Question

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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:

Answer 1

Answer:

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.