Chapter 9 Parallel Lines Eleanor Roosevelt High School

Chapter 9 Parallel Lines Eleanor Roosevelt High School Chin-Sung Lin

ERHS Math Geometry Angles Formed by Intersecting Lines Mr. Chin-Sung Lin

ERHS Math Geometry Transversal A transversal is a line that intersects two coplanar lines at two distinct points k m Line k is a transversal n Mr. Chin-Sung Lin

ERHS Math Geometry Angles of Transversal and Lines Eight angles are formed by the transversal and the two lines k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Interior Angles 3 4 5 and 6 are interior angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Exterior Angles 1 2 7 and 8 are exterior angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Same-Side Interior Angles 3 and 5 are same-side interior angles 4 and 6 are same-side interior angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Alternate Interior Angles 3 and 6 are alternate interior angles 4 and 5 are alternate interior angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Alternate Exterior Angles 1 and 8 are alternate exterior angles 2 and 7 are alternate exterior angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Corresponding Angles 1 and 5, 2 and 6, 3 and 7, 4 and 8 are corresponding angles k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Angles of Transversal and Lines Review: k 1 3 5 7 8 2 4 m 6 n Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Lines Coplanar lines that have no points in common, or have all points in common and, therefore, coincide Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Eight angles are formed by the transversal and the two parallel lines k 1 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Lines and Transversal q Corresponding Angles Postulate q Alternate Interior Angles Theorem q Same-Side Interior Angles Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Parallel Lines and Transversal q Converse of Corresponding Angles Postulate q Converse of Alternate Interior Angles Theorem q Converse of Same-Side Interior Angles Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent If m || n, 1 5, 2 6, 3 7, and 4 8 k 1 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent k 1 If m || n, 3 6, and 4 5 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Alternate Interior Angles Theorem k 1 3 5 7 8 2 4 m 6 n Statements Reasons 1. 2. 3. 4. Mr. Chin-Sung Lin

ERHS Math Geometry Alternate Interior Angles Theorem k 1 3 5 7 8 Statements 2 4 m 6 n Reasons 1. m || n 1. Given 2. 3 7 and 4 8 2. Corresponding angles 3. 6 7 and 5 8 3. Vertical angles 4. 3 6 and 4 5 4. Substitution property Mr. Chin-Sung Lin

ERHS Math Geometry Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of sameside interior angles are supplementary If m || n, 3 and 5, 4 and 6 are supplementary k 1 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Same-Side Interior Angles Theorem k 1 3 5 7 8 2 4 m 6 n Statements Reasons 1. 2. 3. 4. Mr. Chin-Sung Lin

ERHS Math Geometry Same-Side Interior Angles Theorem k 1 3 5 7 8 Statements 2 4 m 6 n Reasons 1. m || n 1. Given 2. 3 7 and 4 8 2. Corresponding angles 3. 6 and 8, and 5 and 7 3. Supplementary angles are supplementary 4. 6 and 4, and 5 and 3 are supplementary 4. Substitution property Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Corresponding Angles Postulate If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel If 1 5, 2 6, 3 7 or 4 8, m || n, k 1 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Alternate Interior Angles Theorem If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel k 1 If 3 6 or 4 5, m || n 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Alternate Interior Angles Theorem k 1 3 5 7 8 2 4 m 6 n Statements Reasons 1. 2. 3. 4. Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Alternate Interior Angles Theorem k 1 3 5 7 8 Statements 2 4 m 6 n Reasons 1. 3 6 or 4 5 1. Given 2. 6 7 and 5 8 2. Vertical angles 3. 3 7 or 4 8 3. Substitution property 4. m || n 4. Converse of corresponding angle postulate Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Same-Side Interior Angles Theorem If two lines are cut by a transversal and the pairs of same-side interior angles are supplementary, then the lines are parallel k 1 If 3 and 5, or 4 and 6 are supplementary, m || n, 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Same-Side Interior Angles Theorem k 1 3 5 7 8 2 4 m 6 n Statements Reasons 1. 2. 3. 4. Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Same-Side Interior Angles Theorem k 1 3 5 7 8 Statements 1. 4 and 6, or 3 and 5 are supplementary 2. 8 and 6, and 7 and 5 are supplementary 3. 4 8 or 3 7 4. m || n 2 4 m 6 n Reasons 1. Given 2. Supplementary angles 3. Supp. angle theorem 4. Converse of corresponding angle postulate Mr. Chin-Sung Lin

ERHS Math Geometry Methods of Proving Lines Parallel q Congruent Corresponding Angles q Congruent Alternate Interior Angles q Same-Side Interior Angles supplementary q Both lines are perpendicular to the same line Mr. Chin-Sung Lin

ERHS Math Geometry Examples Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Classify the following angles into alternate interior angles, same-side interior angles or corresponding angles q 4 3 8 7 p 2 1 r 6 5 Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Alternate interior angles: 3 & 6, 2 & 7 Same-side interior angles: 2 & 3, 6 & 7 Corresponding angles: 1 & 3, 2 & 4, 5 & 7, 6 & 8 q 4 3 8 7 p 2 1 r 6 5 Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of all the angles q r 4 3 125 o 7 p 2 1 6 5 Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Calculate the value of all the angles q 55 o p r 125 o 55 o Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of x q r 4 3 x+50 o 7 p 2 1 6 x– 10 o Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of x x + 50 o + x – 10 o = 180 o 2 x = 140 o x = 70 o q r 4 3 x+50 o 7 p 2 1 6 x– 10 o Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Given: 1 8 Prove: m || n k 1 3 5 7 2 4 m 6 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Given: 1 8 Prove: m || n Statements k 1 3 5 2 4 m 6 7 8 n Reasons 1. 1 8 1. Given 2. 1 4 and 8 5 2. Vertical angles 3. 4 5 3. Substitution property 4. m || n 4. Converse of alternate interior angle theorem Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Given: m 1 + m 6 = 180 m 6 + m 9 =180 Prove: p || n k 1 3 5 7 9 10 11 12 2 p 4 6 m 8 n Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines k 1 Given: m 1 + m 6 = 180 m 6 + m 9 =180 Prove: p || n 3 5 7 9 11 Statements 1. m 1 + m 6 180 m 6 + m 9 180 2. 1 4 and 6 7 3. m 4 + m 6 180 m 7 + m 9 180 4. p || m, m || n 5. p || n 10 12 2 p 4 6 m 8 n Reasons 1. Given 2. Vertical angles 3. Substitution property 4. Converse of same-side interior angle theorem 5. Transitive property Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Given: Quadrilateral ABCD ~ DA, and BC || DA BC = Prove: AB || CD B C A D Mr. Chin-Sung Lin

ERHS Math Geometry Angles Formed by Parallel Lines Given: Quadrilateral ABCD ~ DA, and BC || DA BC = Prove: AB || CD B C A D Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem The sum of the measures of the interior angles of any triangle is 180 Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem The sum of the measures of the interior angles of any triangle is 180. A Draw the graph Given: ∆ ABC Prove: m A + m B B+ m C = 180 C Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem D A 1 Statements 2 B C 1. 2. 3. 4. Reasons Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem D A 1 Statements 1. 2. B 2 C Reasons Let AD be the line through A 1. Use properties of parallel lines and parallel to BC B 1 and C 2 2. Alt. interior angles theorem 3. m 1 + m A + m 2 = 180 3. Def. of straight angle 4. m B + m A + m C = 180 4. Substitution property Mr. Chin-Sung Lin

ERHS Math Geometry Corollaries to the Triangle Angle-Sum Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Corollary A corollary is a statement that follows directly from theorem A corollary is a statement that can be easily proved by applying theorem Mr. Chin-Sung Lin

ERHS Math Geometry Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Proof of Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent Given: A X, and B Y Prove: C Z A Y B X Z C Mr. Chin-Sung Lin

ERHS Math Geometry Y Proof of Corollary 1 B X A Statements Z C Reasons 1. A X, B Y 1. Given 2. m A + m B + m C = 180 2. Triangle sum theorem m X + m Y + m Z = 180 3. m A + m B + m C = 3. Substitution property m X + m Y + m Z 4. m C = m Z or C Z 4. Subtraction property Mr. Chin-Sung Lin

ERHS Math Geometry Corollary 2 The two acute angles of a right triangle are complementary Mr. Chin-Sung Lin

ERHS Math Geometry Proof of Corollary 2 The two acute angles of a right triangle are complementary Given: m C = 90 Prove: A and B are complementary B A C Mr. Chin-Sung Lin

ERHS Math Geometry B Proof of Corollary 2 C A Statements Reasons 1. m C = 90 1. Given 2. m A + m B + m C = 180 2. Triangle sum theorem 3. m A + m B = 180 - 90 = 90 3. Subtraction property 4. A and B are complementary 4. Def. of complementary angles Mr. Chin-Sung Lin

ERHS Math Geometry Corollary 3 Each acute angle of an isosceles right triangle measured 45 o Mr. Chin-Sung Lin

ERHS Math Geometry Proof of Corollary 3 Each acute angle of an isosceles right triangle measured 45 o Given: m C = 90, ∆ABC is isosceles Prove: m A = m B = 45 C A B Mr. Chin-Sung Lin

ERHS Math Geometry C Proof of Corollary 3 A Statements B Reasons 1. m C = 90 1. Given 2. m A + m B + m C = 180 2. Triangle sum theorem 3. m A + m B + 90 = 180 3. Substitution property 4. m A + m B = 90 4. Subtraction property 5. m A = m B 5. Base angle theorem 6. 2 m A = 2 m B = 90 6. Substitution property 7. m A = m B = 45 7. Division property Mr. Chin-Sung Lin

ERHS Math Geometry Corollary 4 Each angle of an equilateral triangle has measure 60 Mr. Chin-Sung Lin

ERHS Math Geometry Proof of Corollary 4 Each angle of an equilateral triangle has measure 60 Given: Equilateral ∆ ABC Prove: m A = m B = m C= 60 C A B Mr. Chin-Sung Lin

ERHS Math Geometry C Proof of Corollary 4 A Statements B Reasons 1. Equilateral ∆ ABC 1. Given 2. m A = m B = m C 2. Euailateral triangle theorem 3. m A + m B + m C = 180 3. Triangle sum theorem 4. m A + m A = 180 4. Substitution property m B + m B = 180 m C + m C = 180 5. m A = m B = m C = 60 5. Division property Mr. Chin-Sung Lin

ERHS Math Geometry Corollary 5 Prove: m A + m B + m C + m D= 360 B A C D Mr. Chin-Sung Lin

ERHS Math Geometry Proof of Corollary 5 B 3 1 A 2 Statements 1. Let AC divides quadrilateral ABCD into two triangles 2. m 1 + m B + m 3 = 180 m 2 + m D + m 4 = 180 3. m 1 + m B + m 3 + m 2 + m D + m 4 = 360 4. m A + m B + m C + m D = 360 C 4 D Reasons 1. Use properties of triangles 2. Triangle-sum theorem 3. Addition property 4. Partition property Mr. Chin-Sung Lin

ERHS Math Geometry Exterior Angle Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Exterior Angle of a Triangle An exterior angle of a triangle is formed when one side of a triangle is extended. The nonstraight angle outside the triangle, but adjacent to an interior angle, is an exterior angle of the triangle C A B Mr. Chin-Sung Lin

ERHS Math Geometry Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles Mr. Chin-Sung Lin

ERHS Math Geometry Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles Given: ∆ ABC Prove: m 1 = m A + m C A C 1 B Mr. Chin-Sung Lin

ERHS Math Geometry C Exterior Angle Theorem 1 A Statements B Reasons 1. m 1 + m B = 180 1. Supplementary angles 2. m A + m B + m C = 180 2. Triangle-sum theorem 3. m A + m B + m C = 3. Substitution property m 1 + m B 4. m 1 = m A + m C 4. Subtraction property Mr. Chin-Sung Lin

ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle Sum Theorem ∆ ABC is an isosceles triangle. The vertex angle , C, exceeds the measure of each base angle by 30 degrees. Find the degree measure of each angle of the triangle. C A B Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle Sum Theorem ∆ ABC is an isosceles triangle. The vertex angle , C, exceeds the measure of each base angle by 30 degrees. Find the degree measure of each angle of the triangle C (x + 30) + x = 180 3 x + 30 = 180 3 x = 150 A x = 50 X+30 X X B Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem Find the values of x and y 5 y 4 x 4 x y 3 y Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem Find the values of x and y 5 y 5 y + 4 x = 180 4 x + 3 y = 180 4 x + 6 y = 180……(1) 8 x + 3 y = 180……(2) (1)/2, 2 x + 3 y = 90. . …. . (3) (2) - (3), 6 x = 90, and then X = 15……(4) Substitutes into (3), 2(15) + 3 y = 90, 3 y = 60, y = 20 4 x y 4 x 3 y So, X =15 Y = 20 Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem Given: m 2 = m C Prove: m 1 = m B A B E 2 1 D C Mr. Chin-Sung Lin

ERHS Math Geometry Triangle Angle-Sum Theorem A B Statements E 2 C 1. m 2 = m C 2. m 1 + m 2 + m A = 180 m B + m C + m A = 180 3. m 1 + m 2 + m A = m B + m C + m A 4. m 1 + m C + m A = m B + m C + m A 5. m 1 = m B 1 D Reasons 1. Given 2. Triangle-sum theorem 3. Substitution property 4. Substitution property 5. Subtraction property Mr. Chin-Sung Lin

ERHS Math Geometry AAS Postulate Mr. Chin-Sung Lin

ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Mr. Chin-Sung Lin

ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Side Congruence (AAS) Mr. Chin-Sung Lin

ERHS Math Geometry Angle-Side Congruence (AAS) If two of corresponding angles and a nonincluded side are equal, then the triangles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry AAS Postulate Given CA is an angle bisector of DCB, and B =~ D ~ Prove ∆ ACD = ∆ ACB D A C B Mr. Chin-Sung Lin

ERHS Math Geometry AAS Postulate Given CA is an angle bisector of DCB, and B =~ D AAS ~ Prove ∆ ACD = ∆ ACB D A C B Mr. Chin-Sung Lin

ERHS Math Geometry AAS Postulate Corollary 1 Two right triangles are congruent if their hypotenuses and one of the acute angles are congruent Given ∆ ABC and ∆ DEF are right triangles ~ D AB = DE, A = ~ Prove ∆ ABC = ∆ DEF A B E C D F Mr. Chin-Sung Lin

ERHS Math Geometry AAS Postulate Corollary 2 If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle Mr. Chin-Sung Lin

ERHS Math Geometry Congruent Right Triangles (HL Postulate) Mr. Chin-Sung Lin

ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Side Congruence (AAS) Mr. Chin-Sung Lin

ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Side Congruence (AAS) Hypotenuse-Leg Postulate (HL) Mr. Chin-Sung Lin

ERHS Math Geometry Hypotenuse-Leg Postulate (HL Postulate) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent A B E C D F Mr. Chin-Sung Lin

ERHS Math Geometry Side-Angle Case (SSA) The condition does not guarantee congruence, because it is possible to have two incongruent triangles. This is known as the ambiguous case Mr. Chin-Sung Lin

ERHS Math Geometry HL Postulate – Example Given BD is the altitude of an isosceles triangle ∆ ABC ~ Prove ∆ ABD = ∆ CBD B A D C Mr. Chin-Sung Lin

ERHS Math Geometry HL Postulate – Example Given BD is the altitude of an isosceles triangle ∆ ABC HL ~ Prove ∆ ABD = ∆ CBD B A D C Mr. Chin-Sung Lin

ERHS Math Geometry HL Postulate – Example B A Statements D 1. ∆ ABC is isosceles triangle C Reasons 1. Given BD is the altitude of ∆ ABC 2. m BDA = 90; m BDC = 90 2. Def. of altitude 3. BA BC 3. Def of isosceles triangle 4. BD 4. Reflexive property 5. ∆ ABD ∆ CBD 5. HL postulate Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem If two sides of a triangle are congruent, then the angles opposite these sides are congruent (Base angles of an isosceles triangle are congruent) Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem If two sides of a triangle are congruent, then the angles opposite these sides are congruent Draw a diagram like the one below Given: Prove: B AB CB A C A C Mr. Chin-Sung Lin

ERHS Math Geometry B Base Angle Theorem A Statements D C Reasons 1. 2. 3. 4. 5. 6. Mr. Chin-Sung Lin

ERHS Math Geometry B Base Angle Theorem A Statements 1. Draw the angle bisector of ABC and let D be the point where it intersects AC 2. ABD CBD 3. AB CB 4. BD 5. ∆ ABD = ∆ CBD 6. A C D C Reasons 1. Any angle of measure less than 180 has exactly one bisector 2. Definition of angle bisector 3. Given 4. Reflexive property 5. SAS Postulate 6. CPCTC Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Base Angle Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Base Angle Theorem If two angles of a triangle are congruent, then the sides opposite these angles are congruent Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Base Angle Theorem If two angles of a triangle are congruent, then the sides opposite these angles are congruent Draw a diagram like the one below Given: A C B Prove: AB CB A C Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Base Angle Theorem B Statements A C D 1. 2. 3. 4. 5. 6. Reasons Mr. Chin-Sung Lin

ERHS Math Geometry Converse of Base Angle Theorem B Statements A D 1. Draw the angle bisector of ABC and let D be the point where it intersects AC 2. ABD CBD 3. A C 4. BD 5. ∆ ABD = ∆ CBD 6. AB CB C Reasons 1. Any angle of measure less than 180 has exactly one bisector 2. Definition of angle bisector 3. Given 4. Reflexive property 5. AAS Postulate 6. CPCTC Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem - Example AO BO and 1 2 AC = BD Given: Prove: A B O C 1 2 D Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem – Example AO BO and 1 2 AC = BD Given: Prove: A B O C 1 2 D Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem - Example A B O C 1 2 D Statements Reasons 1. 2. 3. 4. 5. 6. Mr. Chin-Sung Lin

ERHS Math Geometry Base Angle Theorem - Example A B O C Statements 1. 1 2 2. CO DO 3. AO BO 4. AOC BOD 5. ∆ AOC = ∆ BOD 6. AC BD 1 2 D Reasons 1. Given 2. Converse of Base Angle Theorem 3. Given 4. Vertical Angles 5. SAS Postulate 6. CPCTC Mr. Chin-Sung Lin

ERHS Math Geometry Equilateral and Equiangular Triangles Mr. Chin-Sung Lin

ERHS Math Geometry Equilateral Triangles A equilateral triangle is a triangle that has three congruent sides B A C Mr. Chin-Sung Lin

ERHS Math Geometry Equilateral & Equiangular Triangles If a triangle is an equilateral triangle, then it is an equiangular triangle Mr. Chin-Sung Lin

ERHS Math Geometry Polygons Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Polygons A polygon is a closed plane figure with the following properties: 1. Formed by three or more line segments called sides 2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common end point are colinear Mr. Chin-Sung Lin

ERHS Math Geometry Naming Polygons Each endpoint of a side is a vertex of the polygon A polygon can be named by listing the vertices in consecutive order Polygon can be named as ABCDE, CDEAB, EABCD, or …… A B E D C Mr. Chin-Sung Lin

ERHS Math Geometry Convex & Concave Polygons A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon A polygon that is not convex is called nonconvex or concave Interior Convex Concave Mr. Chin-Sung Lin

ERHS Math Geometry Convex & Concave Polygons A polygon is convex if each of the interior angles measures less than 180 degrees A polygon that is concave if at least one interior angle measures more than 180 degrees < 1800 Convex > 1800 Concave Mr. Chin-Sung Lin

ERHS Math Geometry Identifying Polygons Identify polygons in the following figures and tell whether the figure is a convex or concave polygon Mr. Chin-Sung Lin

ERHS Math Geometry Identifying Polygons Identify polygons in the following figures and tell whether the figure is a convex or concave polygon Concave Convex Concave Mr. Chin-Sung Lin

ERHS Math Geometry Common Polygons • Triangle: a polygon that is the union of three line segments • Quadrilateral: a polygon that is the union of four line segments • Pentagon: a polygon that is the union of five line segments • Hexagon: a polygon that is the union of six line segments • Octagon: a polygon that is the union of eight line segments • Decagon: a polygon that is the union of ten line segments • N-gon: a polygon with n sides Mr. Chin-Sung Lin

ERHS Math Geometry Classifying Polygons -by the Number of Its Sides No. of sides Polygons 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n n-gon Mr. Chin-Sung Lin

ERHS Math Geometry Classifying Polygons -by the Number of Its Sides Mr. Chin-Sung Lin

ERHS Math Geometry Classifying Polygons -by the Number of Its Sides Pentagon Hexagon Quadrilateral Triangle Mr. Chin-Sung Lin

ERHS Math Geometry Classifying Polygons -by the Number of Its Sides Mr. Chin-Sung Lin

ERHS Math Geometry Classifying Polygons -by the Number of Its Sides Decagon Heptagon Dodecagon 16 -gon Mr. Chin-Sung Lin

ERHS Math Geometry Equilateral / Equiangular / Regular Polygons A polygon is equilateral if all sides are congruent A polygon is equiangular if all interior angles of the polygon are congruent A regular polygon is a convex polygon that is both equilateral and equiangular Equilateral Equiangular Regular Mr. Chin-Sung Lin

ERHS Math Geometry Identify Equilateral / Equiangular / Regular Polygons Mr. Chin-Sung Lin

ERHS Math Geometry Identify Equilateral / Equiangular / Regular Polygons Regular Equilateral Equiangular Equilateral Regular Mr. Chin-Sung Lin

ERHS Math Geometry Interior and Exterior Angles of Polygons Mr. Chin-Sung Lin

ERHS Math Geometry Consecutive Angles and Vertices A pair of angles whose vertices are the endpoints of a common side are called consecutive angles The verteices of consecutive angles are called consecutive vertices or adjacent vertices A E B Consecutive Angles: A and B Consecutive Angles: C and D Consecutive Vertices: A and B Non-adjacent Vertices: E and B D C Mr. Chin-Sung Lin

ERHS Math Geometry Diagonals of Polygons A diagonal of a polygon is a line segment whose endpoints are two non-adjacent vertices A Adjacent Vertices of B: A and C E B Non-adjacent Vertices of B: D and E Diagonals with Endpoint B: BD and BE D C Mr. Chin-Sung Lin

ERHS Math Geometry Diagonals of Polygons A diagonal of a polygon is a line segment whose endpoints are two non-adjacent vertices A Adjacent Vertices of B: A and C E B Non-adjacent Vertices of B: D and E Diagonals with Endpoint B: BD and BE D C Mr. Chin-Sung Lin

ERHS Math Geometry Diagonals of Polygons All possible diagonals from a vertex A B F D C A C E A D B G E C B F D C E D Mr. Chin-Sung Lin

ERHS Math Geometry Sum of Interior Angles of Polygons Calculate the sum of intertor angles of each polygon A B F D C A C E A D B G E C B F D C E D Mr. Chin-Sung Lin

ERHS Math Geometry Sum of Interior Angles of Polygons Calculate the sum of intertor angles of each polygon A B F D 2 (1800) C A C E A E 0 D 4 (180 ) B G C B F D 3 (1800) C E 5 (1800) D Mr. Chin-Sung Lin

ERHS Math Geometry Theorem: Sum of Polygon Interior Angles The sum of the measures of the interior angles of a polygon of n sides is 180(n – 2)o A B 180 (6 – 2)0 = 720 o F C E D Mr. Chin-Sung Lin

ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360 o Mr. Chin-Sung Lin

ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360 o Sum of exterior angles = 180 n – Sum of interior angles = 180 n – 180 (n – 2) = 180 n – 180 n + 360 = 360 Mr. Chin-Sung Lin

ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360 o A B F C Sum of exterior angles = 360 o E D Mr. Chin-Sung Lin

ERHS Math Geometry Example: Interior / Exterior Angles The measure of an exterior angle of a regular polygon is 45 o (a) Find the number of sides of the polygon (b) Find the measure of each interior angle (c) Find the sum of interior angles Mr. Chin-Sung Lin

ERHS Math Geometry Example: Interior / Exterior Angles The measure of an exterior angle of a regular polygon is 45 o (a) Find the number of sides of the polygon (b) Find the measure of each interior angle (c) Find the sum of interior angles (a) 3600/450 = 8 sides (b) 1800 – 450 = 1350 (c) 1800 (8 – 2) = 1, 0800 Mr. Chin-Sung Lin

ERHS Math Geometry Example: Interior / Exterior Angles In quadrilateral ABCD, m�A = x, m�B = 2 x – 12, m�C = x + 22, and m�D = 3 x (a) Find the measure of each interior angle (b) Find the measure of each exterior angle Mr. Chin-Sung Lin

ERHS Math Geometry Example: Interior / Exterior Angles In quadrilateral ABCD, m�A = x, m�B = 2 x – 12, m�C = x + 22, and m�D = 3 x (a) Find the measure of each interior angle (b) Find the measure of each exterior angle (a) 500, 880, 720, 1500 (b) 1300, 920, 1080, 300 Mr. Chin-Sung Lin

ERHS Math Geometry The End Mr. Chin-Sung Lin