Download Unit 1 Lines, Angles, and Triangles 1.3 Triangles

Unit 1
Lines, Angles, and Triangles
1.3 Triangles
Definitions and Notation
Triangle
Vertices of a triangle
∆ABC
Equilateral triangle
Isosceles triangle
Vertex angle of an isosceles triangle
Scalene triangle
AB = BC = AC
Equilateral triangle
Acute triangle
Right triangle
Hypotenuse
Obtuse triangle
Acute triangle
A plane closed figure with three line segments as sides
The points at which two sides of a triangle intersect
Triangles with vertices A, B, and C
A triangle with three equal sides
The angle formed by the sides of equal length
The angle formed by the sides of equal length
A triangle with no two sides equal
AB = BC
Isosceles triangle; ∠B is the vertex angle
AB ≠ BC, AB ≠ AC, BC ≠ AC
Scalene triangle
A triangle with each of its angles acute
A triangle with a right angle
The side of a right triangle opposite the right angle
A triangle with an obtuse angle
Right triangle
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Unit 1
Perimeter of a triangle
An altitude of a triangle
h
Base of a triangle
b
Median of a triangle
Lines, Angles, and Triangles
1.3 Triangles
Obtuse triangle
The sum of the lengths of the sides
A line segment from any vertex perpendicular to a line
containing the opposite side
Length of an altitude
The side to which an altitude is perpendicular
Length of the base
A line segment from any vertex to the midpoint of the
side opposite that vertex
F is the midpoint of AE , therefore CF is a median.
B is the midpoint of AC , therefore EB is a median.
D is the midpoint of EC , therefore AD is a median.
(Note: The medians of a triangle intersect in a common point called the centroid of the triangle.)
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
Properties
1. If ∠A, ∠B, and ∠C are angles of a triangle, then m ∠A + m ∠B + m ∠C =180°
2. The Pythagorean Theorem: In a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of the lengths of other two sides.
c2 = b 2 + c2
3. In a right triangle, the acute angles are complimentary.
4. In an equilateral triangle:
a. The measure of each angle is 60°.
b. The altitude equals one-half the length of a side times
c. An altitude bisects the base.
.
a. m∠A = m∠ABC=m∠C=60°
s 3
b. h=
3
s
c. AD=DC=
2
5. In an isosceles triangle:
a. The angles opposite the equal sides are equal.
b. The altitude from the vertex angle bisects the base.
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
a. ∠A = ∠B
b. AD = DB
6. In a right triangle whose acute angles have measures of 45°:
a. The lengths of the sides opposite the acute angles are equal.
b. The length of the hypotenuse is equal to
other sides.
times the length of one of the
a. BC = CA = a
b. c=a 2
7. In a right triangle whose acute angles have measures of 30° and 60°:
a. The length of the hypotenuse is double the length of the short side.
b. The length of the side opposite the angle whose measure is 60° is
the length of the short side.
times
a. c = 2a
b. b=a 3
8. If two angles of a triangle are equal, the sides opposite these angles are equal.
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
Exercises
Use the information given in the figure to find the measure of the indicated angle.
Example: ∠DAB
Solution:
Since, by Property 1, the sum of the measures of the three angles of a triangle equals 180°,
and the measures of the two given angles of ∆DAB have a sum of 120°, the measure of the
third angle, ∆DAB, equals 60°.
1. ∠CED
2. ∠BCD
3. ∠BDE
5. ∠BDC
6. ∠FEC
7. ∠DFE
4. ∠CDE
8. Name the obtuse triangle.
9. Name the 30°-60°-90° triangle.
10. Name the 45°-45°-90° triangles.
11. Name the two acute triangles
12. Name the altitude to base BC of ∆DAB
13. If E is the midpoint of BC , name a median which can be drawn.
14. If G is the midpoint of AB , name a median of ∆ABC which can be drawn.
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
Find the lengths (to the nearest tenth of a unit) of the remaining sides of each triangle as
pictured in the examples below, where the length of one side is given.
Use Table 1 as needed.
Example: ∆ABC, b = 6.2 cm
Solution
By Property 6a, a = b.
Hence, a = 6.2 cm.
By property 6b, c = c= ( 6.2 ) 2 ≈ 8.8 cm
Example: ∆DEF, e = 9.6 ft
Solution
By property 7a, f = 2e.
Hence, f = 2(9.6) = 19.2 ft.
By property 7b, d= ( 9.6 ) 3 ≈ 16.6 cm
Use the triangles above for the following:
15. ∆ABC, a = 4.0 yd
16. ∆ABC,
17. ∆ABC, c = 8.0 yd
18. ∆ABC,
19. ∆DEF, d = 4.0 yd
20. ∆DEF,
22. ∆DEF,
21. ∆DEF, f = 9.0 yd
c = 4.0 yd
b = 8.0 yd
f = 4.0 yd
e = 4.0 yd
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
In exercises 23-26, use the information in the figures to classify each triangle as equilateral,
isosceles, or scalene. Use the following figures for Exercises 23-34.
23. ∆ABC
25. ∆GFE
27. Find m∠ABC
29. Find m∠GFH
31. Find BD
33. Find the perimeter of ∆ABC.
34. Find the perimeter of ∆GFH.
24. ∆ABD
26. ∆FHG
28. Find m∠ABD
30. Find m∠GEF
32. Find GH and HF
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Unit 1
Lines, Angles, and Triangles
1.3 Triangles
Use the information in the figures to find each of the following measures.
Examples:
b. PM
a. PQ
By the Pythagorean theorem,
By Property 2 (Pythagorean theorem),
2
2
2
(PM)2 = (QM)2 + (PQ)2
(PN) = (PQ) + (QN)
(PM)2 = 62 + 82
172 = (PQ)2 + 152
2
(PM)2 = 100
289 = (PQ) + 225
289 - 225 = (PQ)2
PM = 10
64 = (PQ)2
8 = PQ
35. CA
36. EA
37. BD
38. BE
39. The perimeter of ∆BDC
40. The perimeter of ∆BEA
Use the information in the figures to find each of the following measures (to the nearest
tenth of a unit).
41. AC
42. AB
43. RS
44. PR
Vocabulary
Section 1.1
triangle
equilateral
perimeter
altitude
median
sum
triángulo
equilátero
perímetro
altitud
mediana
suma
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Unit 1
difference
bisect
Lines, Angles, and Triangles
1.3 Triangles
diferencia
biseque
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