Chapter 9 Parallel Lines L10_Parallel Lines and Related Angles Chapter 9 Parallel Lines Eleanor Roosevelt High School Chin-Sung Lin

Angles Formed by Intersecting Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Intersecting Lines Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles ERHS Math Geometry Transversal A transversal is a line that intersects two coplanar lines at two distinct points Line k is a transversal k m n Mr. Chin-Sung Lin

Angles of Transversal and Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles of Transversal and Lines Eight angles are formed by the transversal and the two lines n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles ERHS Math Geometry Interior Angles 3 4 5 and 6 are interior angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles ERHS Math Geometry Exterior Angles 1 2 7 and 8 are exterior angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Same-Side Interior Angles L10_Parallel Lines and Related Angles ERHS Math Geometry Same-Side Interior Angles 3 and 5 are same-side interior angles 4 and 6 are same-side interior angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Alternate Interior Angles L10_Parallel Lines and Related Angles ERHS Math Geometry Alternate Interior Angles 3 and 6 are alternate interior angles 4 and 5 are alternate interior angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Alternate Exterior Angles L10_Parallel Lines and Related Angles ERHS Math Geometry Alternate Exterior Angles 1 and 8 are alternate exterior angles 2 and 7 are alternate exterior angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles ERHS Math Geometry Corresponding Angles 1 and 5, 2 and 6, 3 and 7, 4 and 8 are corresponding angles n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Angles of Transversal and Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles of Transversal and Lines Review: n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles 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

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Eight angles are formed by the transversal and the two parallel lines n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Parallel Lines and Transversal L10_Parallel Lines and Related Angles ERHS Math Geometry Parallel Lines and Transversal Corresponding Angles Postulate Alternate Interior Angles Theorem Same-Side Interior Angles Theorem Mr. Chin-Sung Lin

Parallel Lines and Transversal L10_Parallel Lines and Related Angles ERHS Math Geometry Parallel Lines and Transversal Converse of Corresponding Angles Postulate Converse of Alternate Interior Angles Theorem Converse of Same-Side Interior Angles Theorem Mr. Chin-Sung Lin

Corresponding Angles Postulate L10_Parallel Lines and Related Angles 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 n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent If m || n, 3  6, and 4  5 n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Alternate Interior Angles Theorem Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Alternate Interior Angles Theorem Statements 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 n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary If m || n, 3 and 5, 4 and 6 are supplementary n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Same-Side Interior Angles Theorem Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Same-Side Interior Angles Theorem Statements 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 4. Substitution property n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Corresponding Angles Postulate L10_Parallel Lines and Related Angles 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, n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles 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 If 3  6 or 4  5, m || n n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Converse of Alternate Interior Angles Theorem Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Alternate Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Converse of Alternate Interior Angles Theorem Statements 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 n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles 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 If 3 and 5, or 4 and 6 are supplementary, m || n, n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Converse of Same-Side Interior Angles Theorem Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Converse of Same-Side Interior Angles Theorem L10_Parallel Lines and Related Angles ERHS Math Geometry Converse of Same-Side Interior Angles Theorem n m k 1 2 3 4 5 6 7 8 Statements Reasons 1. 4 and 6, or 3 and 5 1. Given are supplementary 2. 8 and 6, and 7 and 5 2. Supplementary angles 3. 4  8 or 3  7 3. Supp. angle theorem 4. m || n 4. Converse of corresponding angle postulate Mr. Chin-Sung Lin

Methods of Proving Lines Parallel L10_Parallel Lines and Related Angles ERHS Math Geometry Methods of Proving Lines Parallel Congruent Corresponding Angles Congruent Alternate Interior Angles Same-Side Interior Angles supplementary Both lines are perpendicular to the same line Mr. Chin-Sung Lin

L10_Parallel Lines and Related Angles ERHS Math Geometry Examples Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Classify the following angles into alternate interior angles, same-side interior angles or corresponding angles r q p 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles 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 r q p 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of all the angles r q p 1 2 3 4 5 6 7 125o Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Calculate the value of all the angles q r 55o p 125o 125o 55o 55o 125o 125o 55o Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of x r q p 1 2 3 4 x–10o 6 7 x+50o Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines If p and q are parallel, calculate the value of x x + 50o + x – 10o = 180o 2x = 140o x = 70o r q p 1 2 3 4 x–10o 6 7 x+50o Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Given: 1  8 Prove: m || n n m k 1 2 3 4 5 6 7 8 Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines n m k 1 2 3 4 5 6 7 8 Given: 1  8 Prove: m || n Statements 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

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Given: m1 + m6 = 180 m6 + m9 =180 Prove: p || n n m k 1 2 3 4 5 6 7 8 9 10 11 12 p Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines n m k 1 2 3 4 5 6 7 8 9 10 11 12 p Given: m1 + m6 = 180 m6 + m9 =180 Prove: p || n Statements Reasons 1. m1 + m6  180 1. Given m6 + m9  180 2. 1  4 and 6  7 2. Vertical angles 3. m4 + m6  180 3. Substitution property m7 + m9  180 4. p || m, m || n 4. Converse of same-side interior angle theorem 5. p || n 5. Transitive property Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Given: Quadrilateral ABCD BC = DA, and BC || DA Prove: AB || CD ~ B C A D Mr. Chin-Sung Lin

Angles Formed by Parallel Lines L10_Parallel Lines and Related Angles ERHS Math Geometry Angles Formed by Parallel Lines Given: Quadrilateral ABCD BC = DA, and BC || DA Prove: AB || CD ~ B C A D Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem 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

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem The sum of the measures of the interior angles of any triangle is 180. Draw the graph Given: ∆ ABC Prove: mA + mB + mC = 180 A B C Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem D A 1 2 B C Statements Reasons 1. 2. 2. 3. 3. 4. 4. Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem D A 1 2 B C Statements Reasons Let AD be the line through A 1. Use properties of parallel lines and parallel to BC 2. 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

Corollaries to the Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollaries to the Triangle Angle-Sum Theorem Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollary A corollary is a statement that follows directly from the theorem A corollary is a statement that can be easily proved by applying the theorem Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem 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

L11_Triangle Angle-Sum Theorem 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 X Z Y A C B Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry A C B X Z Y Proof of Corollary 1 Statements Reasons 1. A  X, B  Y 1. Given 2. mA + mB + mC = 180 2. Triangle angle 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

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollary 2 The two acute angles of a right triangle are complementary Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem 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 A C B Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry A C B Proof of Corollary 2 Statements Reasons 1. mC = 90 1. Given 2. mA + mB + mC = 180 2. Triangle angle 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

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollary 3 Each acute angle of an isosceles right triangle measured 45o Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Proof of Corollary 3 Each acute angle of an isosceles right triangle measured 45o Given: mC = 90, ∆ABC is isosceles Prove: mA = mB = 45 A B C Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Proof of Corollary 3 A B C Statements Reasons 1. mC = 90 1. Given 2. mA + mB + mC = 180 2. Triangle angle 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. 2mA = 2mB = 90 6. Substitution property 7. mA = mB = 45 7. Division property Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollary 4 Each angle of an equilateral triangle has measure 60 Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem 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 A B C Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry A B C Proof of Corollary 4 Statements 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 angle sum theorem 4. mA + mA + mA = 180 4. Substitution property mB + mB + mB = 180 mC + mC + mC = 180 5. mA = mB = mC = 60 5. Division property Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Corollary 5 Prove: mA + mB + mC + mD= 360 A B C D Mr. Chin-Sung Lin

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Proof of Corollary 5 A B C D 2 1 3 4 Statements Reasons 1. Let AC divides quadrilateral 1. Use properties of triangles ABCD into two triangles 2. m1 + mB + m3 = 180 2. Triangle angle-sum theorem m2 + mD + m4 = 180 3. m1 + mB + m3 + 3. Addition property m2 + mD + m4 = 360 4. mA + mB + mC + mD 4. Partition property = 360 Mr. Chin-Sung Lin

Exterior Angle Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Exterior Angle Theorem Mr. Chin-Sung Lin

Exterior Angle of a Triangle L11_Triangle Angle-Sum Theorem 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 A B C Mr. Chin-Sung Lin

Exterior Angle Theorem L11_Triangle Angle-Sum Theorem 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

Exterior Angle Theorem L11_Triangle Angle-Sum Theorem 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 B C 1 Mr. Chin-Sung Lin

Exterior Angle Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Exterior Angle Theorem A B C 1 Statements Reasons 1. m1 + mB = 180 1. Supplementary angles 2. mA + mB + mC = 180 2. Triangle angle-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

L11_Triangle Angle-Sum Theorem ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

Triangle Angle Sum Theorem L11_Triangle Angle-Sum Theorem 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. A B C Mr. Chin-Sung Lin

Triangle Angle Sum Theorem L11_Triangle Angle-Sum Theorem 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 (x + 30) + x + x = 180 3x + 30 = 180 3x = 150 x = 50 A B C X+30 X X Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem Find the values of x and y y 5y 4x 3y Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem Find the values of x and y 5y + y + 4x = 180 4x + 4x + 3y = 180 4x + 6y = 180……(1) 8x + 3y = 180……(2) (1)/2, 2x + 3y = 90..…..(3) (2) - (3), 6x = 90, and then X = 15……(4) (4) Substitutes into (3), 2(15) + 3y = 90, 3y = 60, y = 20 y 5y 4x 3y So, X =15 Y = 20 Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem Given: m2 = mC Prove: m1 = mB A B C D E 1 2 Mr. Chin-Sung Lin

Triangle Angle-Sum Theorem L11_Triangle Angle-Sum Theorem ERHS Math Geometry Triangle Angle-Sum Theorem A B C D E 1 2 Statements Reasons 1. m2 = mC 1. Given 2. m1 + m2 + mA = 180 2. Triangle angle-sum theorem mB + mC + mA = 180 3. m1 + m2 + mA = 3. Substitution property mB + mC + mA 4. m1 + mC + mA = 4. Substitution property 5. m1 = mB 5. Subtraction property Mr. Chin-Sung Lin

L7_Congruent Right Triangles ERHS Math Geometry AAS Postulate Mr. Chin-Sung Lin

Postulates that Prove Congruent Triangles L7_Congruent Right Triangles ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Mr. Chin-Sung Lin

Postulates that Prove Congruent Triangles L7_Congruent Right Triangles ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Mr. Chin-Sung Lin

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

AAS Postulate Given CA is an angle bisector of DCB, and B = D Congruent Triangles 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

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

AAS Postulate Corollary 1 Congruent Triangles 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 AB = DE, A = D Prove ∆ ABC = ∆ DEF ~ ~ A C B D F E Mr. Chin-Sung Lin

AAS Postulate Corollary 2 Congruent Triangles 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

Congruent Right Triangles (HL Postulate) L7_Congruent Right Triangles ERHS Math Geometry Congruent Right Triangles (HL Postulate) Mr. Chin-Sung Lin

Postulates that Prove Congruent Triangles L7_Congruent Right Triangles ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Mr. Chin-Sung Lin

Postulates that Prove Congruent Triangles L7_Congruent Right Triangles ERHS Math Geometry Postulates that Prove Congruent Triangles Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Hypotenuse-Leg Postulate (HL) Mr. Chin-Sung Lin

Hypotenuse-Leg Postulate (HL Postulate) L7_Congruent Right Triangles 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 C B D F E Mr. Chin-Sung Lin

Side-Side-Angle Case (SSA) L7_Congruent Right Triangles ERHS Math Geometry Side-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

L7_Congruent Right Triangles ERHS Math Geometry HL Postulate – Example Given BD is the altitude of an isosceles triangle ∆ ABC Prove ∆ ABD = ∆ CBD ~ A C B D Mr. Chin-Sung Lin

L7_Congruent Right Triangles ERHS Math Geometry HL Postulate – Example Given BD is the altitude of an isosceles triangle ∆ ABC Prove ∆ ABD = ∆ CBD HL ~ A C B D Mr. Chin-Sung Lin

L7_Congruent Right Triangles ERHS Math Geometry HL Postulate – Example A C B D Statements Reasons 1. ∆ ABC is isosceles triangle 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  BD 4. Reflexive property 5. ∆ ABD  ∆ CBD 5. HL postulate Mr. Chin-Sung Lin

L8_Isosceles Triangles ERHS Math Geometry Base Angle Theorem Mr. Chin-Sung Lin

L8_Isosceles Triangles 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

L8_Isosceles Triangles 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: AB  CB Prove: A  C A C B Mr. Chin-Sung Lin

L8_Isosceles Triangles ERHS Math Geometry A C B D Base Angle Theorem Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Mr. Chin-Sung Lin

L8_Isosceles Triangles ERHS Math Geometry A C B D Base Angle Theorem Statements Reasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. AB  CB 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. SAS Postulate 6. A  C 6. CPCTC Mr. Chin-Sung Lin

Converse of Base Angle Theorem L8_Isosceles Triangles ERHS Math Geometry Converse of Base Angle Theorem Mr. Chin-Sung Lin

Converse of Base Angle Theorem L8_Isosceles Triangles 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

Converse of Base Angle Theorem L8_Isosceles Triangles 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 Prove: AB  CB A C B Mr. Chin-Sung Lin

Converse of Base Angle Theorem L8_Isosceles Triangles ERHS Math Geometry Converse of Base Angle Theorem A C B D Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Mr. Chin-Sung Lin

Converse of Base Angle Theorem L8_Isosceles Triangles ERHS Math Geometry Converse of Base Angle Theorem A C B D Statements Reasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. A  C 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. AAS Postulate 6. AB  CB 6. CPCTC Mr. Chin-Sung Lin

Base Angle Theorem - Example L8_Isosceles Triangles ERHS Math Geometry Base Angle Theorem - Example Given: AO  BO and 1  2 Prove: AC = BD A C B D O 1 2 Mr. Chin-Sung Lin

Base Angle Theorem – Example L8_Isosceles Triangles ERHS Math Geometry Base Angle Theorem – Example Given: AO  BO and 1  2 Prove: AC = BD A C B D O 1 2 Mr. Chin-Sung Lin

Base Angle Theorem - Example L8_Isosceles Triangles ERHS Math Geometry Base Angle Theorem - Example A C B D O 1 2 Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Mr. Chin-Sung Lin

Base Angle Theorem - Example L8_Isosceles Triangles ERHS Math Geometry Base Angle Theorem - Example A C B D O 1 2 Statements Reasons 1. 1  2 1. Given 2. CO  DO 2. Converse of Base Angle Theorem 3. AO  BO 3. Given 4. AOC  BOD 4. Vertical Angles 5. ∆ AOC = ∆ BOD 5. SAS Postulate 6. AC  BD 6. CPCTC Mr. Chin-Sung Lin

Equilateral and Equiangular Triangles L8_Isosceles Triangles ERHS Math Geometry Equilateral and Equiangular Triangles Mr. Chin-Sung Lin

Equilateral Triangles L8_Isosceles Triangles ERHS Math Geometry Equilateral Triangles A equilateral triangle is a triangle that has three congruent sides A C B Mr. Chin-Sung Lin

Equilateral & Equiangular Triangles L8_Isosceles Triangles ERHS Math Geometry Equilateral & Equiangular Triangles If a triangle is an equilateral triangle, then it is an equiangular triangle Mr. Chin-Sung Lin

L8_Isosceles Triangles ERHS Math Geometry Polygons Mr. Chin-Sung Lin

Definition of Polygons L5_Polygons ERHS Math Geometry Definition of Polygons A polygon is a closed plane figure with the following properties: Formed by three or more line segments called sides 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

Naming Polygons Each endpoint of a side is a vertex of the polygon L5_Polygons 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 E B D C Mr. Chin-Sung Lin

Convex & Concave Polygons L5_Polygons 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 Interior Convex Concave Mr. Chin-Sung Lin

Convex & Concave Polygons L5_Polygons 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 > 1800 Convex Concave Mr. Chin-Sung Lin

L5_Polygons 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

L5_Polygons ERHS Math Geometry Identifying Polygons Identify polygons in the following figures and tell whether the figure is a convex or concave polygon Convex Concave Convex Concave Mr. Chin-Sung Lin

L8_Isosceles Triangles 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

Classifying Polygons -- by the Number of Its Sides L5_Polygons ERHS Math Geometry Classifying Polygons -- by the Number of Its Sides No. of sides Polygons 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon No. of sides Polygons 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon Mr. Chin-Sung Lin

Classifying Polygons -- by the Number of Its Sides L5_Polygons ERHS Math Geometry Classifying Polygons -- by the Number of Its Sides Mr. Chin-Sung Lin

Classifying Polygons -- by the Number of Its Sides L5_Polygons ERHS Math Geometry Classifying Polygons -- by the Number of Its Sides Pentagon Hexagon Quadrilateral Triangle Mr. Chin-Sung Lin

Classifying Polygons -- by the Number of Its Sides L5_Polygons ERHS Math Geometry Classifying Polygons -- by the Number of Its Sides Mr. Chin-Sung Lin

Classifying Polygons -- by the Number of Its Sides L5_Polygons ERHS Math Geometry Classifying Polygons -- by the Number of Its Sides Decagon Heptagon Dodecagon 16-gon Mr. Chin-Sung Lin

Equilateral / Equiangular / Regular Polygons L5_Polygons 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

Identify Equilateral / Equiangular / Regular Polygons L5_Polygons ERHS Math Geometry Identify Equilateral / Equiangular / Regular Polygons Mr. Chin-Sung Lin

Identify Equilateral / Equiangular / Regular Polygons L5_Polygons ERHS Math Geometry Identify Equilateral / Equiangular / Regular Polygons Regular Equilateral Equiangular Equilateral Regular Mr. Chin-Sung Lin

Interior and Exterior Angles of Polygons L8_Isosceles Triangles ERHS Math Geometry Interior and Exterior Angles of Polygons Mr. Chin-Sung Lin

Consecutive Angles and Vertices L5_Polygons 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 Consecutive Angles: A and B Consecutive Angles: C and D E B Consecutive Vertices: A and B Non-adjacent Vertices: E and B D C Mr. Chin-Sung Lin

L5_Polygons ERHS Math Geometry Diagonals of Polygons A diagonal of a polygon is a line segment whose endpoints are two non-adjacent vertices A B C D E Adjacent Vertices of B: A and C Non-adjacent Vertices of B: D and E Diagonals with Endpoint B: BD and BE Mr. Chin-Sung Lin

L5_Polygons ERHS Math Geometry Diagonals of Polygons A diagonal of a polygon is a line segment whose endpoints are two non-adjacent vertices A B C D E Adjacent Vertices of B: A and C Non-adjacent Vertices of B: D and E Diagonals with Endpoint B: BD and BE Mr. Chin-Sung Lin

Diagonals of Polygons All possible diagonals from a vertex L5_Polygons ERHS Math Geometry Diagonals of Polygons All possible diagonals from a vertex A B C D F A B C D E F A B C D E G A B C D E Mr. Chin-Sung Lin

Sum of Interior Angles of Polygons L5_Polygons ERHS Math Geometry Sum of Interior Angles of Polygons Calculate the sum of intertor angles of each polygon A B C D F A B C D E F A B C D E G A B C D E Mr. Chin-Sung Lin

Sum of Interior Angles of Polygons L5_Polygons ERHS Math Geometry Sum of Interior Angles of Polygons Calculate the sum of intertor angles of each polygon A B C D F A B C D E F A B C D E G A B C D E 2 (1800) 4 (1800) 5 (1800) 3 (1800) Mr. Chin-Sung Lin

Theorem: Sum of Polygon Interior Angles L5_Polygons 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 F A B C D E 180 (6 – 2)0 = 720o Mr. Chin-Sung Lin

Theorem: Sum of Polygon Exterior Angles L5_Polygons ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360o Mr. Chin-Sung Lin

Theorem: Sum of Polygon Exterior Angles L5_Polygons ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360o Sum of exterior angles = 180n – Sum of interior angles = 180n – 180 (n – 2) = 180n – 180n + 360 = 360 Mr. Chin-Sung Lin

Theorem: Sum of Polygon Exterior Angles L5_Polygons ERHS Math Geometry Theorem: Sum of Polygon Exterior Angles The sum of the measures of the exterior angles of a polygon is 360o F A B C D E Sum of exterior angles = 360o Mr. Chin-Sung Lin

Example: Interior / Exterior Angles L5_Polygons ERHS Math Geometry Example: Interior / Exterior Angles The measure of an exterior angle of a regular polygon is 45o (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

Example: Interior / Exterior Angles L5_Polygons ERHS Math Geometry Example: Interior / Exterior Angles The measure of an exterior angle of a regular polygon is 45o (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

Example: Interior / Exterior Angles L5_Polygons ERHS Math Geometry Example: Interior / Exterior Angles In quadrilateral ABCD, mA = x, mB = 2x – 12, mC = x + 22, and mD = 3x (a) Find the measure of each interior angle (b) Find the measure of each exterior angle Mr. Chin-Sung Lin

Example: Interior / Exterior Angles L5_Polygons ERHS Math Geometry Example: Interior / Exterior Angles In quadrilateral ABCD, mA = x, mB = 2x – 12, mC = x + 22, and mD = 3x (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

L10_Parallel Lines and Related Angles ERHS Math Geometry The End Mr. Chin-Sung Lin