LESSON 2-7
Focus on Mathematical Content
As you learned in Chapter 1, a segment can be measured, and measures can be used in
calculations because they are real numbers. The Ruler Postulate states that the points on any
line or line segment can be paired with real numbers so that, given any two points A and B on a
line, A corresponds to 0, and B corresponds to a positive real number. That number is the length
of the segment. Another postulate states that if point B lies between points A and C on the same
line, AB + BC = AC. The converse statement holds true as well.
AB + BC = AC
The Reflexive, Symmetric, and Transitive Properties of Equality can be used to write proofs about
segment congruence. The theorem resulting from the proofs states that congruence of segments
is reflexive, symmetric, and transitive.
1. Prove that if AB ≅ CD, then AC ≅ BD
Proof:
Statements
Reasons
1. AB ≅ CD
2. AB = CD
3. BC = BC
4. AB + BC = AC
5. CD + BC = AC
6. CD + BC = BD
7. AC = BD
8. AC ≅ BD
2. Segment Congruence
Jamie is designing a badge for her club. The length of the top edge of the badge is equal to the
length of the left edge of the badge. The top edge of the badge is congruent to the right edge of
the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove
that the bottom edge of the badge is congruent to the left edge of the badge.
Given:
WY = YZ
YZ ≅ XZ
XZ ≅ WX
Prove: WX ≅ WY
LESSON 2-7
Proof:
Statements
Reasons
1. WY = YZ
YZ ≅ XZ
XZ ≅ WX
2. WY ≅ YZ
3. YZ ≅ WX
4. WX ≅ WY
DISTANCE Martha and Laura live 1400 meters apart. A library is opened between them and is
500 meters from Martha.
How far is the library from Laura?
Complete this proof.
Given: C is the midpoint of ̅̅̅̅
𝐵𝐷 and ̅̅̅̅
𝐴𝐸 .
Prove: AB = DE
Statements
Reasons
1. C is the midpoint of ̅̅̅̅
𝐵𝐷 and ̅̅̅̅
𝐴𝐸 .
1. Given
2. BC = CD and _____________
2.
3. AC = AB + BC,
CE = CD + DE
3.
4. AB = AC – BC
4.
5.
5. Substi. Prop. of =
6. DE = CE – CD
6.
LESSON 2-8
Focus on Mathematical Content
This lesson introduces postulates and theorems about angle relationships.
The Protractor Postulate states,
 The Protractor Postulate states, “Given
and a number r between 0 and 180,
there is exactly one ray with end-point A, extending on either side of
such
that the measure of the angle formed is r.”
 The Angle Addition Postulate states that if R is in the interior of ∠PQS, then
m∠PQR + m∠RQS = m∠PQS. The converse is also true.
What do the angles of rectangles and squares have in common? ___________________
If a square and a rectangle are adjacent, what is the sum of the measure of any two
adjacent angles?
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Supplementary and Complementary Angles
I. Using a protractor, a construction worker measures that the angle a beam makes
with the ceiling is 42°. What is the measure of the angle the beam makes with the
wall?
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II. At 4 o'clock, the angle between the hour and minute hands of a clock is 120°.
When the second hand bisects the angle between the hour and minute hands,
what are the measures of the angles between the minute and second hands and
between the second and hour hands?
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LESSON 2-8
Congruent and Right Angles
III. In the figure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180.
Prove that ∠3 and ∠4 are congruent.
Statements
Reasons
1. m∠3 + m∠1 = 180;
∠1 and ∠4 form a linear pair.
2. ∠1 and ∠4 are supplementary.
3. ∠3 and ∠1 are supplementary.
4. ∠3 ≅ ∠4
IV. If ∠1 and ∠2 are vertical angles and m∠1 = d − 32 and m∠2 = 175 − 2d, find m∠1
and m∠2. Justify each step.
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Statements
1. ∠1 and ∠2 are vertical angles.
2. ∠1 ≅ ∠2
3.
4.
5.
6.
m∠1 = m∠2
d − 32 = 175 − 2d
3d = 207
d = 69
7. m∠1 = 37
8. m∠2 = 37
Reasons