Special Topics Eleanor Roosevelt High School Chin-Sung Lin

Similar Triangles Mr. Chin-Sung Lin ERHS Math Geometry

Definition of Similar Triangles Mr. Chin-Sung Lin Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional (The number represented by the ratio of similitude is called the constant of proportionality) ERHS Math Geometry

Example of Similar Triangles Mr. Chin-Sung Lin  A   X,  B   Y,  C   Z AB = 6, BC = 8, and CA = 10 XY = 3, YZ = 4 and ZX = 5 Show that  ABC~  XYZ X Y Z A B C ERHS Math Geometry

Example of Similar Triangles Mr. Chin-Sung Lin  A   X,  B   Y,  C   Z AB BC CA 2 XY YZ ZX 1 Therefore  ABC~  XYZ = = = ERHS Math Geometry X Y Z A B C

Example of Similar Triangles Mr. Chin-Sung Lin The sides of a triangle have lengths 4, 6, and 8. Find the sides of a larger similar triangle if the constant of proportionality is 5/ ? ? ? ERHS Math Geometry

Example of Similar Triangles Mr. Chin-Sung Lin Assume x, y, and z are the sides of the larger triangle, then x 5 y 5 z x = 10 z = 15 y = 20 ERHS Math Geometry = = =

Example of Similar Triangles Mr. Chin-Sung Lin In  ABC, AB = 9, BC = 15, AC = 18. If  ABC~  XYZ, and XZ = 12, find XY and YZ ERHS Math Geometry X Y Z ? ? A B C

Example of Similar Triangles Mr. Chin-Sung Lin Since  ABC~  XYZ, and XZ = 12, then XY YZ X Y Z A B C ERHS Math Geometry = =

Example of Similar Triangles Mr. Chin-Sung Lin In  ABC, AB = 4y – 1, BC = 8x + 2, AC = 8. If  ABC~  XYZ, and XZ = 6, find XY and YZ ERHS Math Geometry X Y Z ? ? 6 4y – 1 8x A B C

Example of Similar Triangles Mr. Chin-Sung Lin Since  ABC~  XYZ, and XZ = 6, then XY YZ 6 4y–1 8x+28 X Y Z 3y–¾ 6x+ 3 / 2 6 4y – 1 8x A B C ERHS Math Geometry = =

Prove Similarity Mr. Chin-Sung Lin ERHS Math Geometry

Angle-Angle Similarity Theorem (AA~) Mr. Chin-Sung Lin If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar Given:  ABC and  XYZ with A  X, and C  Z Prove:  ABC~  XYZ X Y Z A B C ERHS Math Geometry

Example of AA Similarity Theorem Mr. Chin-Sung Lin Given: m  A = 45 and m  D = 45 Prove:  ABC~  DEC 45 o A B C D E ERHS Math Geometry

Example of AA Similarity Theorem Mr. Chin-Sung Lin StatementsReasons 1. mA = 45 and mD = Given 2. A  D 2. Substitution property 3. ACB  DCE3. Vertical angles 4.  ABC~  DEC 4. AA similarity theorem 45 o A B C D E ERHS Math Geometry

Side-Side-Side Similarity Theorem (SSS~) Mr. Chin-Sung Lin Two triangles are similar if the three ratios of corresponding sides are equal Given: AB/XY = AC/XZ = BC/YZ Prove:  ABC~  XYZ ERHS Math Geometry X Y Z A B C

Side-Angle-Side Similarity Theorem (SAS~) Mr. Chin-Sung Lin Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent Given:  A   X, AB/XY = AC/XZ Prove:  ABC~  XYZ X Y Z A B C ERHS Math Geometry

Example of SAS Similarity Theorem Mr. Chin-Sung Lin Prove:  ABC~  DEC Calculate: DE 16 A B C D E ? ERHS Math Geometry

Example of SAS Similarity Theorem Mr. Chin-Sung Lin Prove:  ABC~  DEC Calculate: DE 16 A B C D E ERHS Math Geometry

Triangle Proportionality Theorem Mr. Chin-Sung Lin ERHS Math Geometry

Triangle Proportionality Theorem Mr. Chin-Sung Lin If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally Given: DE || BC Prove: AD AE DB EC = D E A B C ERHS Math Geometry

Triangle Proportionality Theorem Mr. Chin-Sung Lin If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally DE || BC AD AE DB EC AD AE DE AB AC BC D E A B C = = ERHS Math Geometry =

Converse of Triangle Proportionality Theorem Mr. Chin-Sung Lin If the points at which a line intersects two sides of a triangle divide those sides proportionally, then the line is parallel to the third side Given: AD AE DB EC Prove: DE || BC = D E A B C ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, AD = 4, BD = 3, AE = 6 Calculate: CE and BC ? D E A B C ? ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, AD = 4, BD = 3, AE = 6 Calculate: CE and BC D E A B C 14 ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12 Calculate: EC and AD 8 4 ? 6 ? D E A B C 12 ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12 Calculate: EC and AD D E A B C 12 ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12 Calculate: AE and AB 8 5 ? 10 ? D E A B C 12 ERHS Math Geometry

Example of Triangle Proportionality Theorem Mr. Chin-Sung Lin Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12 Calculate: AE and AB / 3 D E A B C 12 ERHS Math Geometry

Pythagorean Theorem Mr. Chin-Sung Lin ERHS Math Geometry

Pythagorean Theorem Mr. Chin-Sung Lin A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides  ABC, m  C = 90 if and only if a 2 + b 2 = c 2 ERHS Math Geometry A C B a b c

Pythagorean Example - Distance Mr. Chin-Sung Lin Find the distance between A and B. ERHS Math Geometry A (5, 3) B(2, 1) ?

Pythagorean Example - Distance Mr. Chin-Sung Lin Find the distance between A and B. ERHS Math Geometry A (5, 3) B(2, 1) C (5, 1) | 3 – 1 | = 2 | 5 – 2 | = 3 ?

Pythagorean Example - Distance Mr. Chin-Sung Lin Find the distance between A and B. ERHS Math Geometry A (5, 3) B(2, 1) √13 C (5, 1) | 3 – 1 | = 2 | 5 – 2 | = 3

Parallelograms Mr. Chin-Sung Lin ERHS Math Geometry

Theorems of Parallelogram Mr. Chin-Sung Lin Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles ERHS Math Geometry

Criteria for Proving Parallelograms Mr. Chin-Sung Lin Parallel opposite sides Congruent opposite sides Congruent & parallel opposite sides Congruent opposite angles Supplementary consecutive angles Bisecting diagonals ERHS Math Geometry

Rectangles Mr. Chin-Sung Lin ERHS Math Geometry

Rectangles Mr. Chin-Sung Lin A rectangle is a parallelogram containing one right angle A B C D ERHS Math Geometry

Properties of Rectangle Mr. Chin-Sung Lin The properties of a rectangle  All the properties of a parallelogram  Four right angles (equiangular)  Congruent diagonals A B C D ERHS Math Geometry

Proving Rectangles Mr. Chin-Sung Lin To show that a quadrilateral is a rectangle, by showing that the quadrilateral is equiangular or a parallelogram  that contains a right angle, or  with congruent diagonals If a parallelogram does not contain a right angle, or doesn’t have congruent diagonals, then it is not a rectangle ERHS Math Geometry

Rhombuses Mr. Chin-Sung Lin ERHS Math Geometry

Rhombus Mr. Chin-Sung Lin A rhombus is a parallelogram that has two congruent consecutive sides A B C D ERHS Math Geometry

Properties of Rhombus Mr. Chin-Sung Lin The properties of a rhombus  All the properties of a parallelogram  Four congruent sides (equilateral)  Perpendicular diagonals  Diagonals that bisect opposite pairs of angles A B C D ERHS Math Geometry

Proving Rhombus Mr. Chin-Sung Lin To show that a quadrilateral is a rhombus, by showing that the quadrilateral is equilateral or a parallelogram  that contains two congruent consecutive sides  with perpendicular diagonals, or  with diagonals bisecting opposite angles If a parallelogram does not contain two congruent consecutive sides, or doesn’t have perpendicular diagonals, then it is not a rectangle ERHS Math Geometry

Application Example Mr. Chin-Sung Lin ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13 Prove: ABCD is a rhombus AB DC 2x+1 3x-11 x+13 ERHS Math Geometry

Application Example Mr. Chin-Sung Lin ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13 Prove: ABCD is a rhombus x = 12 AB = AD = 25 ABCD is a rhombus AB DC 2x+1 3x-11 x+13 ERHS Math Geometry

Application Example ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A B C D ERHS Math Geometry Mr. Chin-Sung Lin

Application Example ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus x = 4 AB = BC = 10 ABCD is a rhombus A B C D ERHS Math Geometry Mr. Chin-Sung Lin

Squares Mr. Chin-Sung Lin ERHS Math Geometry

Squares Mr. Chin-Sung Lin A square is a rectangle that has two congruent consecutive sides A B CD ERHS Math Geometry

Squares Mr. Chin-Sung Lin A square is a rectangle with four congruent sides (an equilateral rectangle) ERHS Math Geometry A B CD

Squares Mr. Chin-Sung Lin A square is a rhombus with four right angles (an equiangular rhombus) ERHS Math Geometry A B CD

Squares Mr. Chin-Sung Lin A square is an equilateral quadrilateral A square is an equiangular quadrilateral ERHS Math Geometry A B CD

Squares Mr. Chin-Sung Lin A square is a rhombus A square is a rectangle ERHS Math Geometry A B CD

Properties of Square Mr. Chin-Sung Lin The properties of a square  All the properties of a parallelogram  All the properties of a rectangle  All the properties of a rhombus A B CD ERHS Math Geometry

Proving Squares Mr. Chin-Sung Lin To show that a quadrilateral is a square, by showing that the quadrilateral is a  rectangle with a pair of congruent consecutive sides, or  a rhombus that contains a right angle ERHS Math Geometry

Application Example ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y A B CD ERHS Math Geometry Mr. Chin-Sung Lin

Application Example ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y 4x – 30 = 90 x = 30 y = 25 A B CD ERHS Math Geometry Mr. Chin-Sung Lin

Trapezoids Mr. Chin-Sung Lin ERHS Math Geometry

Trapezoids Mr. Chin-Sung Lin A trapezoid is a quadrilateral that has exactly one pair of parallel sides The parallel sides of a trapezoid are called bases. The nonparallel sides of a trapezoid are the legs A B CD Upper base Lower base Leg ERHS Math Geometry

Isosceles Trapezoids Mr. Chin-Sung Lin A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid ERHS Math Geometry A B CD Upper base Lower base Leg

Properties of Isosceles Trapezoids Mr. Chin-Sung Lin The properties of a isosceles trapezoid  Base angles are congruent  Diagonals are congruent The property of a trapezoid  Median is parallel to and average of the bases ERHS Math Geometry

Proving Trapezoids Mr. Chin-Sung Lin To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel To prove that a quadrilateral is not a trapezoid, show that both pairs of opposite sides are parallel or that both pairs of opposite sides are not parallel ERHS Math Geometry

Proving Isosceles Trapezoids Mr. Chin-Sung Lin To prove that a trapezoid is an isosceles trapezoid, show that one of the following statements is true:  The legs are congruent  The lower/upper base angles are congruent  The diagonals are congruent ERHS Math Geometry

Numeric Example of Trapezoids Mr. Chin-Sung Lin Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y A B C D 2x o xoxo 3y o ERHS Math Geometry

Numeric Example of Trapezoids Mr. Chin-Sung Lin Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y x = 60 y = 20 A B C D 2x o xoxo 3y o ERHS Math Geometry

Q & A Mr. Chin-Sung Lin ERHS Math Geometry

The End Mr. Chin-Sung Lin ERHS Math Geometry