Question

I need help on this ASAP please help if you can

Answer 1

Recall the following definitions regarting the geometry of angles:

When two angles ∠A and ∠B have the same measure:

`m\angle A=m\angle B`

we say that they are congruent:

`\angle A\cong\angle B\Leftrightarrow m\angle A=m\angle B`

If the measures of two angles add up to 90 degrees, we say that they are complementary angles.

If the measures of two angles add up to 180 degrees, we say that they are supplementary angles.

If the measure of an angle is 90 degrees, we call it a right angle.

A pair of vertical angles is such that they appear at both opposing sides of a vertex:

If two angles are vertical angles (in this case, ∠M and ∠L), then they have the same measure:

`m\angle L=m\angle M`

When two angles form a right angle, this means that the sum of their measures is equal to 90 degrees, which means that those angles are complementary angles.

Congruent complement theorem:

Let A, B and C be angles such that ∠A is complementary to ∠B and ∠B is complementary to ∠C. Then, ∠A and ∠C are congruent.

Proof:

`\begin{gathered} m\angle A+m\angle B=90 \\ m\angle B+m\angle C=90 \end{gathered}`

Substract both equations:

`m\angle A-m\angle C=0`

Therefore:

`\begin{gathered} m\angle A=m\angle C \\ \therefore\angle A\cong\angle C \end{gathered}`

Congruent supplement theorem:

Let A, B and C be angles such that ∠A is supplementary to ∠B and ∠B is supplementary to ∠C. Then, ∠A and ∠C are congruent.

Proof: the procedure is the same as in the congruent complement theorem.

To complete the proof in question 10, notice the following reasons:

1.- Given

2.- ∠1 and ∠2 form a linear pair.

3.- ∠1 and ∠2 are supplementary (definition of supplementary angles).

4.- Given

5.-∠1 and ∠3 are supplementary (definition of supplementary angles).

6.- m∠1+m∠2=180 and m∠1+m∠3=180

7.- m∠1+m∠2=m∠1+m∠3, substract m∠1 from both sides.

8.- Definition of congruence.