Download Chapter 10: Introducing Geometry

Chapter 10: Introducing Geometry
10.4
More About Triangles
10.4.1. Congruent Triangles
10.4.1.1. Definition of congruent triangles – Two triangles are congruent if and only if,
for some correspondence between the two triangles, each pair of corresponding
sides are congruent and each pair of corresponding angles are congruent
10.4.1.2. Triangle congruence postulates
10.4.1.2.1. SSS – if all of the corresponding pairs of sides of a triangle are congruent,
then the two triangles are congruent
10.4.1.2.2. SAS – If two sides and the included angle of the corresponding pairs of
sides and angles of a triangle are congruent, then the two triangles are
congruent
10.4.1.2.3. ASA – If two angles and the shared side of the corresponding pairs of
angles and sides of a triangle are congruent, then the two triangles are
congruent
10.4.1.2.4. AAS – If two angles and a non-shared side of the corresponding pairs of
angles and sides of a triangle are congruent, then the two triangles are
congruent
10.4.1.2.5. For Right Triangles ONLY –
10.4.1.2.5.1.
HA – If the hypotenuse and one angle of the corresponding pairs of
angles and sides of a right triangle are congruent, then the two right
triangles are congruent
HL – If the hypotenuse and one leg of the corresponding pairs of
10.4.1.2.5.2.
sides of a right triangle are congruent, then the two right triangles are
congruent
10.4.1.2.5.3.
What congruence postulates for any triangle make these rules for
right triangles work?
10.4.1.3. T-proof
Statement
AB ≅ DC
AC ≅ DB
EC ≅ EB
AE ≅ DE
∆ACE ≅ ∆DBE
Reason
Given
Added BC to AB and CB to DC
Given
Given
SSS
10.4.1.3.1.
10.4.2. Similar Triangles
10.4.2.1. Definition of a Similar Triangle – two triangles are similar, if and only if, for some
correspondence between the two triangles, each pair of corresponding angles are
congruent and the ratios of the corresponding sides are equal
10.4.2.2. AA Similarity Postulate – if two angles of one triangle are congruent,
respectively, to two angles of another triangle, then the two triangles are similar
10.4.2.3. Right Triangle Similarity Theorem – if an acute angle of one right triangle is
congruent to an acute angle of another right triangle, then the triangles are similar
10.4.2.4. The SAS Similarity Theorem – if an angle of one triangle is congruent to an
angle of another triangle and the lengths of the sides including these angles are
proportional, then the triangles are similar
10.4.2.5. The SSS Similarity Theorem – if the corresponding sides of two triangles are
proportional, then the triangles are similar
10.4.3. The Pythagorean Theorem
10.4.3.1. used for right triangles ONLY
10.4.3.2. a 2 + b 2 = c 2
10.4.3.3. a and b are legs of a right triangle
10.4.3.4. c is ALWAYS the hypotenuse of the right triangle
10.4.3.5. Pythagorean triples –
10.4.3.5.1. 3, 4, 5
10.4.3.5.2. 5, 12, 13
10.4.3.5.3. ???
10.4.4. Special Right Triangles
10.4.4.1. 45°, 45°, 90°
10.4.4.1.1. c = a 2
OR
10.4.4.1.2. c = b 2
10.4.4.2. 30°, 60°, 90°
10.4.4.2.1. c = 2a
where a is the shorter leg
10.4.4.2.2. b = a 3
10.4.5. Problems and Exercises p. 628
10.4.5.1. Home work: 1, 3, 5, 6, 12, 13, 16, 17, 19, 24, 28, 31, 35, 37, 43-45