1.6B Special Lines in a Triangle

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1. Angle bisector of a triangle: An angle bisector of a triangle is a segment or ray that bisects an angle and extends to the opposite side.

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Thus BD, the angle bisector of /B in Fig. 1-31, bisects /B, making /1 > /2.

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2. Median of a triangle: A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

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Thus BM, the median to AC, in Fig. 1-32, bisects AC, making AM MC.

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Fig. 1-31

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Fig. 1-32

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3. Perpendicular bisector of a side: A perpendicular bisector of a side of a triangle is a line that bisects and is perpendicular to a side.

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Thus PQ, the perpendicular bisector of AC in Fig. 1-32, bisects AC and is perpendicular to it.

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4. Altitude to a side of a triangle: An altitude of a triangle is a segment from a vertex perpendicular to the opposite side.

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Thus BD, the altitude to AC in Fig. 1-33, is perpendicular to AC and forms right angles 1 and 2. Each angle bisector, median, and altitude of a triangle extends from a vertex to the opposite side.

CHAPTER 1 Lines, Angles, and Triangles

5. Altitudes of obtuse triangle: In an obtuse triangle, the altitude drawn to either side of the obtuse angle falls outside the triangle.

Thus in obtuse triangle ABC (shaded) in Fig. 1-34, altitudes BD and CE fall outside the triangle. In each case, a side of the obtuse angle must be extended.

Fig. 1-33

Fig. 1-34

SOLVED PROBLEMS

1.11 Naming a triangle and its parts In Fig. 1-35, name (a) an obtuse triangle, and (b) two right triangles and the hypotenuse and legs of each. (c) In Fig. 1-36, name two isosceles triangles; also name the legs, base, and vertex angle of each.

Fig. 1-35

Fig. 1-36

Solutions

(a) Since /ADB is an obtuse angle, /ADB or n II is obtuse. (b) Since/C is a right angle, n I and n ABC are right triangles. In n I, AD is the hypotenuse and AC and CD are the legs. In n ABC, AB is the hypotenuse and AC and BC are the legs. (c) Since AD AE, n ADE is an isosceles triangle. In n ADE, AD and AE are the legs, DE is the base, and /A is the vertex angle. Since AB AC, n ABC is an isosceles triangle. In n ABC, AB and AC are the legs, BC is the base, and /A is the vertex angle.

1.12 Special lines in a triangle Name the equal segments and congruent angles in Fig. 1-37, (a) if AE is the altitude to BC; (b) if CG bisects /ACB; (c) if KL is the perpendicular bisector of AD; (d) if DF is the median to AC.

Fig. 1-37

CHAPTER 1 Lines, Angles, and Triangles

Solutions

(a) Since AE ' BC, /1 > /2. (b) Since CG bisects /ACB, /3 > /4. (c) Since LK is the ' bisector of AD, AL (d) Since DF is median to AC, AF FC. LD and /7 > /8.

1.7 Pairs of Angles

1.7A Kinds of Pairs of Angles

1. Adjacent angles: Adjacent angles are two angles that have the same vertex and a common side between them.

Thus, the entire angle of c in Fig. 1-38 hasS been cut into two adjacent angles of a and b . These adjacent angles have the same vertex A, and a common side AD between them. Here, a b c.

Fig. 1-38

Fig. 1-39

2. Vertical angles: Vertical angles are two nonadjacent angles formed by two intersecting lines.

Thus, /1 and /3 in Fig. 1-39 are vertical angles formed by intersecting lines AB and CD. Also, /2 and /4 are another pair of vertical angles formed by the same lines.

3. Complementary angles: Complementary angles are two angles whose measures total 90 .

Thus, in Fig. 1-40(a) the angles of a and b are adjacent complementary angles. However, in (b) the complemenb 90 . Either of two complementary angles is said to be the comtary angles are nonadjacent. In each case, a plement of the other.

Fig. 1-40

Fig. 1-41