/7. ABC and DEF are two isosceles trian- gles on bases BC and EF respectively. If AB : BC = DE : EF, prove that the two triangles are similar. %3D "Exercise 4.14 #10: If the hypotenuse and a leg of one right triangle are proportional to the hypotenuse and a leg of another, then the two triangles are similar. A4 c' a' T' c' - Given: b b' - Prove: AT ~ AT' Statements Reasons Statements Reasons 1. (see above) 1. Given a2 is b2 5. Subtraction a'2 %| b'2 Transformation 2. If 4 quantities are in proportion, then like powers are in proportion. c2 c'2 a2 b2 6. Alternation %3D b2 b'2 b'2 Transformation 3. c2 = a2 + b2; c'2 = a'2 + b'2 7. If 4 quantitie proportion, the are in proportic b. 3. Pythagorean Theorem b' a' a?+b2 4. b2 a'2+b'2 %3D b'2 4. Substitution (step 3 -> 2) 9. AT ~ AT' 9. 1.1. 6. 7. 2.
About proof (statements and reasons)-similar
*Maybe* you will have to use some of these reasons to prove the statements true. (Those below are the ideas for you to use.)
Def. of ~
Def. of perimeter
Def. of ~ triangles
Def. of median
Def. of midpoint
Multiplication
Division
If 4 quantities are in proportion, then like powers are in proportion.
Subtraction Transformation
Alternation Transformation
Pythagorean Theorem
If 2
In a series of = ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.
In any proportion the product of the means equals the product of the extremes.
Angles inscribed in the same segment or equal segments are equal.
If a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle.
Two isosceles triangles are similar if any angle of one equals the corresponding angle of the other.
C.A.S.T.E. - corresponding angles of similar triangles are equal
C.S.S.T.P. - corresponding sides of similar triangles are proportional
Theorem 57- If two triangles have the three angles of one equal respectively to the three angles of the other, then the triangles are similar
Corollary 57-1 If two angles of one triangle are equal respectively to two angles of another, then the triangles are similar. (a.a.)
Corollary 57-2 Two right triangles are similar if an acute angle of one is equal to an acute angle of the other.
Theorem 58-If two triangles have two pairs of sides proportional and the included angles equal respectively, then the two triangles are similar. (s.a.s.)
Corollary 58-1 If the legs of one right triangle are proportional to the legs of another, the triangles are similar. (l.l.)
Theorem 59- If two triangles have their sides respectively proportional, then the triangles are similar. (s.s.s.)
Theorem 60- If two parallels are cut by three or more transversals passing through a common point, then the corresponding segments of the parallels are proportional.
Theorem 61-If in a right triangle the perpendicular is drawn from the vertex of the right angle to the hypotenuse
Corollary 61-1 If a perpendicular is dropped from any point on a
Theorem 62-The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
Corollary 62-1 The difference of the square of the hypotenuse and the square of one leg equals the square of the other leg.
Theorem 63- If two chords intersect within a circle, then the product of the segments of one is equal to the product of the segments of the other.
Corollary 63-1 The product of the segments of any chord through a fixed point within a circle is constant.
Theorem 64-If from a point outside a circle, a tangent and a secant are drawn to the circle, then the tangent is the mean proportional between the secant and its external segment.
Corollary 64-1 The product of any secant from a fixed point outside the circle and its external segment is constant.
Corollary 64-2 If two or more secants are drawn to a circle from a fixed point outside the circle, then the product of one secant and its external segment is equal to the product of any other secant and its external segment.
Theorem 65- The perimeters of two similar polygons are to each other as any two corresponding sides.
Theorem 66-If two polygons are similar, then they may be decomposed into the same number of triangles similar each to each and similarly placed.
Theorem 67-If two polygons are composed of the same number of triangles similar each to each and similarly placed, then they are similar. (Converse of Theorem 66.)
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