Warm Up Angle Sum Theorem The sum of

  • Slides: 17
Download presentation
Warm Up Angle Sum Theorem The sum of the measures of the interior angles

Warm Up Angle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. “a + b + c = 180 o” Given: The figure Prove: a + b + c = 180 o alternate interior b b a c alternate interior c

Lesson 2 -6 Write two-column proofs

Lesson 2 -6 Write two-column proofs

Addition Property Definition of congruent seg. Subtraction Given Property Conclusion Definition of congruent <s

Addition Property Definition of congruent seg. Subtraction Given Property Conclusion Definition of congruent <s Multiplication Property Definition of Linear Pair <s Division Property Definition of right <s Reflexive Property Definition of supplementary <s Symmetric Property Transitive Property • Definitions • Postulates Definition of complementary <s • Properties • Theorems Definition of angle bisector Substitution Property Segment addition postulate Distributive Property Angle addition postulate

Definition of congruent <s If “two angles are congruent”, then “they have the same

Definition of congruent <s If “two angles are congruent”, then “they have the same measure” A D “m 40 o ABD = m DBC” 40 o B C Definition of Linear pair If “two angles form a linear pair”, then “they are supplementary” D “m 150 o A B 30 o C ABD + m DBC = 180 o”

Definition of right <s If “an angle is a right angle”, then “it measures

Definition of right <s If “an angle is a right angle”, then “it measures 90 o” Definition of supplementary <s If “two angles are supplementary”, then “their sum is 180 o” Definition of complementary <s If “two angles are complementary”, then “their sum is 90 o” Definition of < bisector If “a ray bisects an angle”, then “it divides it into two equal halves” A 40 o 80 o 40 o B D “m C ABD = m DBC”

Given: <A and <B are supplementary and m<A = 45 o Prove: m<B =

Given: <A and <B are supplementary and m<A = 45 o Prove: m<B = 135 o 45 o A Statements B Reasons 1) <A and <B are supplementary 1) Given 2) m<A = 45 o 2) Given 3) m<A + m<B = 180 o 3) Def. of supp. <s 4) 45 + m<B = 180 4) Substitution property 5) 45 + m<B – 45 = 180 – 45 5) Subtraction property 6) m<B = 135 o 6) Simplification

Given: <1 and <2 are complementary and m<1 = m<3 Prove: <3 and <2

Given: <1 and <2 are complementary and m<1 = m<3 Prove: <3 and <2 are comp. 3 Statements 1 Reasons 1) <1 and <2 are complementary 1) Given 2) m<1 = m<3 2) Given 3) m<1 + m<2 = 90 o 2 3) Def. of comp. <s 4) m<3 + m<2 = 90 4) Substitution property 5) <3 and <2 are complementary 5) Def. of comp. <s

Given: <BAC is a right angle and m<2 = m<3 Prove: m<1 + m<3

Given: <BAC is a right angle and m<2 = m<3 Prove: m<1 + m<3 = 90 o B 3 2 1 A Statements Reasons C 1) <BAC is a right angle 1) Given 2) m<2 = m<3 2) Given 3) m<BAC = 90 o 3) Def. of right. <s 4) m<1 + m<2 = m<BAC 4) Angle addition postulate 5) m<1 + m<2 =90 o 5) Substitution of step 3 in 4 6) m<1 + m<3 =90 o 6) Substitution of steps 2 in 5

Given: <2 <3 Prove: m<1 + m<3 = 180 o 1 Statements 1) <2

Given: <2 <3 Prove: m<1 + m<3 = 180 o 1 Statements 1) <2 <3 2 3 Reasons 1) Given 2) m<2 = m<3 2) Definition of congruent <s 3) m<1 + m<2 = 180 o 3) Definition of Linear Pair 4) m<1 + <3 =180 o 4) Substitution of step 2 in 3

X Given: BX bisects <ABC and m<XBC = 45 o A Prove: m<ABC =

X Given: BX bisects <ABC and m<XBC = 45 o A Prove: m<ABC = 90 o 45 o B Statements C Reasons 1) BX bisects <ABC 1) Given 2) m<XBC = 45 o 2) Given 3) m<ABX = m<XBC 3) Def. of < bisector. 4) m<ABX = 45 o 4) Substitution property 5) m<ABX + m<XBC = m<ABC 5) Angle addition postulate 6) 45 o + 45 o = m<ABC 6) Substitution of steps 7) m<ABC = 90 o 7) Simplification & symmetric 2 & 4 in 5

Given: <1 and <2 are supplementary <3 and <4 are supplementary m<2 = m<3

Given: <1 and <2 are supplementary <3 and <4 are supplementary m<2 = m<3 Prove: m<1 = m<4 1 Statements 4 2 3 Reasons 1) m<1 + m<2 = 180 o 1) Definition of Supp. <s 2) m<3 + m<4 = 180 o 2) Definition of Supp. <s 3) m<1 + m<2 = m<3 + m<4 3) Substitution property 4) m<1 + m<3 = m<3 + m<4 4) Substitution property 5) m<1 = m<4 5) Subtraction property

Given: <BAC is a right angle & m<2 = m<3 Prove: m<1 + m<3

Given: <BAC is a right angle & m<2 = m<3 Prove: m<1 + m<3 = 90 o B 3 2 1 A Statements Reasons C 1) <BAC is a right angle 1) Given 2) m<2 = m<3 2) Given 3) m<BAC = 90 o 3) Def. of right. <s 4) m<1 + m<2 = m<BAC 4) Angle addition postulate 5) m<1 + m<2 =90 o 5) Substitution of step 3 in 4 6) m<1 + m<3 =90 o 6) Substitution of steps 2 in 5

1 Statements 48 o Reasons 1) m<1 + 48 o = 180 o 1)

1 Statements 48 o Reasons 1) m<1 + 48 o = 180 o 1) Definition of linear pair 2) m<1 = 138 o 2) Subtraction property

(4 n + 5)o (8 n – 5)o Statements Reasons 1) 4 n +

(4 n + 5)o (8 n – 5)o Statements Reasons 1) 4 n + 5 + 8 n – 5 = 180 1) Definition of linear pair 2) 12 n = 180 2) Simplification 3) n = 15 3) Division

2 63 o Statements Reasons 1) m<2 + 63 o = 90 o 1)

2 63 o Statements Reasons 1) m<2 + 63 o = 90 o 1) Definition of comp. <s 2) m<2 = 27 o 2) Subtraction property

4 x (3 x + 6)o Statements Reasons 1) 4 x + 3 x

4 x (3 x + 6)o Statements Reasons 1) 4 x + 3 x + 6 = 90 1) Definition of Comp. <s 2) 7 x + 6 = 90 2) Simplification 3) 7 x = 84 3) Subtraction 4) x = 12 4) Division

Two angles are complementary. The measure of one angle is 10 o less than

Two angles are complementary. The measure of one angle is 10 o less than the measure of the other angle. Find the measure of each angle. Statements Reasons 1) x + y = 90 o 1) Definition of comp. <s 2) x = y – 10 2) Translation 3) y – 10 + y = 90 o 3) Substitution 4) 2 y – 10 = 90 o 4) simplification 5) 2 y = 80 o 5) Addition property 6) y = 40 o 6) Division property 7) x = 50 o 7) Substitution property

Triangle Sum Theorem Exterior Angle Theorem Work inTriangle Sum Theorem Exterior Angle Theorem Work in
Triangle Sum Theorem Exterior Angle Theorem Work inTriangle Sum Theorem Exterior Angle Theorem Work in
Severe Angle Blocking Severe Angle Blocking Severe AngleSevere Angle Blocking Severe Angle Blocking Severe Angle
Angle and Angle Measure Angles and Angle MeasureAngle and Angle Measure Angles and Angle Measure
ANGLE MODULATION Angle Modulation Introduction Types of AngleANGLE MODULATION Angle Modulation Introduction Types of Angle
Reference Angle Reference angle the positive acute angleReference Angle Reference angle the positive acute angle
RIGHT ANGLE THEOREM Lesson 4 3 Theorem 23RIGHT ANGLE THEOREM Lesson 4 3 Theorem 23
RIGHT ANGLE THEOREM Lesson 4 3 Theorem 23RIGHT ANGLE THEOREM Lesson 4 3 Theorem 23
Parallel Lines and the Triangle Angle Sum TheoremParallel Lines and the Triangle Angle Sum Theorem
Triangle Angle Sum Theorem Proof Mr Erlin GeometryTriangle Angle Sum Theorem Proof Mr Erlin Geometry
Triangle Angle Sum Theorem Proof Mr Erlin GeometryTriangle Angle Sum Theorem Proof Mr Erlin Geometry
SHAPE SPACE Pythagorean Theorem Pythagorean Theorem Pythagorean TheoremSHAPE SPACE Pythagorean Theorem Pythagorean Theorem Pythagorean Theorem
LEMIs Theorem Lamis Theorem Statement Lamis theorem isLEMIs Theorem Lamis Theorem Statement Lamis theorem is
VARIGNONS THEOREM What is Varigons Theorem The theoremVARIGNONS THEOREM What is Varigons Theorem The theorem
Thvenins Theorem Thevenins Theorem Thevenins theorem simplifies theThvenins Theorem Thevenins Theorem Thevenins theorem simplifies the
LEMIs Theorem Lamis Theorem Statement Lamis theorem isLEMIs Theorem Lamis Theorem Statement Lamis theorem is