SOLVED: The proof of Theorem 5.6.3 requires the use of Theorem 5.6.1. THEOREM 5.6.3 (The Angle-Bisector Theorem): If a ray bisects one angle of a triangle, then it divides the opposite side into segme

Angle bisector theorem proof | Special properties and parts of triangles | Geometry | Khan Academy
Angle bisector theorem proof | Special properties and parts of triangles | Geometry | Khan Academy

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The proof of Theorem 5.6.3 requires the use of Theorem 5.6.1.
THEOREM 5.6.3 (The Angle-Bisector Theorem): If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.
GIVEN:
Triangle ABC in Figure 5.51(a), in which CD bisects angle ACB.
AD and DB are the segments formed by the bisecting ray.
PROVE: AC/CB
PROOF: We begin by extending BC beyond C (there is only one line through B and C) to meet the line drawn through A parallel to DC. [See Figure 5.51(b).] Let E be the point of intersection. (These lines must intersect; otherwise, AE would have two parallels, BC and CD, through point C.) Because CD bisects angle ACB, we have:
EC/AD = CB/DB (by Theorem 5.6.1)
Now, angle L1 = angle L2 because CD bisects angle ACB. Angle L2 = angle L4 (corresponding angles for parallel lines), and angle L3 = angle L4 (alternate interior angles for parallel lines). By the Transitive Property, angle L1 = angle L3. Therefore, triangle LACE is isosceles with EC = AC.
Using substitution, the starred proportion becomes:
AC/CB = AD/DB (by inversion)
AC/CB = AD/DB
Figure 5.51

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Transcript

In the given question, we have been given triangle a b c in which line c d bisects the angle, a c b. We need to prove that a d divided by a c equal to d b, divided by c b. So to solve this question, we first extend the line b c and parallel to c d. So this is the extension helinand. We also extend a line from point a that is parallel to c r, so both the lines meet at point e. Suppose this angle, a a c, is 3, so to prove this equation, we have to use the concept of parallel lines so because, as we know, that c d is parallel to a so. We can write that e c, divided by a d equal to c b. Divided by d v, as we know that angle 1 is congruent to angle 2, because c d, bisect angle, a c band, angle, 1 and 3 is also corresponding…

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You are watching: SOLVED: The proof of Theorem 5.6.3 requires the use of Theorem 5.6.1. THEOREM 5.6.3 (The Angle-Bisector Theorem): If a ray bisects one angle of a triangle, then it divides the opposite side into segme. Info created by THVinhTuy selection and synthesis along with other related topics.

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