Please Help! 1.
Given: ΔABC is a right triangle
∠A is a right angle
m∠B = 45˚
Prove: ∠B ≅ ∠C
STATEMENT: REASON:

2.
Given: m∠X = 4a + 2
m∠Y = 21a + 3
∠X and ∠Y are a linear pair
Prove: ∠Y is an obtuse angle
STATEMENT: REASON:

3.
Given: ∠A and ∠B are complementary angles
∠B and ∠C are complementary angles
Prove: ∠A ≅ ∠C
STATEMENT: REASON:

4.
Given: ΔMNP is a right triangle
∠M is a right angle
Prove: ∠N and ∠P are complementary angles
STATEMENT: REASON:

1. ∠C ≅ ∠B  by definition of congruency

2. m∠Y by definition of obtuse angle

3. ∠C ≅ ∠A by definition of congruency

4. ∠N and ∠P  are  complementary angles by definition of complementary angles

Step-by-step explanation:

1. The given parameters are;

Statement                                                Reason

ΔABC is a right triangle                           Given

∠A is a right angle                                    Given

m∠B = 45°                                                  Given

m∠A + m∠B + m∠C = 180°                         Sum of interior angles of a triangle

m∠C = 180° - (m∠A + m∠B)                      

m∠C = 180° - (90° + 45°) = 45°                   Substitution property

m∠C = 45° = m∠B                                      Substitution property

∠C ≅ ∠B                                                      Definition of congruency

2. Statement                                              Reason

m∠X = 4·a + 2                                              Given

m∠Y = 21·a + 3                                            Given

∠X and ∠Y are linear pair                          Given

m∠X + m∠Y = 180°                                      Sum of angles of linear pair                  

4·a + 2 + 21·a + 3 = 180°                               Substitution property      

25·a + 5 = 180°                                              

a = (180 - 5)/25 = 7

m∠Y = 21 × 7 + 3 = 150°                               Substitution property      

m∠Y = 150° >  90

m∠Y is an obtuse angle                             Definition of obtuse angle

3. Statement                                                 Reason

∠A and ∠B are complementary angles        Given

∠A + ∠B = 180°                                         Definition of complementary angles

∠B and ∠C are complementary angles        Given

∠B + ∠C = 180°                                         Definition of complementary angles

∠B + ∠C = ∠A + ∠B                                 Transitive property

∠C = ∠A                                            Reverse of addition property of equality

∠C ≅ ∠A                                            Definition of congruency

4. Statement                                        Reason

ΔMNP is a right triangle                        Given

∠M is a right angle                               Given

∠M + ∠N + ∠P = 180°                             Sum of interior angles of a triangle

∠N + ∠P = 180° - ∠M                              Subtraction property of equality

∠N + ∠P = 180° - 90° = 90°                    Substitution property of equality

∠N + ∠P = 90°                                        Substitution property of equality

∠N and ∠P  are  complementary angles    Definition of complementary angles.

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range:

what