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5. / A 5 2BD 6. / P 5 2AD / P 5 1(AD 2
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CHAPTER 16 Proofs of Important Theorems
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8c. An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
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Given: /P formed by tangents PA and PD intersecting at P, a point outside circle O. 1 AFD) To Prove: /P 5 2(AED Plan: When chord AD is drawn, /1 becomes an exterior angle of ^ADP, of which /P and /2 are nonadjacent interior angles.
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PROOF:
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Statements 1. Draw AD. 2. m/P m/2 m/1
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Reasons 1. A straight line may be drawn between two points. 2. The measure of an exterior angle of a triangle equals the sum of the measures of the nonadjacent interior angles. 3. Subtraction Postulate. 4. An angle formed by a tangent and a chord is measured by one-half its intercepted arc. 5. Substitution Postulate
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1 4. / 1 5 AED, / 2 5 2AFD
5. / P 5 2 AED / P 5 ( AED
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9. If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar.
Given: ^ABC and ^A B C /A > /A , /B > /B , /C > /C To Prove: ^ABC ~ ^A B C Plan: To prove the triangles similar, it must be shown that corresponding sides are in proportion. This is done by placing the triangles so that a pair of congruent angles coincide, and then repeating this so that another pair of congruent angles coincide.
PROOF:
Statements 1. /A > /A 2. Place ^A B C on ^ABC so that /A coincides with /A. 3. /B > /B 4. BrCr y BC 5. ArBr AB ArCr AC
Reasons 1. Given 2. A geometric figure may be moved without change of its size or shape. Equal angles may be made to coincide. 3. Given 4. Two lines are parallel if their corresponding angles are congruent. 5. A line parallel to one side of a triangle divides the other two sides proportionately. 6. Reasons 1 to 5
6. In like manner, by placing ^A B C on ^ABC so that /B coincides BrCr ArBr with /B, show that AB BC
(contd.)
CHAPTER 16 Proofs of Important Theorems
PROOF:
Statements ArCr BrCr ArBr 7. AB AC BC 8. ^A B C ~ ^ABC
Reasons 7. Things (ratios) equal to the same thing are equal to each other. 8. Two polygons are similar if their corresponding angles are congruent and their corresponding sides are in proportion.
10. If the altitude is drawn to the hypotenuse of a right triangle, then (a) the two triangles thus formed are similar to the given triangle and to each other, and (b) each leg of the given triangle is the mean proportional between the hypotenuse and the projection of that leg upon the hypotenuse.
Given: ^ABC with a right angle at C, altitude CD to hypotenuse AB To Prove: (a) ^ADC ~ ^CDB ~ ^ACB (b) c:a a:p, c:b b:q Plan: The triangles are similar since they have a right angle and a pair of congruent acute angles. The proportions follow from the similar triangles.
PROOF:
Statements 1. /C is a right angle. 2. CD is the altitude to AB . 3. CD ' AB 4. /CDB and /CDA are right angles. 5. /A > /A, /B > /B 6. ^ADC ~ ^ACB, ^BDC ~ ^BCA 7. ^ADC ~ ^CDB 8. c:a a:p, c:b b:q
Reasons 1. Given 2. Given 3. An altitude to a side of a triangle is perpendicular to that side. 4. Perpendiculars form right angles. 5. Reflexive property 6. Right triangles are similar if an acute angle of one is congruent to an acute angle of the other. 7. Triangles similar to the same triangle are similar to each other. 8. Corresponding sides of similar triangles are in proportion.
11. The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
Given: Right ^ABC, with a right angle at C. Legs have lengths a and b, and hypotenuse has length c. To Prove: c2 a2 b2 Plan: Draw CD ' AB and apply Theorem 10.
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