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Learning Target
I can use theorems, postulates or definitions to prove that…
a. vertical angles are congruent.
b. When a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and same-side interior angles are supplementary.
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Proving Vertical Angle Theorem
THEOREM
Vertical Angles Theorem
Vertical angles are congruent
1 3, 2 4
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Proving Vertical Angle Theorem
PROVE 5 7
GIVEN 5 and 6 are a linear pair,6 and 7 are a linear pair
1
2
3
Statements Reasons
5 and 6 are a linear pair, Given6 and 7 are a linear pair
5 and 6 are supplementary, Linear Pair Postulate6 and 7 are supplementary
5 7 Congruent Supplements Theorem
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Third Angles Theorem
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Goal 1
The Third Angles Theorem below follows from the Triangle Sum Theorem.
THEOREM
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.If A D and B E, then C F.
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PROPERTIES OF PARALLEL LINES
POSTULATE
POSTULATE 15 Corresponding Angles Postulate
1
2
1 2
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
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PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.4 Alternate Interior Angles
3
4
3 4
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
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PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.5 Consecutive Interior Angles
5
6
m 5 + m 6 = 180°
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
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PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.6 Alternate Exterior Angles
7
8
7 8
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
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PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.7 Perpendicular Transversal
j k
If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.
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Proving the Alternate Interior Angles Theorem
Prove the Alternate Interior Angles Theorem.
SOLUTION
GIVEN p || q
p || q Given
Statements Reasons
1
2
3
4
PROVE 1 2
1 3 Corresponding Angles Postulate
3 2 Vertical Angles Theorem
1 2 Transitive property of Congruence
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Using Properties of Parallel Lines
SOLUTION
Given that m 5 = 65°, find each measure. Tellwhich postulate or theoremyou use.
Linear Pair Postulatem 7 = 180° – m 5 = 115°
Alternate Exterior Angles Theoremm 9 = m 7 = 115°
Corresponding Angles Postulatem 8 = m 5 = 65°
m 6 = m 5 = 65° Vertical Angles Theorem
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Using Properties of Parallel Lines
Use properties ofparallel lines to findthe value of x.
SOLUTION
Corresponding Angles Postulatem 4 = 125°
Linear Pair Postulatem 4 + (x + 15)° = 180°
Substitute.125° + (x + 15)° = 180°
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Subtract.x = 40°
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Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel.
When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that
Estimating Earth’s Circumference: History Connection
m 2 150 of a circle
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Estimating Earth’s Circumference: History Connection
m 2 150 of a circle
Using properties of parallel lines, he knew that
m 1 = m 2
He reasoned that
m 1 150 of a circle
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The distance from Syene to Alexandria was believed to be 575 miles
Estimating Earth’s Circumference: History Connection
m 1 150 of a circle
Earth’s circumference
150 of a circle
575 miles
Earth’s circumference 50(575 miles)
Use cross product property
29,000 miles
How did Eratosthenes know that m 1 = m 2 ?
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Estimating Earth’s Circumference: History Connection
How did Eratosthenes know that m 1 = m 2 ?
SOLUTION
Angles 1 and 2 are alternate interior angles, so
1 2
By the definition of congruent angles,
m 1 = m 2
Because the Sun’s rays are parallel,
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Example
Using the Third Angles Theorem
Find the value of x.
SOLUTIO
NIn the diagram, N R and L S.From the Third Angles Theorem, you know that M T.
So, m M = m T.From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚.
m M = m T
60˚ = (2 x + 30)˚
30 = 2 x
15 = x
Third Angles Theorem
Substitute.
Subtract 30 from each side.
Divide each side by 2.
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Goal 2
SOLUTIO
NParagraph ProofFrom the diagram, you are given that all three corresponding sides are congruent.
, NQPQ ,MNRP QMQR and
Because P and N have the same measures, P
N.By the Vertical Angles Theorem, you know that PQR
NQM.By the Third Angles Theorem, R M.
Decide whether the triangles are congruent. Justify your reasoning.
So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, .
PQR NQM
Proving Triangles are CongruentLearningTarget
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Example
Proving Two Triangles are Congruent
A B
C D
E
|| , DCAB ,
DCAB E is the midpoint of BC and AD.
Plan for Proof Use the fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.
GIVEN
PROVE .AEB DEC
Prove that .AEB DEC
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Example
Proving Two Triangles are Congruent
Statements Reasons
EAB EDC, ABE DCE
AEB DEC
E is the midpoint of AD,E is the midpoint of BC
,DEAE CEBE
Given
Alternate Interior Angles Theorem
Vertical Angles Theorem
Given
Definition of congruent triangles
Definition of midpoint
|| ,DCAB DCAB
SOLUTION
AEB DEC
A B
C D
E
Prove that .AEB DEC
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Goal 2
You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.
THEOREM
Theorem 4.4 Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
D
E
F
A
B
C
J K
L
Every triangle is congruent to itself.Symmetric Property of Congruent Triangles
Transitive Property of Congruent Triangles
If , then .ABC DEF DEF ABC
If and , then .JKLABC DEF DEF ABC JKL
Proving Triangles are Congruent
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1
Using the SAS Congruence Postulate
Prove that AEB DEC.
2
3 AEB DEC SAS Congruence Postulate
21
AE DE, BE CE Given
1 2 Vertical Angles Theorem
Statements
Reasons
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D
GA R
Proving Triangles Congruent
MODELING A REAL-LIFE SITUATION
PROVE DRA DRG
SOLUTION
ARCHITECTURE You are designing the window shown in the drawing. Youwant to make DRA congruent to DRG. You design the window so that DR AG and RA RG.
Can you conclude that DRA DRG ?
GIVEN DR AG
RA RG
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2
3
4
5
6 SAS Congruence Postulate DRA DRG
1
Proving Triangles Congruent
GivenDR AG
If 2 lines are , then they form 4 right angles.
DRA and DRGare right angles.
Right Angle Congruence Theorem DRA DRG
GivenRA RG
Reflexive Property of CongruenceDR DR
Statements Reasons
D
GA R
GIVEN
PROVE DRA DRG
DR AG
RA RG
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Congruent Triangles in a Coordinate Plane
AC FH
AB FGAB = 5 and FG = 5
SOLUTION
Use the SSS Congruence Postulate to show that ABC FGH.
AC = 3 and FH = 3
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Congruent Triangles in a Coordinate Plane
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 3 2 + 5
2
= 34
BC = (– 4 – (– 7)) 2 + (5 – 0 )
2
d = (x 2 – x1 ) 2 + ( y2 – y1 )
2
= 5 2 + 3
2
= 34
GH = (6 – 1) 2 + (5 – 2 )
2
Use the distance formula to find lengths BC and GH.
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Congruent Triangles in a Coordinate Plane
BC GH
All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.
BC = 34 and GH = 34