1. Chapter 9
Geometry
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Chapter 9: Geometry
9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles 9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5 Space Figures, Volume, and Surface Area 9.6 Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry
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Chapter 1
Section 9-4
The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem
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The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem
Congruent Triangles
Similar Triangles
The Pythagorean Theorem
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Congruent Triangles
Triangles that are both the same size and same shape are called congruent triangles. B E A D C F
The corresponding sides are congruent and corresponding angles have equal measures. Notation:
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6. Congruence Properties - SAS
Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.
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7. Congruence Properties - ASA
Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.
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8. Congruence Properties - SSS
Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.
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9. Example: Proving Congruence (SAS)
Given: CE = ED
AE = EB
Prove: C B E A
Proof D
STATEMENTS
REASONS
1. CE = ED
1. Given
2. AE = EB
2. Given 3.
3. Vertical Angles are equal 4.
4. SAS property
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10. Example: Proving Congruence (ASA)
Given:
Prove: B C A D
Proof
STATEMENTS
REASONS 1.
1. Given 2.
2. Given
3. DB = DB
3. Reflexive property 4.
4. ASA property
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11. Example: Proving Congruence (SSS)
Given: AD = CD
AB = CB
Prove: B
Proof A C D
STATEMENTS
REASONS
1. AD = CD
1. Given
2. AB = CB
2. Given
3. BD = BD
3. Reflexive property 4.
4. SSS property
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12. Important Statements About Isosceles Triangles
If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold.
1. The base angles A and C are equal.
2. Angles ABD and CBD are equal.
3. Angles ADB and CDB are both right
angles. B A D C
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Similar Triangles
Similar Triangles are pairs of triangles that are exactly the same shape, but not necessarily the same size. The following conditions must hold.
1. Corresponding angles must have the same
measure.
2. The ratios of the corresponding sides must be
constant; that is, the corresponding sides are
proportional.
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14. Angle-Angle (AA) Similarity Property
If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar.
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15. Example: Finding Side Length in Similar Triangles8 B F
Find the length of side DF. 16 24 D
Solution C 32
Set up a proportion with corresponding sides: A
Solving, we find that DF = 16.
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Pythagorean Theorem
If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then
That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.
hypotenuse c
leg a
leg b
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17. Example: Using the Pythagorean Theorem
Find the length a in the right triangle below. 39 a
Solution 36
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18. Converse of the Pythagorean Theorem
If the sides of lengths a, b, and c, where c is the length of the longest side, and if
then the triangle is a right triangle.
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19. Example: Applying the Converse of the Pythagorean Theorem
Is a triangle with sides of length 4, 7, and 8, a right triangle?
Solution ? ?
No, it is not a right triangle.
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