Theorem 57 - If 2 triangles have the 3 angles of one = respectively to the 3 angles of the other, then the triangles are ~.
Corollary 57-1 - If 2 angles of one triangle are = respectively to 2 angles of another, then the triangles are ~.
Corollary 57 -2 - 2 right triangles are similar if an acute angle of one is = to an acute angle of the other.
Theorem 58 - If 2 triangles have 2 pairs of sides proportional and the included angles = respectively, then the 2 triangles are ~. (s.a.s.)
Corollary 58-1 - If the legs of one triangle are proportional to the legs of another, the triangles are similar. (l.l.)
Theorem 59 - If 2 triangles have their sides respectively proportional, then the triangles are ~. (s.s.s.)
*Read through the proofs for T's and c's in your text before moving on to the exercises.
Source: ABEKA Plane Geometry 2006
Prove that 2 isosceles triangles are similar if any angle of one equals the corresponding angle of the other.
case 1: equal vertex angles
Given: Isosceles triangle ABC, isosceles triangle DEF
AC=BC; DF=EF; <C=<F
Prove: triangle ABC ~ triangle DEF
Draw 2 isosc. triangles, different sizes for your diagrams.
statements for case 1 proof:
2. <A = <D
3. triangle ABC ~ triangle DEF
case 2: equal base angles
statements for case 2 proof:
2. <A = <B; <D = <E
3. <B = <E
4. triangle ABC ~ triangle DEF
Source: ABEKA Plane Geometry 2006
case 1:
2. isosc. triangles with = vertex angles have = base angles also.
3. a.a.
case 2:
2. in any isosc. triangle, the angles opposite the = sides are =.
3. quantities that are = to = quantities are =.
4. a.a.
Source: ABEKA Plane Geometry 2006
If AC = 9, AE = 3, AB = 12, AD = 4, prove: triangle ABC ~ triangle ADE
*diagram in text
given: triangle ABC; DE; AC = 9; AE = 3; AB = 12; AD = 4
prove: triangle ABC ~ triangle ADE
statements for proof:
2. 12:4 = 9:3
3. AB:AD = AC:AE
4. <A = <A
5. triangle ABC ~ triangle ADE
Source: ABEKA Plane Geometry 2006
2. definition of proportion
3. substitution
4. identity
5. S.A.S.
Source: ABEKA Plane Geometry 2006
Prove that if a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle
*look at what the problem tells you, and draw the needed diagram. Remember, the line is drawn parallel to the base of the triangle, this line will be the transversal to the sides of the triangle. If 2 parallel lines are cut by a transversal, what do you know about the corresponding angles?
given: triangle ABC; DE parallel AB
prove: triangle DEC ~ triangle ABC
statements for proof:
2. <CDE = <A; <CED = <B
3. triangle ABC ~ triangle DEC
Source: ABEKA Plane Geometry 2006
2. if 2 parallel lines are cut by a transversal, then the corresponding angles are =.
3. a.a.
Source: ABEKA Plane Geometry 2006
In the triangle ABC, AE is the altitude on BC and CD is the altitude on AB. Prove: triangle CBD ~ triangle AEB
given: triangle ABC; AE altitude to BC; CD altitude to AB
*draw your diagram
statements for proof:
2. <B = <B
3. AE perpendicular to BC; CD perpendicular to AB
4. <AEB, <CDB are right angles
5. triangle CDB and triangle AEB are right triangles
6. triangle CDB ~ triangle AEB
Source: ABEKA Plane Geometry 2006
