample proof is shown.
Statements
8. Sample proof is shown.
Reasons
1. M is the midpoint 1. Given
of LN.
2. LM ≅ MN
2. Def. of mdpt.
3. LM = MN
3. Def. of ≅ segments
4. LM = 8
4. Given
5. MN = 8
5. Trans. Prop.
6. LN = LM + MN
6. Segment Addition
Postulate
7. LN = 8 + 8 = 16 7. Substitution
264
1.
2. ∠2 ≅ ∠3
3.
7. Given: M is the midpoint of LN; LM = 8.
ADAPT
Check students’ answers to the Lesson
Practice to ensure that they understand
the basics of two-column proofs. Call
students’ attention to the Math Tip about
drawing a diagram if one is not
provided. For students who struggle,
provide additional practice with placing
statements and reasons in logical order
and have them work with another
student.
Reasons
Construct viable arguments. Write a two-column proof for Items 7–9.
5. m∠1 + m∠2 = 5. Def. of
compl. ∠
90°
7. Def. of
7. ∠1 is
compl. ∠
complementary
to ∠3.
E
Statements
uur
1. BE bisects ∠DBC.
6. m∠1 + m∠3 = 90°
7.
4. Given
4. ∠1 is
complementary
to ∠2.
6. m∠1 + m∠3 = 6. Substitution
90°
2
3
C
LESSON 15-2 PRACTICE
Statements
1. BE bisects
∠DBC.
1
Statements
1. BD bisects ∠ABC.
Reasons
1. Given
2. ∠ABD ≅ ∠DBC
2. Def. of ∠ bisector
3. m∠ABD = m∠DBC
3. Def. of ≅ ∠
4. m∠DBC = 90°
4. Given
5. m∠ABD = 90°
5. Trans. Prop.
6. m∠ABC = m∠ABD + m∠DBC 6. ∠ Add. Post.
7. m∠ABC = 90° + 90° = 180° 7. Substitution Property
8. ∠ABC is a straight angle.
SpringBoard® Integrated Mathematics I, Unit 3 • Lines, Segments, and Angles
8. Definition of straight ∠
© 2017 College Board. All rights reserved.
ASSESS
© 2017 College Board. All rights reserved.
ACTIVITY 15 Continued