LUOA geometry Module 1

m∠WYX=(4x−8)° and m∠WYZ=10x°. If ∠WYX and ∠WYZ are complementary, what is the measure of each angle?

m∠WYX = 20°; m∠WYZ = 70°

JK¯¯¯¯¯ has endpoints J(−5,4) and K(7,−10). Identify the coordinates of the midpoint of JK¯¯¯¯¯.

(1, −3)

B is the midpoint of AC¯¯¯¯¯. A has coordinates (−5,8), and B has coordinates (0,0). Identify the coordinates of C.

(5, −8)

Find the coordinates of the image of Q(3, 4) after the translation (x, y)→(x+3, y−5).

(6, −1)

D is the midpoint of RJ¯¯¯¯¯. R has coordinates (−4,7), and D has coordinates (2,−1). Identify the coordinates of J.

(8, −9)

Identify the correct justification for each step, given that ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. The figure shows four rays with a common endpoint B. Ray B A is directed to the upper left corner. Ray B D is directed vertically upward. Ray B E is directed to the upper right corner. Ray B C is directed horizontally to the right. Rays B A and B D form angle 1. Rays B D and B E form angle 2. Rays B E and B C form angle 3. 1.∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary 2. m∠2+m∠3=90° 3. m∠2+m∠3=90° 4. m∠1+m∠2=m∠2+m∠3 5. m∠1=m∠3

1. Given 2. Def. of Comp. ∠s 3. Def. of Comp. ∠s 4. Trans. Prop. of = 5. Subt. Prop. of =

m∠T=(3y−36)°. Find the measure of the supplement of ∠T.

(216 − 3y)°

Identify the coordinates of the midpoint of RS¯¯¯¯¯, which has endpoints R(6,−1) and S(4,3).

(5, 1)

Find the coordinates of the image of J(−7, −3) after the translation (x, y)→(x−4, y+6).

(−11, 3)

Identify the coordinates of the midpoint of RS¯¯¯¯¯ with endpoints R(4,8) and S(−8,−12).

(−2, −2)

Find the coordinates of the image of Z(−3,−1) after the translation (x,y)→(x−2,y−7).

(−5, −8)

V is the midpoint of GH¯¯¯¯¯¯. G has coordinates (5,−3), and V has coordinates (−1,3). Identify the coordinates of H.

(−7, 9)

Solve the equation. Write a justification for each step. −11=p−2/3

-31=p

Find the next item in the pattern. 0.3, 0.6, 0.9, 0.12, ...

.15

Find the circumference and area of a circle with diameter 22 ft. Express your answer in terms of π.

22π ft; 121π ft2

m∠F=109°. Find the measure of the supplement of ∠F.

71

Identify the property that justifies the statement. ∠XYZ≅∠PDQ and ∠PDQ≅∠ABC, so ∠XYZ≅∠ABC.

trans prop

Determine whether the statement is true or false. If false, give a counter example. If 127° is a right angle, then it is an angle.

true

D is in the interior of ∠ABC, m∠ABD=63°, and m∠DBC=23°. Find m∠ABC.

86

Write the definition as a biconditional statement. A triangle which has no congruent sides is a scalene triangle.

A triangle is scalene if and only if it has no sides that are congruent.

Classify ∠OMN as acute, right, straight, or obtuse.

acute

Tell whether this statement is sometimes, always, or never true. Identify the answer with a sketch. If B is a point between A and C, then AB+BC=AC.

always true;

Identify the converse and a biconditional statement for the conditional. If an angle is a straight angle, then its measure is 180°

converse: If an angle measures 180°, then the angle is a straight angle. biconditional: An angle is a straight angle if and only if its measure is 180°.

S is in the interior of ∠PQR, m∠PQS=87°, and m∠SQR=33°. Find m∠PQR.

120

Find the circumference and area of a circle with radius 6 cm. Express your answer in terms of π.

12π cm; 36π cm2

Find the next item in the pattern. 3, 5, 7, 9, 11, ...

13

Identify the distance, to the nearest tenth, between F(3,−4) and G(−8,3).

13.0

The area of a rectangle is 108.35 in2 and the length is 7.8 in. Find the width.

13.89

Identify the distance, to the nearest tenth, between S(−3,4) and T(10,−1).

13.9

Find the circumference and area of a circle with radius 7 cm. Express your answer in terms of π.

14π cm; 49π cm2

Find the area and perimeter. The figure shows a square. One side of the square is equal to 3.9 centimeters.

15.21 cm2 ; 15.6 cm

Identify the distance, to the nearest tenth, between D(7,9) and E(−2,−5).

16.6

Find the area and perimeter.

18x2 + 63x; 22x + 14

The coordinates of the vertices of △ABC are A(−2,2), B(5,−3), and C(−4,−1). Identify the perimeter of △ABC. Round each side length to the nearest tenth before adding.

21.4

The coordinates of the vertices of △WXY are W(4,−6), X(−3,−2), and Y(4,3). Identify the perimeter of △WXY. Round each side length to the nearest tenth before adding.

25.7

The coordinates of the vertices of △HIJ are H(−2,2), I(8,−2), and J(−4,−3). Identify the perimeter of △HIJ to the nearest tenth.

28.2

Find the lengths of LM¯¯¯¯¯¯ and PQ¯¯¯¯¯ and determine whether they are congruent.

35-√; 35-√; yes

Solve the equation. Write a justification for each step. −3(−x−2)=5x−9

7.5=x

The area of a rectangle is 92.43 in2 and the width is 11.7 in. Find the length.

7.9 in.

Write the converse, inverse, and contrapositive of the conditional statement. Find the truth value of each. If Julie scores 90 or above, then she receives an A.

Converse: If Julie receives an A, then her score is 90 or above. Inverse: If Julie does not score 90 or above, then she will not get an A. Contrapositive: If Julie does not get an A, then her score is not 90 or above.

The doctor has advised Catherine to take aspirin only after meals. Catherine takes aspirin, so she must have just eaten. Does the conclusion use inductive or deductive reasoning?

deductive reasoning

Determine if this conjecture is true. If not, give a counterexample. The square of an even integer is odd.

false; 122 = 144

Determine whether the biconditional statement is true or false. If false, give a counterexample. An angle is a straight angle if and only if it is not acute.

false; m∠1 = 102°, ∠1 is not a straight angle

Determine if this conjecture is true. If not, give a counterexample. Two complementary angles are not congruent.

false; m∠1 = m∠2 = 45°

Determine whether this statement is true or false. If false, identify the correct explanation. If an angle is obtuse, then its supplement is greater than a right angle.

false; supplement is less than a right angle

Determine whether the statement is true or false. If false, give a counterexample. If t2=289, then t=−17.

false; t = 17

Determine whether the biconditional statement is true or false. If false, give a counterexample. x − y is positive if and only if |x| > y.

false; x = −1, y = 0

Determine if this conjecture is true. If not, give a counterexample. The difference of two negative numbers is a negative number.

false; −11 − (−13) = 2

Dyann takes a bus to work every day. On Monday the trip took 34 minutes. On Tuesday the trip took 36 minutes. On Wednesday the trip took 35 minutes. On Thursday the trip took 34 minutes. Dyann determines that the trip to work on Friday will take about 35 minutes. Does the conclusion use inductive or deductive reasoning?

inductive reasoning

Identify the plane containing X, Y, and Z.

plane E

A figure has vertices at L(3, 3), M(2, 4), and N(−1, 0). After a transformation, the image of the figure has vertices at L′(−6, 3), M′(−5, 4), and N′(−2, 0). Identify the preimage, the image and the transformation.

reflection Triangle L M N with vertices at (3, 3), (2, 4), and (-1, 0), respectively, is graphed on a coordinate plane. A transformation shows triangle L prime M prime N prime at (-6, 3), (-5, 4), and (-2, 0), respectively.

Identify the transformation suggested by the movement of the minute hand of a clock in fifteen minutes.

rotation

A figure has vertices at L(3,3), M(2,4), and N(−1,0). After a transformation, the image of the figure has vertices at L′(2,−5), M′(3,−4), and N′(−1,−1). Identify the preimage, the image and the transformation.

rotation Triangle L M N with vertices at (3, 3), (2, 4), and (-1, 0), respectively, is graphed on a coordinate plane. A transformation shows triangle L prime M prime N prime at (2, -5), (3, -4), and (-1, -1), respectively.

Determine whether the statement is true or false. If false, give a counter example. If b3=−27, then b=−3.

true

Determine whether this statement is true or false. If false, identify the correct explanation. If a ray divides an angle into two supplementary angles, then the original angle is a straight angle.

true

Determine if the following conjecture is valid. Given: An American citizen who is at least 16 years old, is eligible to get a driver's license. Julian is a 25-year-old American citizen. Conclusion: Julian is eligible to get a driver's license.

valid

Determine if the following conjecture is valid. Given: If x is an odd number, then x is an integer. If x is an integer, then x is a rational number. 3 is an odd number. Conclusion: 3 is a rational number.

valid

PQ¯¯¯¯¯ bisects ST¯¯¯¯¯ at R. SR=3x+3 and ST=30. Find x.

x=4

GH¯¯¯¯¯¯ bisects LM¯¯¯¯¯¯ at K. LK=5x+2 and LM=64. Find x.

x=6

Solve the equation. Write a justification for each step. −4=n+7/6

−4=n+7/6 −24=n+7 −31=n GivenMult. Prop. of = Subt. Prop. of =

Find the area and perimeter. The figure shows a triangle. One side is given by 4 x minus 3 units. The second side is given by x plus 9 units. The third side is equal to 8 units. The height from the third side is given by 3 x units. Hint: The height of a non-right triangle is the length of the segment of a line that is perpendicular to the base and that contains the opposite vertex.

12x; 5x + 14

m∠W=73°. Find the measure of the complement of ∠W.

17

Solve the equation. Write a justification for each step. 2(x−4)=−4x+10

2(x−4)=−4x+10 2x−8=−4x+10 6x−8=10 6x=18 x=3 Given Distrib. Prop. of = Add. Prop. of = Add. Prop of= Div. Prop of=

Identify the intersection of plane R and plane Q.

AB←→

B is between A and C, AB=13.7 and BC=8.3. Find AC.

AC = 22

Identify a point on AD←→ .

B

Identify a point on FE←→.

B

Write the converse, inverse, and contrapositive of the conditional statement. Find the truth value of each. If John is on vacation, then he is in London.

Converse: If John is in London, then he is on vacation. Inverse: If John is not on vacation, then he is not in London. Contrapositive: If John is not in London, he is not on vacation.

Identify the hypothesis and conclusion of the conditional. A quadrilateral with congruent angles and sides is called a square.

H: A quadrilateral has congruent angles and sides. C: The quadrilateral is a square.

Identify the hypothesis and conclusion of the conditional. In a right triangle, (hypotenuse)2= (adjacent side)2 + (opposite side)2.

H: A triangle is a right triangle. C: (hypotenuse)2 = (adjacent side)2 + (opposite side)2

Given points U(−4,4) and V(5,1), identify UV¯¯¯¯¯ and its reflection across the x-axis.

Line segment U V with endpoints (-4, 4) and (5, 1) is graphed on a coordinate plane. A transformation shows line segment U prime V prime with endpoints at (-4, -1) and (5, -4).

Identify two opposite rays.

OB−→− and OD−→−

Identify the intersection of plane A and plane B.

PQ←→

Identify the property that justifies the statement. ∠LMN≅∠LMN

Reflexive Prop. of ≅

Identify the property that justifies the statement. OP¯¯¯¯¯≅RS¯¯¯¯¯, so RS¯¯¯¯¯≅OP¯¯¯¯¯

Sym. Prop. of ≅

Identify the converse and a biconditional statement for the conditional. If two lines are parallel, then they are equidistant everywhere.

converse: If two lines are equidistant everywhere, then they are parallel lines. biconditional: Two lines are parallel if and only if they are equidistant everywhere.

Identify the converse and a biconditional statement for the conditional. If two lines intersect, then they are not parallel.

converse: If two lines are not parallel, then they intersect. biconditional: Two lines intersect if and only if they are not parallel.

Solve the equation. Write a justification for each step. m−17/11=8

m=105 mult prop add prop

m∠ABC=(6x+8)° and m∠DEF=(12x−8)°. If ∠ABC and ∠DEF are supplementary, what is the measure of each angle?

m∠ABC = 68°; m∠DEF = 112°

m∠WYX=(2x−1)° and m∠WYZ=(4x+1)°. If ∠WYX and ∠WYZ are complementary, what is the measure of each angle?

m∠WYX = 29°; m∠WYZ = 61°

Determine if the following conjecture is valid. Given: If x is an odd number, then x is an integer. If x is an integer, then x is a rational number. 19.132 is a rational number. Conclusion: 19.132 is an odd number.

not valid

Classify ∠CBE as acute, right, straight, or obtuse.

obtuse

Identify the correct justification for each step, given that m∠1≅m∠2, m∠3=110°, and ∠2 and ∠DBA are supplementary. The figure shows four rays with a common endpoint B. Rays B A and B C are opposite. They are directed horizontally to the left and to the right, respectively. Ray B E is directed to the upper left corner. Ray B D is directed to the upper right corner. Rays B A and B E form angle 1. Rays B D and B C form angle 2. Rays B E and B D form angle 3. 1. ∠2 and ∠DBA are supplementary 2. m∠2+m∠DBA=180° 3. m∠1+m∠3=m∠DBA 4. m∠2+m∠1+m∠3=180° 5. m∠1≅m∠2 6. m∠2+m∠2+m∠3=180° 7. m∠3=110° 8. m∠2+m∠2+110°=180° 9. 2(m∠2)+110°=180° 10. 2(m∠2)=70° 11. m∠2=35°

1. Given 2. Def. of Supp. ∠s 3. ∠ Add. Post. 4. Subst. Prop. of = 5. Given 6. Subst. Prop. of = 7. Given 8. Subst. Prop. of = 9. Simplify 10. Subt. Prop. of = 11. Div. Prop. of =

Identify the correct justification for each step, given that m∠1+m∠2=90° and m∠2+m∠3=90° The figure shows four rays with a common endpoint B. Ray B A is directed to the upper left corner. Ray B D is directed vertically upward. Ray B E is directed to the upper right corner. Ray B C is directed horizontally to the right. Rays B A and B D form angle 1. Rays B D and B E form angle 2. Rays B E and B C form angle 3. 1. m∠1+m∠2=90° m∠2+m∠3=90° 2. m∠1+m∠2=m∠2+m∠3 3. m∠1=m∠3

1. Given 2. Trans. Prop. of = 3. Subt. Prop. of =

Identify the correct justification for each step, given that m∠1=25° and m∠3=110°. The figure shows four rays with a common endpoint B. Rays B A and B C are opposite. They are directed horizontally to the left and to the right, respectively. Ray B E is directed to the upper left corner. Ray B D is directed to the upper right corner. Rays B A and B E form angle 1. Rays B D and B C form angle 2. Rays B E and B D form angle 3. 1. m∠1=25°, m∠3=110° 2. m∠1+m∠3=m∠ABD 3. ∠2 and ∠ABD are supplementary 4. m∠2+m∠ABD=180° 5. m∠2+m∠1+m∠3=180° 6. 25°+110°+m∠3=180° 7. 135°+m∠3=180° 8. m∠3=45°

1. Given 2. ∠ Add. Post. 3. Lin. Pair Thm. 4. Def. of Supp. ∠s 5. Subst. Prop. of = 6. Subst. Prop. of = 7. Simplify 8. Subt. Prop. of =

Identify the correct justification for each step, given that m∠1=50° and m∠2=50°. The figure shows four rays with a common endpoint B. Rays B A and B C are opposite. They are directed horizontally to the left and to the right, respectively. Ray B E is directed to the upper left corner. Ray B D is directed to the upper right corner. Rays B A and B E form angle 1. Rays B D and B C form angle 2. Rays B E and B D form angle 3. 1. m∠1=50°, m∠2=50° 2. m∠2+m∠3=m∠CBE 3. ∠1 and ∠CBE are supplementary 4. m∠1+m∠CBE=180° 5. m∠1+m∠2+m∠3=180° 6. 50°+50°+m∠3=180° 7. 100°+m∠3=180° 8. m∠3=80°

1. Given 2. ∠ Add. Post. 3. Lin. Pair Thm. 4. Def. of Supp. ∠s 5. Subst. Prop. of = 6. Subst. Prop. of = 7. Simplify 8. Subt. Prop. of =

Identify the correct two-column proof for this plan. Given: QP¯¯¯¯¯≅MP¯¯¯¯¯¯ and NP¯¯¯¯¯¯≅RP¯¯¯¯¯ Prove: MN=QR. The figure shows two segments Q P and M N. These segments intersect at point P. Plan: Use the definition of congruent segments to write the given information in terms of lengths. Then use the Segment Addition Postulate to show that MN=QR.

1. QP¯¯¯¯¯≅MP¯¯¯¯¯¯, NP¯¯¯¯¯¯≅RP¯¯¯¯¯ (Given)2. QP=MP, NP=RP (Defn. of ≅)3. QP+PR=QR (Seg. Add. Post.)4. MP+PN=QR (Subst. Prop. of =)5. MP+PN=MN (Seg. Add. Post.)6. MN=QR (Trans. Prop. of =)

Identify the correct two-column proof for this plan. Given: ∠1≅∠3 Prove: ∠2 and ∠3 are supplementary. The figure shows a compound object that consists of two parts. The left part of the object shows three rays with a common endpoint O. Ray O C is directed horizontally to the right. Ray O A is directed horizontally to the left. Ray O B is directed to the upper right corner. Rays O B and O C form angle 1. Rays O B and O A form angle 2. The right part of the object shows two rays with a common endpoint D. Ray D E is directed horizontally to the right. Ray D F is directed to the lower right corner. Rays D E and D F form angle 3. Plan: Use the Linear Pair Theorem to show that ∠1 and ∠2 are supplementary. Then use the definition of supplementary angles, congruence, and substitution.

1. ∠1 and ∠2 are supp. (Lin. Pair. Thm.) 2. m∠1 + m∠2 = 180° (Defn. of supp. ∠s) 3. ∠1 ≅ ∠3 (Given) 4. m∠1 = m∠3 (Defn. of ≅) 5. m∠3 + m∠2 = 180° (Subst. Prop. of = ) 6. ∠2 and ∠3 are supp (Defn. of supp. ∠s)

Write the definition as a biconditional statement. An equilateral triangle is a triangle with three congruent sides.

A triangle is equilateral if and only if it has three congruent sides.

Identify the flowchart proof for the two-column proof. Given: m∠a=18° and m∠b=18° Prove: ∠a≅∠b Two-Column Proof 1. m∠a=18° (Given) 2. m∠b=18° (Given) 3. m∠a=m∠b (Trans. Prop. of =) 4. ∠a≅∠b (Def. of ≅∠s) The figure shows a slant line that intersects two parallel rays. An angle between the slant line and the first parallel line is labeled as a. An angle between the slant line and the second parallel line is labeled as b.

Four boxes, two in each row. In the top left box, the measure of angle a equals 18 degrees and the reason is Given. There is an arrow pointing to the bottom left box. In the top right box, the measure of angle b equals 18 degrees and the reason is Given. There is an arrow pointing to the bottom left box. In the bottom left box, the measure of angle a equals the measure of angle b and the reason is the Transitive Property of Equality. There is an arrow pointing to the bottom right box. In the bottom right box, angle a is congruent to angle b and the reason is the Definition of Congruence.

Convert the following statement into a conditional. A quadrilateral with 4 congruent sides and 4 congruent angles is a square.

If a quadrilateral has 4 congruent sides and 4 congruent angles, then it is a square.

Identify the correct justification for each step, given that AC¯¯¯¯¯ intersects BE¯¯¯¯¯, m∠AFE=40° and m∠CFD=50°. The figure shows two lines A C and B E. These lines intersect at point F. Ray F D separates angle E F C into two smaller angles. It is given that AC¯¯¯¯¯ intersects BE¯¯¯¯¯. By the ________, ∠AFE and ∠BFC are vertical ∠s. By the ________, ∠AFE≅∠BFC. By the ________, m∠AFE=m∠BFC. It is given that m∠AFE=40°. So, m∠BFC=40° by the ________. By the ________, m∠BFD=m∠BFC+m∠CFD. It is given that m∠CFD=50°. So, m∠BFD=40°+50° by the ________. Simplify to get m∠BFD=90°. Thus, ∠BFD is a right ∠ by ________.

It is given that AC¯¯¯¯¯ intersects BE¯¯¯¯¯. By the definition of vertical ∠s−−−−−−−−−−−−−−−−−−, ∠AFE and ∠BFC are vertical ∠s. By the Vertical ∠s Thm.−−−−−−−−−−−−−−,∠AFE≅∠BFC. By the definition of ≅ ∠s−−−−−−−−−−−−−−−, m∠AFE=m∠BFC. It is given that m∠AFE=40°. So, m∠BFC=40° by the Subst. Property of =−−−−−−−−−−−−−−−−. By the ∠ Addition Post.−−−−−−−−−−−−−, m∠BFD=m∠BFC+m∠CFD. It is given that m∠CFD=50°. So, m∠BFD=40°+50° by the Subst. Property of =−−−−−−−−−−−−−−−−. Simplify to get m∠BFD=90°. Thus, ∠BFD is a right ∠ by the definition of right ∠s

Identify the correct paragraph proof for this two-column proof. Given: AC¯¯¯¯¯≅DC¯¯¯¯¯ and AB¯¯¯¯¯≅DE¯¯¯¯¯ Prove: CB=CE Two-Column proof 1. AC¯¯¯¯¯≅DC¯¯¯¯¯, AB¯¯¯¯¯≅DE¯¯¯¯¯ (Given) 2. AC=DC, AB=DE (Def. of ≅) 3. AC+CB=AB (Seg. Add. Post.) 4. DC+CE=DE (Seg. Add. Post.) 5. AC+CE=DE (Subst. Prop. of =) 6. AC+CB=DE (Subst. Prop. of =) 7. CB=CE (Trans. Prop. of =) The figure shows two line segments A B and D E that intersect at point C.

It is given that line segments AC and DC are congruent, as well as line segments AB and DE. So, by the definition of congruence, the lengths of line segments AC and DC are equal, as well as AB and DE. By the Segment Addition Postulate, AC + CB = AB and DC + CE = DE. It follows that by the Substitution Property of Equality, AC + CE = DE since AC = DC and AC + CB = DE since AB = DE. Therefore, by the Transitive Property of Equality, CB = CE.

Identify the paragraph proof for the two-column proof. Given: m∠f=115° and m∠g=115° Prove: ∠f≅∠g Two-Column Proof 1. m∠f=115° (Given) 2. m∠g=115° (Given) 3. m∠f=m∠g (Trans. Prop. of = ) 4. ∠f≅∠g (Def. of ≅∠s) The figure shows two parallel lines. A slant line intersects these parallel lines. An angle between the slant line and the first parallel line is labeled as f. An angle between the slant line and the second parallel line is labeled as g.

It is given that m∠f = 115° and m∠g = 115°. By the Transitive Property of Equality, m∠f = m∠g. By the definition of congruent angles, ∠f ≅ ∠g.

Given points D(−3,−1), F(1,3), and G(−3,3), identify △DFG and its reflection across the line y=x.

Triangle D F G with vertices at (-3, -1), (1, 3), and (-3, 3), respectively, is graphed on a coordinate plane. A transformation shows triangle D prime F prime G prime at (-1, -3), (3, 1), and (3, -3), respectively.

Identify the paragraph proof for the two-column proof. Given: m∠t=4⋅m∠s Prove: m∠s=36° Two-Column Proof 1. m∠t=4⋅m∠s (Given) 2. ∠t and ∠s are supplementary (Lin. Pair Thm.) 3. m∠t+m∠s=180° (Def. of Supp. ∠s) 4. 4⋅m∠s+m∠s=180° (Subst. Prop. of =) 5. 5⋅m∠s=180° (Simplify.) 6. m∠s=36° (Div. Prop. of =) The figure shows a straight angle that consists of two smaller angles. The first small angle is labeled as s, and the second one is labeled as t.

It is given that m∠t = 4 ⋅ m∠s. ∠t and ∠s are supplementary angles by the Linear Pair Theorem. By definition of supplementary angles, m∠t + m∠s = 180°. By Substitution Property of Equality, 4 ⋅ m ∠s + m∠s = 180°. By simplification, 5 ⋅ m ∠s = 180°. By Division Property of Equality, m∠s = 36°.

Identify the paragraph proof for the two-column proof. Given: m∠t=5⋅m∠s Prove: m∠s=30° Two-Column Proof 1. m∠t=5⋅m∠s (Given) 2. ∠t and ∠s are supplementary (Lin. Pair Thm.) 3. m∠t+m∠s=180° (Def. of Supp. ∠s) 4. 5⋅m∠s+m∠s=180° (Subst. Prop. of =) 5. 6⋅m∠s=180° (Simplify.) 6. m∠s=30° (Div. Prop. of =) The figure shows a straight angle that consists of two smaller angles. The first small angle is labeled as s, and the second one is labeled as t.

It is given that m∠t = 5 ⋅ m∠s. ∠t and ∠s are supplementary angles by the Linear Pair Theorem. By the definition of supplementary angles, m∠t + m∠s = 180°. By the Substitution Property of Equality, 5 ⋅ m ∠s + m∠s = 180°. By simplification, 6 ⋅ m ∠s = 180°. By the Division Property of Equality, m∠s = 30°.

If p→q and q→r and r→s, then p→s. This is most similar to which Law?

Law of Syllogism

Find the next item in the pattern. The figure shows a pattern. The first item of the pattern is a left up right arrow. The second item is an up right down arrow. The third item is a left down right arrow.

Three connected arrows that point directly down, left, and up.