geometry section 4-3
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Congruent TrianglesTRANSCRIPT
SECTION 4-3Congruent Triangles
Thursday, February 2, 2012
ESSENTIAL QUESTIONS
How do you name and use corresponding parts of congruent polygons?
How do you prove triangles congruent using the definition of congruence?
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VOCABULARY
1. Congruent:
2. Congruent Polygons:
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:
Thursday, February 2, 2012
VOCABULARY
1. Congruent: Two figures with exactly the same size and shape
2. Congruent Polygons:
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:
Thursday, February 2, 2012
VOCABULARY
1. Congruent: Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:
Thursday, February 2, 2012
VOCABULARY
1. Congruent: Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon
3. Corresponding Parts: The matching parts between two polygons; Corresponding parts have the same position in each polygon
Theorem 4.3 - Third Angles Theorem:
Thursday, February 2, 2012
VOCABULARY
1. Congruent: Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are congruent to matching parts of another polygon
3. Corresponding Parts: The matching parts between two polygons; Corresponding parts have the same position in each polygon
Theorem 4.3 - Third Angles Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles must be congruent
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G ∠E ≅ ∠F
Thursday, February 2, 2012
EXAMPLE 1Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G ∠E ≅ ∠F
Since all corresponding parts are congruent, ABCDE ≅ JIHGF
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EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
∠P ≅ ∠O
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9 IT ≅ GN
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9 x − 2 y = 7.5 IT ≅ GN
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9 x − 2 y = 7.5 IT ≅ GN
x − 2(9) = 7.5
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9 x − 2 y = 7.5 IT ≅ GN
x − 2(9) = 7.5
x − 18 = 7.5
Thursday, February 2, 2012
EXAMPLE 2In the diagram, ∆ITP ≅ ∆GNO. Find the values of x and y.
6 y − 14 = 40 ∠P ≅ ∠O
6 y = 54
y = 9 x − 2 y = 7.5 IT ≅ GN
x − 2(9) = 7.5
x − 18 = 7.5
x = 25.5
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles3. ∠M ≅ ∠O
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles3. ∠M ≅ ∠O 3. Third Angle Theorem
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles3. ∠M ≅ ∠O 3. Third Angle Theorem
4. LMN ≅PON
Thursday, February 2, 2012
EXAMPLE 3Write a two-column proof.
Given: ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON
Prove: LMN ≅PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN , MN ≅ ON 1. Given2. ∠LNM ≅ ∠PNO 2. Vertical Angles3. ∠M ≅ ∠O 3. Third Angle Theorem
4. LMN ≅PON 4. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
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CHECK YOUR UNDERSTANDING
p. 256 #1-8
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PROBLEM SET
Thursday, February 2, 2012
PROBLEM SET
p. 257 #9-23 odd, 29, 36, 39, 49, 53, 55
“I've always tried to go a step past wherever people expected me to end up.” - Beverly Sills
Thursday, February 2, 2012