Question

NO LINKS!! Please help me with the Angle Proofs Part 3​

Answer 1

1. Given.

2. Definition of complementary angles.

3. Given.

4. Definition of congruent angles.

5. Vertical Angles Theorem.

6. Definition of congruent angles.

7. Substitution Property of Equality.

8. Definition of complementary angles.

Step-by-step explanation:

Statement 1

∠1 and ∠2 are complementary.

Reason: Given.

Statement 2

m∠1 + m∠2 = 90°

Reason: Definition of complementary angles.

Complementary angles are two angles that sum to 90°. Therefore, if ∠1 and ∠2 are complementary, then their measures must sum to 90°.

Statement 3

∠1 ≅ ∠4

Reason: Given.

Statement 4

m∠1 = m∠4

Reason: Definition of congruent angles.

Congruent angles are angles with exactly the same measure. As angle 1 is congruent to angle 4, then their measures are equal.

Statement 5

∠2 ≅ ∠3

Reason: Vertical Angles Theorem.

Vertical angles are always congruent.

Statement 6

m∠2 = m∠3

Reason: Definition of congruent angles.

Congruent angles are angles with exactly the same measure. As angle 2 is congruent to angle 3, then their measures are equal.

Statement 7

m∠4 + m∠3 = 90°

Reason: Substitution Property of Equality.

For all numbers a and b, if a=b, then a may be replaced by b in any equation or expression.

As m∠2 = m∠3 and m∠1 = m∠4, replacing them in m∠1 + m∠2 = 90° gives m∠4 + m∠3 = 90°.

Statement 1

∠3 and ∠4 are complementary.

Reason: Definition of complementary angles.

Complementary angles are two angles whose sum is 90°. Therefore, if the sum of m∠4 and m∠3 is 90°, this means that ∠4 and ∠3 must be complementary.

Answer 2

Answers in bold

1. Given
2. Definition of complementary
3. Given
4. Definition of congruence
5. Vertical angles
6. Definition of congruence
7. Substitution property
8. Definition of complementary

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Step-by-step explanation:

Always start with what is given. The last statement is the thing we want to prove.

Complementary angles add to 90 degrees. They form a right angle.

When "definition of congruence" is mentioned, we go from two angles being congruent to having them be of the same measure. It's a slight change of notation.

Vertical angles form whenever we cross two lines to form an X shape. The vertical angles are opposite one another, and always congruent.

The substitution property is the idea of replacing one thing with something it is equivalent to. In other words, we plug in the item.

Notice on line 2, if we replace "angle 1" with "angle 4" (valid because the two angles are congruent), and then replace "angle 2" with "angle 3", then we get the result on line 7.