Download Lesson 5-2: Congruent Triangles

Triangles
4-3
5-2 Congruent
Congruent
Triangles
Warm Up
Lesson Presentation
Class Practice
HoltGeometry
Geometry
Holt
5-2 Congruent Triangles
4-3
Warm Up
1. Name all sides and angles of ∆FGH.
FG, GH, FH, F, G, H
2. What is true about K and L? Why?
 ;Third s Thm.
3. What does it mean for two segments to
be congruent?
They have the same length.
Holt Geometry
5-2 Congruent Triangles
4-3
Objectives
Use properties of congruent triangles.
Prove triangles congruent by using the definition
of congruence.
Holt Geometry
5-2 Congruent Triangles
4-3
Vocabulary
• corresponding angles
• corresponding sides
• congruent polygons
Holt Geometry
5-2 Congruent Triangles
4-3
What makes two figures congruent?
Geometric figures are congruent if they are the
same size and shape.
Corresponding angles and corresponding sides
are in the same position in polygons with an equal
number of sides.
Two polygons are congruent polygons if and
only if their corresponding sides are congruent.
Thus triangles that are the same size and shape
are congruent.
Holt Geometry
5-2 Congruent Triangles
4-3
Holt Geometry
5-2 Congruent Triangles
4-3
Helpful Hint
Two vertices that are the endpoints
of a side are called consecutive
vertices.
For example, P and Q are
consecutive vertices.
Holt Geometry
5-2 Congruent Triangles
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To name a polygon, write the vertices
in consecutive order. For example, you
can name polygon PQRS as QRSP or
SRQP, but not as PRQS.
In a congruence statement, the order
of the vertices indicates the
corresponding parts.
Holt Geometry
5-2 Congruent Triangles
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Helpful Hint
When you write a statement such as
ABC  DEF, you are also stating
which parts are congruent.
Holt Geometry
5-2 Congruent Triangles
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Example 1: Naming Congruent Corresponding Parts
Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.
Angles: P  S, Q  T, R  W
Sides: PQ  ST, QR  TW, PR  SW
Holt Geometry
5-2 Congruent Triangles
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Check It Out!
If polygon LMNP  polygon EFGH, identify all
pairs of corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH
Holt Geometry
5-2 Congruent Triangles
4-3
“I Do”
Given: ∆ABC  ∆DBC.
Find the value of x.
Holt Geometry
5-2 Congruent Triangles
4-3
Example 2A: Using Corresponding Parts of Congruent
Triangles
Given: ∆ABC  ∆DBC.
Find the value of x.
BCA and BCD are rt. s.
Def. of  lines.
BCA  BCD
Rt.   Thm.
mBCA = mBCD
Def. of  s
(2x – 16)° = 90°
2x = 106
x = 53
Holt Geometry
Substitute values for mBCA and
mBCD.
Add 16 to both sides.
Divide both sides by 2.
5-2 Congruent Triangles
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“We Do”
Given: ∆ABC  ∆DBC.
Find mDBC.
Holt Geometry
5-2 Congruent Triangles
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Example 2B: Using Corresponding Parts of Congruent
Triangles
Given: ∆ABC  ∆DBC.
Find mDBC.
mABC + mBCA + mA = 180° ∆ Sum Thm.
Substitute values for mBCA and
mABC + 90 + 49.3 = 180
mA.
mABC + 139.3 = 180 Simplify.
mABC = 40.7
DBC  ABC
Subtract 139.3 from both
sides.
Corr. s of  ∆s are  .
mDBC = mABC Def. of  s.
mDBC  40.7°
Holt Geometry
Trans. Prop. of =
5-2 Congruent Triangles
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“You Do With a Partner”
Given: ∆ABC  ∆DEF
Find the value of x.
Holt Geometry
5-2 Congruent Triangles
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Check It Out! Example 2a
Given: ∆ABC  ∆DEF
Find the value of x.
AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
2x – 2 = 6
2x = 8
x=4
Holt Geometry
Substitute values for AB and DE.
Add 2 to both sides.
Divide both sides by 2.
5-2 Congruent Triangles
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“We Do”
Given: ∆ABC  ∆DEF
Find mF.
Holt Geometry
5-2 Congruent Triangles
4-3
“You Do with a Partner”
Given: ∆ABC  ∆DEF
Find mF.
Holt Geometry
5-2 Congruent Triangles
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Check It Out! Example 2b
Given: ∆ABC  ∆DEF
Find mF.
mEFD + mDEF + mFDE = 180°
ABC  DEF
Corr. s of  ∆ are .
mABC = mDEF
Def. of  s.
mDEF = 53°
Transitive Prop. of =.
mEFD + 53 + 90 = 180
mF + 143 = 180
mF = 37°
Holt Geometry
∆ Sum Thm.
Substitute values for mDEF
and mFDE.
Simplify.
Subtract 143 from both sides.
5-2 Congruent Triangles
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“You Do Alone”
,X= 20
degrees.
Holt Geometry
5-2 Congruent Triangles
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Holt Geometry
5-2 Congruent Triangles
4-3
Holt Geometry
5-2 Congruent Triangles
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Prove It!
Given: AD bisects BE.
BE bisects AD.
AB  DE, A  D
Prove: ∆ABC  ∆DEC
Holt Geometry
5-2 Congruent Triangles
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Statements
Reasons
1. A  D
1. Given
2. BCA  DCE
2. Vertical s are .
3. ABC  DEC
3. Third s Thm.
4. AB  DE
4. Given
5. AD bisects BE,
5. Given
BE bisects AD
6. BC  EC, AC  DC
6. Def. of bisector
7. ∆ABC  ∆DEC
7. Def. of  ∆s
Holt Geometry
5-2 Congruent Triangles
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Example 4: Engineering Application
The diagonal bars across a gate give it
support. Since the angle measures and the
lengths of the corresponding sides are the
same, the triangles are congruent.
Given: PR and QT bisect each other.
PQS  RTS, QP  RT
Prove: ∆QPS  ∆TRS
Holt Geometry
5-2 Congruent Triangles
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Example 4 Continued
Statements
1. QP  RT
2. PQS  RTS
3.
4.
5.
6.
7.
1.
2.
PR and QT bisect each other. 3.
QS  TS, PS  RS
4.
QSP  TSR
5.
QSP  TRS
6.
∆QPS  ∆TRS
7.
Holt Geometry
Reasons
Given
Given
Given
Def. of bisector
Vert. s Thm.
Third s Thm.
Def. of  ∆s
5-2 Congruent Triangles
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Lesson Quiz
1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50.
Find x and AB. 31, 74
Given that polygon MNOP  polygon QRST,
identify the congruent corresponding part.
RS
P
2. NO  ____
3. T  ____
4. Given: C is the midpoint of BD and AE.
A  E, AB  ED
Prove: ∆ABC  ∆EDC
Holt Geometry
5-2 Congruent Triangles
4-3
Lesson Quiz
4.
Statements
Reasons
1. A  E
1. Given
2. C is mdpt. of BD and AE
2. Given
3. AC  EC; BC  DC
3. Def. of mdpt.
4. AB  ED
4. Given
5. ACB  ECD
5. Vert. s Thm.
6. B  D
6. Third s Thm.
7. ABC  EDC
7. Def. of  ∆s
Holt Geometry