L21_Ratios and Proportions Eleanor Roosevelt High School Chin-Sung Lin Special Topics Eleanor Roosevelt High School Chin-Sung Lin

Similar Triangles ERHS Math Geometry Mr. Chin-Sung Lin L22_Similar Triangles ERHS Math Geometry Similar Triangles Mr. Chin-Sung Lin

Definition of Similar Triangles L22_Similar Triangles ERHS Math Geometry Definition of Similar Triangles 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) Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles A  X, B  Y, C  Z AB = 6, BC = 8, and CA = 10 XY = 3, YZ = 4 and ZX = 5 Show that ABC~XYZ A B C 6 8 10 X Y Z 3 4 5 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles A  X, B  Y, C  Z AB BC CA 2 XY YZ ZX 1 Therefore ABC~XYZ = = = A B C 6 8 10 X Y Z 3 4 5 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles 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 5/2 ? 4 6 8 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles Assume x, y, and z are the sides of the larger triangle, then x 5 y 5 z 5 4 2 8 2 6 2 = = = 4 6 8 y = 20 x = 10 z = 15 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles In ABC, AB = 9, BC = 15, AC = 18. If ABC~XYZ, and XZ = 12, find XY and YZ 9 15 18 A B C X Y Z ? 12 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles Since ABC~XYZ, and XZ = 12, then XY YZ 12 9 15 18 = = 9 15 18 A B C X Y Z 6 10 12 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles In ABC, AB = 4y – 1, BC = 8x + 2, AC = 8. If ABC~XYZ, and XZ = 6, find XY and YZ 4y – 1 8x + 2 8 A B C X Y Z ? 6 Mr. Chin-Sung Lin

Example of Similar Triangles L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles Since ABC~XYZ, and XZ = 6, then XY YZ 6 4y–1 8x+2 8 = = 4y – 1 8x + 2 8 A B C X Y Z 3y–¾ 6x+ 3/2 6 Mr. Chin-Sung Lin

Prove Similarity ERHS Math Geometry Mr. Chin-Sung Lin L22_Similar Triangles ERHS Math Geometry Prove Similarity Mr. Chin-Sung Lin

Angle-Angle Similarity Theorem (AA~) L22_Similar Triangles ERHS Math Geometry Angle-Angle Similarity Theorem (AA~) 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 Mr. Chin-Sung Lin

Example of AA Similarity Theorem L22_Similar Triangles ERHS Math Geometry Example of AA Similarity Theorem Given: mA = 45 and mD = 45 Prove: ABC~DEC 45o A B C D E Mr. Chin-Sung Lin

Example of AA Similarity Theorem L22_Similar Triangles ERHS Math Geometry Example of AA Similarity Theorem 45o A B C D E Statements Reasons 1. mA = 45 and mD = 45 1. Given 2. A  D 2. Substitution property 3. ACB  DCE 3. Vertical angles 4. ABC~DEC 4. AA similarity theorem Mr. Chin-Sung Lin

Side-Side-Side Similarity Theorem (SSS~) L22_Similar Triangles ERHS Math Geometry Side-Side-Side Similarity Theorem (SSS~) Two triangles are similar if the three ratios of corresponding sides are equal Given: AB/XY = AC/XZ = BC/YZ Prove: ABC~XYZ A B C X Y Z Mr. Chin-Sung Lin

Side-Angle-Side Similarity Theorem (SAS~) L22_Similar Triangles ERHS Math Geometry Side-Angle-Side Similarity Theorem (SAS~) 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 A B C X Y Z Mr. Chin-Sung Lin

Example of SAS Similarity Theorem L22_Similar Triangles ERHS Math Geometry Example of SAS Similarity Theorem Prove: ABC~DEC Calculate: DE 16 A B C D E 10 12 8 6 ? Mr. Chin-Sung Lin

Example of SAS Similarity Theorem L22_Similar Triangles ERHS Math Geometry Example of SAS Similarity Theorem Prove: ABC~DEC Calculate: DE 16 A B C D E 10 12 8 6 5 Mr. Chin-Sung Lin

Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem Mr. Chin-Sung Lin

Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem 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 = Mr. Chin-Sung Lin

Triangle Proportionality Theorem L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Proportionality Theorem 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 = = = Mr. Chin-Sung Lin

Converse of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Converse of Triangle Proportionality Theorem 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 = Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, AD = 4, BD = 3, AE = 6 Calculate: CE and BC 8 3 4 6 ? D E A B C Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, AD = 4, BD = 3, AE = 6 Calculate: CE and BC 8 3 4 6 4.5 D E A B C 14 Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem 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 Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12 Calculate: EC and AD 8 4 6 3 D E A B C 12 Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem 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 Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12 Calculate: AE and AB 8 5 15 10 20/3 D E A B C 12 Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Pythagorean Theorem Mr. Chin-Sung Lin

L24_Right Triangle Altitude Theorem ERHS Math Geometry Pythagorean Theorem 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 a2 + b2 = c2 A C B a b c Mr. Chin-Sung Lin

Pythagorean Example - Distance L24_Right Triangle Altitude Theorem ERHS Math Geometry Pythagorean Example - Distance Find the distance between A and B. A (5, 3) ? B(2, 1) Mr. Chin-Sung Lin

Pythagorean Example - Distance L24_Right Triangle Altitude Theorem ERHS Math Geometry Pythagorean Example - Distance Find the distance between A and B. A (5, 3) ? | 3 – 1 | = 2 B(2, 1) C (5, 1) | 5 – 2 | = 3 Mr. Chin-Sung Lin

Pythagorean Example - Distance L24_Right Triangle Altitude Theorem ERHS Math Geometry Pythagorean Example - Distance Find the distance between A and B. A (5, 3) √13 | 3 – 1 | = 2 B(2, 1) C (5, 1) | 5 – 2 | = 3 Mr. Chin-Sung Lin

L21_Ratios and Proportions ERHS Math Geometry Parallelograms Mr. Chin-Sung Lin

Theorems of Parallelogram L14_Properties of a Parallelogram ERHS Math Geometry Theorems of Parallelogram Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles Mr. Chin-Sung Lin

Criteria for Proving Parallelograms L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Criteria for Proving Parallelograms Parallel opposite sides Congruent opposite sides Congruent & parallel opposite sides Congruent opposite angles Supplementary consecutive angles Bisecting diagonals Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Rectangles Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Rectangles A rectangle is a parallelogram containing one right angle A B C D Mr. Chin-Sung Lin

Properties of Rectangle L17_Rectangles Rhombuses and Squares ERHS Math Geometry Properties of Rectangle The properties of a rectangle All the properties of a parallelogram Four right angles (equiangular) Congruent diagonals A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Proving Rectangles 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 Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Rhombuses Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Rhombus A rhombus is a parallelogram that has two congruent consecutive sides A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Properties of Rhombus A B C D The properties of a rhombus All the properties of a parallelogram Four congruent sides (equilateral) Perpendicular diagonals Diagonals that bisect opposite pairs of angles Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Proving Rhombus 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 Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Application Example ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13 Prove: ABCD is a rhombus A B 2x+1 x+13 D 3x-11 C Mr. Chin-Sung Lin

L15_Proving Quadrilaterals Are Parallelograms ERHS Math Geometry Application Example 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 A B 2x+1 x+13 D 3x-11 C Mr. Chin-Sung Lin

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 Mr. Chin-Sung Lin

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 x = 4 AB = BC = 10 ABCD is a rhombus A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares A square is a rectangle that has two congruent consecutive sides A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares A square is a rectangle with four congruent sides (an equilateral rectangle) A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares A square is a rhombus with four right angles (an equiangular rhombus) A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares A square is an equilateral quadrilateral A square is an equiangular quadrilateral A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Squares A square is a rhombus A square is a rectangle A B C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Properties of Square 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 C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Proving Squares 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 Mr. Chin-Sung Lin

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 C D Mr. Chin-Sung Lin

ERHS Math Geometry 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 C D Mr. Chin-Sung Lin

L17_Rectangles Rhombuses and Squares ERHS Math Geometry Trapezoids Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Trapezoids 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 C D Upper base Lower base Leg Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Isosceles Trapezoids A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid A B C D Upper base Lower base Leg Mr. Chin-Sung Lin

Properties of Isosceles Trapezoids L18_Trapezoids ERHS Math Geometry Properties of Isosceles Trapezoids 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 Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Proving Trapezoids 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 Mr. Chin-Sung Lin

Proving Isosceles Trapezoids L18_Trapezoids ERHS Math Geometry Proving Isosceles Trapezoids 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 Mr. Chin-Sung Lin

Numeric Example of Trapezoids L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y A B C D 2xo xo 3yo Mr. Chin-Sung Lin

Numeric Example of Trapezoids L18_Trapezoids ERHS Math Geometry Numeric Example of Trapezoids Isosceles Trapezoid ABCD, AB || CD and AD  BC Solve for x and y x = 60 y = 20 A B C D 2xo xo 3yo Mr. Chin-Sung Lin

L21_Ratios and Proportions ERHS Math Geometry Q & A Mr. Chin-Sung Lin

L21_Ratios and Proportions ERHS Math Geometry The End Mr. Chin-Sung Lin