Congruent Triangles.pptx
7/16/15
Congruent Triangles OBJECTIVES: USE PROPERTIES OF CONGRUENCE AND PROVE THEM CONGRUENT VIA DEFINITION (4.4)
Vocabulary Corresponding angles Corresponding sides Congruent polygons Third Angle Theorem Transitive Property
What does congruent mean? 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.
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7/16/15
How to name a polygon 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. P
Helpful Hint
Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices.
S
Q
P
R
Example 1: Naming Congruent Corresponding Parts
Given: ∆PQR ≅ ∆STW Identify all pairs of corresponding congruent parts.
Q
Example 2:
S
T
Using Corresponding Parts of Congruent Triangles
Given: ∆ABC ≅ ∆DBC. Find the value of x.
Angles: ∠P ≅ ∠S,
Sides: PQ ≅ ST,
∠Q ≅ ∠T,
QR ≅ TW,
∠R ≅ ∠W
PR ≅ SW
∠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
Substitute values for m∠BCA and m∠BCD. Add 16 to both sides. Divide both sides by 2.
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Example 3: Proving Triangles Congruent
Statements 1. ∠YWX and ∠YWZ are rt. ∠s.
1. Given
Given: ∠YWX and ∠YWZ are right angles.
2. ∠YWX ≅ ∠YWZ
2. Rt. ∠ ≅ Thm.
YW bisects ∠XYZ. W is the midpoint of XZ. XY ≅ YZ.
3. YW bisects ∠XYZ
3. Given
4. ∠XYW ≅ ∠ZYW
4. Def. of bisector
Prove: ∆XYW ≅ ∆ZYW
Reasons
5. ∠X ≅ ∠Z
5. Third ∠s Thm.
6. W is mdpt. of XZ
6. Given
7. XW ≅ ZW
7. Def. of mdpt.
8. YW ≅ YW 9. XY ≅ YZ
8. Reflex. Prop. of ≅ 9. Given
10. ∆XYW ≅ ∆ZYW
10. Def. of ≅ ∆
End (SLIDES FROM HOLT MCDOUGAL GEOMETRY USED WITH PERMISSION)
3
Congruent Triangles OBJECTIVES: USE PROPERTIES OF CONGRUENCE AND PROVE THEM CONGRUENT VIA DEFINITION (4.4)
Vocabulary Corresponding angles Corresponding sides Congruent polygons Third Angle Theorem Transitive Property
What does congruent mean? 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.
1
7/16/15
How to name a polygon 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. P
Helpful Hint
Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices.
S
Q
P
R
Example 1: Naming Congruent Corresponding Parts
Given: ∆PQR ≅ ∆STW Identify all pairs of corresponding congruent parts.
Q
Example 2:
S
T
Using Corresponding Parts of Congruent Triangles
Given: ∆ABC ≅ ∆DBC. Find the value of x.
Angles: ∠P ≅ ∠S,
Sides: PQ ≅ ST,
∠Q ≅ ∠T,
QR ≅ TW,
∠R ≅ ∠W
PR ≅ SW
∠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
Substitute values for m∠BCA and m∠BCD. Add 16 to both sides. Divide both sides by 2.
2
7/16/15
Example 3: Proving Triangles Congruent
Statements 1. ∠YWX and ∠YWZ are rt. ∠s.
1. Given
Given: ∠YWX and ∠YWZ are right angles.
2. ∠YWX ≅ ∠YWZ
2. Rt. ∠ ≅ Thm.
YW bisects ∠XYZ. W is the midpoint of XZ. XY ≅ YZ.
3. YW bisects ∠XYZ
3. Given
4. ∠XYW ≅ ∠ZYW
4. Def. of bisector
Prove: ∆XYW ≅ ∆ZYW
Reasons
5. ∠X ≅ ∠Z
5. Third ∠s Thm.
6. W is mdpt. of XZ
6. Given
7. XW ≅ ZW
7. Def. of mdpt.
8. YW ≅ YW 9. XY ≅ YZ
8. Reflex. Prop. of ≅ 9. Given
10. ∆XYW ≅ ∆ZYW
10. Def. of ≅ ∆
End (SLIDES FROM HOLT MCDOUGAL GEOMETRY USED WITH PERMISSION)
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